Exponent: The exponent says what the equation will do at the given intercept.
- If the exponent is linear (raised to the first power), then the function will cross the x-axis at that intercept
- If the exponent is quadratic (raised to the second power), then the function will “bounce” off the x-axis at that intercept
- If the exponent is cubic (raised to the third power), then the function will “slide” at the x-axis (it will cross, but not in a linear fashion. Steeper further from the intercept and shallower close to the intercept)
- There will be a unique intercept for every unique value of x that could set the equation to 0. This is usually found by looking at the constants in the binomials and seeing if there are any numbers repeated in the parenthesis and which have the same sign. If the sign and constant are repeated, then this does not create an additional intercept.
- “Real Solutions” with Quadratic Equations
When a question presents a quadratic equation and asks about “real solutions” or simply “solutions”, there is a two step process to finding the answer the question is looking for.
Step 1: Try to Factor
See if the quadratic equation factors cleanly. If the quadratic equation can be factor into two binomials with whole numbers, then that will always be the quickest and easiest way to find “solutions” to that equation. Because this is the easier way to solve quadratic equations, questions that are designed for this method will often come earlier in the math sections. They also will often be the problems that mention just “solutions” rather than “real solutions”. Non-real solutions tend to show up more during the second step.
Step 2: Look at the Discriminant
$$\frac$$ For these questions, it’s necessary to use the quadratic equation. The most important portion of that equation is called “the discriminant”, which refers to the portion of the equation that is underneath the square root: b 2 − 4ac.
The discriminant determines if there is a negative, positive, or zero number underneath the square root. This then determines how many “real” answers there are to the equation.
A negative number under a square root means that there are only non-real answers (answers that include i (. Zero under the square root means that the equation simplest to just $$\frac$$, which since a and b are constants, which will lead to just one answer. And finally a positive under the square root leads to a positive number, which will then be added and subtracted from − b, which then creates two answers.
| Discriminant | # of Real Answers |
|---|
| b 2 = 4ac | 1 |
|---|
| b 2 < 4ac | 0 |
|---|
| b 2 > 4ac | 2 |
|---|
Just like with linear equations, questions about real answers for quadratic equations will either ask a student to find how many real answers there are for a given equation (unlike with linear equations, there is just one quadratic equation, since quadratics can have multiple solutions on their own) or they will ask a student to create an equation that either CAN or CANNOT be true if the equation is to have a given number of real solutions.
11. Area/Perimeter of basic shapes
On the ACT, students may be asked to calculate the areas of certain common shapes, including triangles, rectangles, and circles. The test will provide students with these formulas, but it’s good to be comfortable with using them before the test, so that questions involving these concepts are just about automatic!
The area of a triangle is = $$\frac\text$$ where b is the length of the triangle’s base and h is the height of the triangle. For right triangles the height will be equal to the vertical side coming up from the triangle. For non-right triangles, the height is equal to the distance from the base of the triangle to its opposite corner.
The area of a rectangle is equal to the base times the height.
The area of a circle is = πr 2 where r = radius of the circle.
With circles, it’s worth noting that some questions might ask students to use a portion of the area of a circle. This means students just need to use the internal angle measurement of the sector in question to convert the total circle area into the sector area.
12. Negatives
The easiest simple mistake to make on an ACT math section is to make a sign error. Whether it’s addition, subtraction, division, or multiplication, negatives make it easy to switch a sign (or to forget to switch a sign) and lose points on a question that a student would otherwise have gotten right.It’s important for students to be comfortable with how negatives are affected or not affected by the normal math operators: this starts with addition and subtraction.
Negative Operators
Adding negative numbers is just like adding positive numbers. The sum of two negative numbers has the same magnitude as the sum of the same two positive numbers.
The notation used here is worth paying attention to. Since − b is technically being “added” to − a in the second equation, it has to be put in parenthesis, since two operator signs cannot go next to each other. This can be helpful in preventing mistakes, as whenever students see a parenthesis between an operator and a negative number, that’s a great time to slow down and double check work.
Subtracting numbers is a bit different than adding numbers, since subtracting a negative number means taking away something less than zero, which actually translates to adding something. The phrase that many people are familiar with is: two negatives make a positive.
Multiplication can either be like addition or subtraction depending on how many negative numbers are being multiplied. If there is an odd number of negative numbers being multiplied by any number of even numbers, then the total product will be negative.
− 5 • 2 • − 4 • 3 • − 2 = − 240
Any two pairs of negative numbers will cancel out their negatives and produce a positive, and any positive numbers added will just keep the product positive. Therefore, if there is an even number of negative numbers being multiplied together with any number of even numbers, then the total product will be positive.
Division is the same as multiplication, but where the pairs of negative numbers have to be in the numerator and denominator to cancel out signs with each other. If both numbers in a fraction are negative, then their signs will divide out and the final product will be positive. If only one of the numbers (in either the numerator or the denominator) is negative, then the final product will be negative.
Negative Tricks
The most common way to make a negative sign mistake on any given problem is by not distributing the negative properly when multiple terms are multiplied by the same negative. For example:
− 5(a+4) = − 5a − 20 Since the -5 is right next to the a, it’s easy (on a timed test when a student is rushing) to think that the -5 only distributes to the variable a and to leave the + 4 alone. But since +4 is inside the parentheses, it needs to be multiplied by the -5 when it is distributed.
13. SOHCAHTOA
SOHCAHTOA is a pneumonic device used to remind students of the three trigonometric functions and their equivalents on a right triangle. Problems utilizing SOHCAHTOA will often have students solve for a missing side length on a triangle, or occasionally find a formula to express a side length (where there may only be variables or a mixture of numbers and variables, but where a final number cannot be solved for).
One tip for remembering the letters themselves is that each of the side labels (adjacent, opposite, and hypotenuse) is used twice. There are two Os, two As, and two Hs. Remember either that Tan doesn’t use hypotenuse or that Sin and Cos do, and the rest of the relationships will fall into order.
It’s also worth remembering that these identities are true for both acute angles of a right triangle. And since the labels of “adjacent” and “opposite” are relative to which side is being used, there is some overlap between these identities.
Sin and Cos Overlap
The Sine of the 1st acute angle of a triangle is equal to the Cos of the other acute angle, and vice versa. In our examples above, sinA = cosB and sinB = cosA.
Word Problems
SOHCAHTOA is most common on problems that look like our image above, where there is a triangle on its own and the problem will look to have students solve for a given angle or side. But SOHCAHTOA can also be used to solve word problems.
The triangle may already be drawn in (like the example below) or the question may simply describe a situation that creates a right triangle. The key thing to remember is that: right triangle + trig functions = SOHCAHTOA.
The example below is from the July 2019 ACT. The problem references the following information provided above the question itself: the maximum altitude during Tour C is 1,000 feet.
The angle 37 is used in all of the answer choices, and so the answer will involve SOHCAHTOA, the 37-degree angle, and the height of 1,000 feet.
The question asks about the length of the hypotenuse of the triangle and the question gives the length of the side opposite the angle. Therefore, this is a situation where SOH ($$sin\ = \ \frac>>)$$will be relevant.
$$sin37\ = \ \frac$$ A little algebra then leads to: $$c\ = \frac$$. F is our answer.
Extremely Likely (> 80% chance):
14. Function shifts
Vertical Shifts
Vertical function shifts are highly intuitive. They occur from either a coefficient placed in front of the variable(s) in the function or because of a constant term added to the function equation.
Sample Function: y = 2(x+1) 2 − 5
Here the function is shifted down by 5 units, and is multiplied by a factor of 2. The directions of these changes are intuitive, as they are the same size and direction as the numbers in the equation.
Horizontal Shifts
Horizontal shifts can be a bit more counterintuitive. For example, in the sample equation above the x variable has a + 1 next to it. Following the same rule as for vertical shifts, it would make sense that the x-intercept would then move one unit in the positive direction on the x-axis (to the right).
However, it’s actually the opposite. (X+1) means a horizontal shift one unit to the left. This is also true for leading coefficients. (2X) does not expand a graph in the horizontal direction, it shrinks it.
15. Average sum trick
The average sum trick simply refers to the idea that if a question gives the average of a data set and the number of data points in that data set, then that also allows a student to calculate the sum of all of the data points in the set.
This is important to know for questions about adding or subtracting data points.
5 tests, average is 80. After the 6th test, the average is 82. What was the 6th test score?
The new average is going to be equal to: (the old sum + the new number)/the updated # of data points
This works the same for subtracting a data point. The only difference is that the denominator is reduced by 1 instead of increased by 1, and the removed test score is subtracted rather than added to the original sum.
16. MPH
Miles per hour is a way to express speed: distance covered over a time interval. Questions using MPH often involve some kind of unit conversion. This can be a conversion of distance units (miles to kilometers) or time units (hours to minutes).
MPH also shows up in word problems very often, where a student might need to create an equation with a variable for “hours” (the variable choice is based off of the units that speed is originally given in or that the student is trying to solve towards).
17. Median
The median is the middle number of a data set when the data set is arranged in numerical order. Unlike the mean, it will always be one of the actual data points in the set itself. The set can be arranged as ascending or descending order: the median will still always be in the middle.
When a student wants to find the middle/median number in a data set, they must take the number of values in a data set (n = 23 for example) and divide it by two.
Median of Odd-Numbered Data Sets
In the example listed above, n= 23/2 = 11.5. Because there is an odd number of values in the data set, dividing this number by two results in a nonwhole number. For an odd-numbered data set, we will take this partial number and round it up, in this example to 12. This tells us that the middle number, in other words, the median, is the 12th number in the data set.
Median of Even-Numbered Data Sets
For even-numbered data sets, there is not one number in the middle of the data set. Instead, there are two middle values, and the median is the average (mean) of those two numbers. This is why if there was a sample of 24 data points (instead of the 23 from above), the same operation from above leads to a whole number: 24/2 = 12. The 12th data point is the first of the two median numbers. To get the 2nd number, round up to the next whole number, which tells us that the 13th data point is the second of the two median numbers. The median is then the average of the 12th and 13th numbers.
Remember, for this trick the product of the equations is not equal to the value of the median itself, but merely its “place in line” in the data set. It says where to look, but not what the median is.
How It Appears on the Test
Median questions are similar to average/mean questions. Students will normally be asked to do one of two things. Either they will be asked to calculate the median of a given data set (displayed through a table or graph) or they will be asked to describe how a change in the data set will or won’t change the median (or what change(s) would need to be made to the data set to lead to a certain change in the median).
18. Radicals
Radicals (also known as “roots”) are the opposite of exponents. The most common radical that students are familiar with is the square root or root-2, which is why the default radical symbol is also the square root symbol: . A square root basically says: to get the number under the square root, what number would need to be multiplied by itself to get that number?
Roots above a square root will have a subscript next to the root to show what “degree” the root is taken to. Again, this is the same as exponents where X 3 means x cubed. In the same way, $$\sqrt[3]$$ is the cube root of x. The first expression says what happens when x is multiplied by itself 3 times. The second expression asks what would need to be multiplied by itself three times to get x.
Radical Operations
Cancel Out
Because radicals and exponents express opposite operations, they can cancel out with each other under certain circumstances. For example, a square root and an exponent of 2 will cancel out with each other.
= x The same thing is true for any evenly balanced roots and exponents
Multiplication
Two separate square roots multiplied by each other are equivalent to the square root of the products of those numbers
Simplification
The rule above can also be used to help simplify square roots. Instead of combining square roots though, it will involve separating them.
For example, take the square root of 48. This is equal to the product of the roots of its factors: sqrt(48) = sqrt(4⋅12). The square root of 12 is not a whole number, but the square root of 4 is.
Therefore, the expression can be simplified as 2sqrt(12). There are still only whole numbers here, but now if there was a or a 2 in the denominator below , it would be clear that these would cancel out and that the expression could be simplified further.
Whenever a root appears with a whole number underneath it, students should ask themselves: “is there a way to factor this number such that I can simplify the expression?”
This also works when there are variables under the root.
Division
Division with roots is the opposite of multiplication. The ratio of the square roots of two numbers $$\frac<><>$$ is equal to the square root of the ratio of those two numbers.
Fractional Exponents
As mentioned in the exponents section, roots can be expressed as fractional exponents. If a number has a fractional exponent, then the denominator (bottom half) of that fractional exponent is the root that that number is being taken to.
$$x^> =$$ This doesn’t affect or change exponents in the denominator. Those remain in the normal exponent spot, even when the fractional exponent is converted to the root form:
19. System of Equations
System of Equations involves taking a series (usually two) of equations and finding their “solution”, meaning variable values that make all equations in the series true. Another way to think of this is to imagine graphing those equations, and a system of equations problems asking students to find the intersection points between the given equations.
Two Methods for Systems of Equations: Elimination and Substitution
Substitution
The substitution method of system of equations involves isolating a variable in one of the equations, and then plugging in that expression for the variable into the other equation.
For example, if a question gives the following system of equations:
Then to substitute, a student must choose one equation and one variable to isolate. Isolating the x variable in the bottom equation gives x = 2y + 8. That expression is then substituted in for the x variable in the top equation: 2(2y+8) + 3y = 2. This then leads to 7y= − 14 or y = − 2.
Elimination
Just like with any algebraic equation of coure, one can either work with variables individually or with the equation as a whole. With systems of equations, questions are generally looking for solutions that make both equations true. Since that means the equations must be true statements, the two equations can be multiplied by, divided by, subtracted from, or added to each other, just like any other true statements.
The reason the word “elimination” is used to refer to this is because the most common way to make the two equations in a system of equations interact directly is to subtract one equation from another. Specifically, a student should look to subtract one equation from another such that one of the two variables in those equations is eliminated.
To do this, the variable should have the same coefficient but opposite signs in each equation. This may already be true when the equations are first stated in the problem, or one equation may need to be manipulated to make this work.
In the example above, neither variable has the same coefficient but opposite signs, so a little work would need to be done.
The bottom equation could be multiplied by -2, which would lead to the x variables having same coefficients and opposite signs. A student could also look to eliminate the y variable, but this would require either multiplying one equation by a non-whole number or manipulating both equations. With the ACT, speed is always of the essence, so students should try to choose the quickest/most efficient path!
Therefore, the bottom equation is multiplied by -2: − 2x + 4y = − 16
The two equations are then added together. Technically either side can be added together, but it’s easiest to add the variable sides and constant sides together.
7y = -16 is our sum, or y = − 2. Same answer as our previous method, but just done a little differently.
Deciding Between Substitution and Elimination
Students should do what they’re most comfortable and confident with.
That being said, there are some general guidelines for choosing between these two methods, if a student is comfortable with either approach.
Look at the Numbers in the Equation
First, look at the numbers in the equations. As the following questions:
- Are there any shared coefficients and opposite signs in front of the variables?
If yes, that means no work is needed and elimination will be the easiest route.
- Are the constants and coefficients factors of each other? Would they divide into each other evenly or cleanly?
If yes, then it could be easy to manipulate the equation and utilize elimination.
If no, then it could be difficult to eliminate numbers easily. substitution may be best.
Look at the Question
What is the question asking for?
- Does it want the value of a single variable or of multiple variables or the product/sum of two variables?
If it is asking for a single variable, which cannot be easily reached through elimination, then substitution is often helpful.
If it is asking for the product or sum of multiple variables, especially if that product or sum can be reached through elimination or by leaving an equation intact.
Word Problems
Sometimes problems will involve systems of equations, but the equations will not be directly given by the question. Instead, the problem might be given as a word problem, but where the information presented would clearly lead to the creation of a pair of equations. With these equations there should be two variables, which can be used to calculate two different things: time and distance, for example.
Here is an example question from the April 2021 ACT.
20 = A + B where A and B are the number of courses Susan received an A or B in, respectively
$$3.15\ = \ \frac$$. Again, we can use either substitution or elimination here.
3 = A Susan received 3 A’s
One important note: since the question asked for the value of A, B was the variable that the student cancelled out. If A had been cancelled out, then that would have added an extra (and unnecessary) step.
Cancel out or substitute in for the variable that you’re not solving for: this will leave just the variable you’re solving for in the equation.
20. Angle chasing
How Many Degrees in ______ ?
How Many Degrees in a Line
There are 180 Degrees in a Line. This means that if there are two angles across a line, then the sum of those two angles will be 180 degrees. These are known as supplementary angles.
How Many Degrees in Common Shapes
There are a number of common shapes that it is useful to know the number of degrees within.
- Triangle: 180 degrees
- Square: 360 degrees
- Circle: 360 degrees (sometimes also expressed as 2π radians)
How Many Degrees in Any Shape
While normally the common shapes listed above are all that students need to know. Occasionally there are weird shapes in angle problems, and knowing the number of degrees within those shapes may be crucial for answering that question.
Thankfully, there is a simple equation for knowing the number of degrees in any polygon.
For example, this is why a triangle has 180 degrees:180 (3−1) = 180 • 1 = 180
Vertical/Opposite Angles
Vertical Angles are angles opposite each other at the intersection of two different lines. The lines can have any slope (they do not have to hit at a right angle or anything special). As long as they are straight lines, opposite/vertical angles will be congruent (have the same measure).
In the illustration below, angle a = angle b
Angles at Intersections with Parallel Lines
When a line intersects a pair of parallel lines, there are a number of set relationships between the angles created by those lines. These are useful to know, because they can be massive time savers. You can save a full minute on an angle question just by going beyond just the supplementary angles rule (which can be tedious to work out with many lines or many shapes).
Depending on which pair of angles are being compared, the angles will either be congruent (equal) or supplementary (adding to 180 degrees), as follows:
21. Time
Time Conversions
Be able to convert between different units of time. The most common ones being between hours and minutes and between minutes and seconds.
Time Variables
Some equation questions might have variables that stand in for time, especially as an exponent. The question may then ask for how to maintain the same equation using a different variable for time.
For example, taking the equation T = 100(1.05) t where t is the time in hours. If a question wanted this converted to minutes, but without changing the actual equation, then there will need to be compensation for the unit change.
Going from hours to minutes means that the exponent would become larger if there was no compensation. As a result, the new variable will need to be divided by 60 (as this is the number of minutes in an hour).
Therefore, the new equation would look like: $$T = 100(1.05)^>$$where m is the number of minutes.
The key for keeping equations consistent while changing exponent variables is that if you’re subbing in a bigger variable, then it will need to be divided by something, and if you’re subbing in a smaller variable, then it will need to be multiplied by something.
22. Pythagorean Theorem
This is one of the most fundamental concepts in math, and as a result it is something that will be applicable on every single ACT. The pythagorean theorem applies to right triangles and states that the square of the hypotenuse is equal to the sum of the squares of the other sides.
A question may simply show a triangle directly and ask for either the side lengths or a calculation that requires knowing all of the side lengths (such as perimeter). This may also be usable in a word problem.
Be on the lookout for if a word problem explicitly states that something happens at a 90-degree angle or if there is a perpendicular line or tangent line. These all lead to 90 degree angles and mean that a right triangle may be created, and so the pythagorean theorem is valid.
23. Apply formula
Applying formula problems will provide a student with a particular formula/equation, and will then ask a student to properly plug in numbers into that equation to calculate something. This will often be done in the context of a word problem that describes the real-world application of the equation. For these equations the main tricks are to:
- Clearly identify the dependent and independent variables. The independent variables are what is being changed in the equation (in the cause/effect chain, they are the cause). The dependent variables are the outputs of the equation. THey are the result of the change made.
- Be sure to remain consistent with units. The original formula might be given in hours, but then the number that has to be plugged in is in minutes. Remember that the original formula is the guide, and that everything should be kept consistent with that guide.
24. Composite function
Composite functions are where one function is put inside of another. This can look intimidating at first, but these problems are quite simple once a student gets the hang of it.
Inside Out
It’s easiest to think of composite functions as a priority list or an order of operations. Work from the inside to the outside.
Here is an example from the June 2020 ACT:
Working from the inside out. g(−1) = (−1) 2 − 1 = 1 − 1 = 0. Since g(−1) = 0, g(−1) can be replaced with 0, as if it was a variable that we had solved for and could now plug into an equation.
Now the next layer: f(0) = 2(0) + 3 = 3. The answer is 3!
25. Matrices
A matrix is a series of rows and columns used to display numbers. ACT questions about matrices will generally be about performing fundamental math operations such as addition, subtraction, or multiplication between matrices.
Is It Possible?
A key note is that not all matrices can be added, subtracted, or multiplied.
- Two matrices can only be added or subtracted if they have the same dimensions
- Two matrices can only be multiplied if the number of columns in the first matrix is equal to the number of rows in the second matrix
Adding and Subtracting
Adding and subtracting matrices is quite simple. Since the matrices have to have the same dimensions, each number has a “twin” that shares the same space in the opposite matrix.
Matrix addition means adding each pair of numbers in the same places in their respective matrices. Subtraction means subtracting pairs.
The resulting matrix will have the same dimensions as each of the original matrices.
Multiplying
Multiplying matrices is a bit more complicated. Matrix multiplication is used to find the dot product. This is sometimes the term used on the ACT to imply matrix multiplication.
As stated above, matrices can only be multiplied if the number of columns in the first matrix is equal to the number of rows in the second matrix. The top left number in the dot product is equal to the sum of the numbers in the first row of the first matrix multiplied, in order, by the numbers in the first column of the second matrix.
The next number down in the dot product’s first column is equal to the numbers from the second row of the first matrix multiplied, in order, by the numbers in the first column of the second matrix.
This is then repeated until there are no more rows left in the first matrix. The process is then repeated again for the next column in the second matrix.
The final dot product will have the same number of rows as the first matrix and the same number of columns as the second matrix.
26. Midpoint
Midpoint questions involve working with two ordered (x,y) pairs and finding their midpoint. This is done using the following equation:
$$<(x>_,y_) = (\frac>_<> + x_>,\frac>_<> + y_>)$$, where (xM, yM) is the midpoint and the x and y values on the right side of the equation are from the ordered pairs of that midpoint.
Occasionally these questions might give the midpoint and one of the ordered pairs, in which case a student must use normal algebra on the above equation to find the missing pair. Each variable will need to be dealt with separately for that algebra to work (it cannot be done simultaneously).
Very Likely (> 50% chance):
27. Special Right Triangles
There are certain special triangles that students might be expected to know on the math section of the ACT. Like the formulas for the area of a circle or rectangle, these identities are normally presented to students at the beginning of the test, and so the challenge is knowing how to identify problems where these identities are relevant and how to use them to solve relevant problems.
The most common special triangles are: 45-45-90 and 30-60-90 triangles.
Because these length and angle relationships are set, any time these angles appear or these side lengths appear, they can give us useful information for solving a problem. Problems involving these triangles often utilize words/drawings or the xy-plane. They are also often relevant on problems involving triangles and points of tangency. For example:
Since the square touches both the center and the perimeter of the triangle, a line going from one corner of the square to the other corner will be equal to the radius of the circle. This straight line divides the square perfectly in half, which means that the angles on each side of the line are = 45 degrees. That means that this square can be defined as follows:
The question gives us the area of the shaded region, which equals 36π − 18. From the image, it’s clear that this is equal to the area of the circle minus the area of the square. In other words:
πr 2 − s 2 = 36π − 18 where r is the radius of the circle and s is the side lengths of the square. The triangle above shows that r = x. Plugging this into the formula above gives:
x was equal to the side length of the square and the question asks for the perimeter of the square. Therefore, the answer is: 4(3) = 12 and so C is the correct answer.
28. Absolute Value
|x| (absolute value of x) is the magnitude (the non-negative value) of x, regardless of its original sign.
The concept of an absolute value revolves around negative and positive numbers. A positive number doesn’t change when placed in an absolute value, but a negative value does change: it becomes a positive. Therefore, if a problem involves an absolute value, it’s a good idea to pay attention to any negative numbers in the problem (or to use a negative number as a test value like in one of the above practice questions).
29. Multi Step conversion
Multi-step conversions will involve a mixture of ratios and/or equations. Questions will either go from a proportional ratio to an overall total, or they will ask for a conversion of one ratio to another (usually in the form of a unit conversion).
One of the most common versions of this question will involve mph and gas costs in miles/gallon.
For example, a problem might say that a person can drive their car, which gets 50 miles-per-gallon, for a maximum of 1,340 miles. If gas costs $$3.28/gallon, how much would it cost to fill the car’s tank?
Since the cost is based on the number of gallons, the mpg and mileage must be used to calculate the size of the car’s tank. If they can drive a maximum of 1,340 miles getting 50 miles-per-gallon, then that means the car has a tank that holds 26.8 gallons.
Therefore, the cost of filling the tank is: 26.8 * $$3.28 = $$87.90.
30. Remainders
Remainder problems will generally either be: simple or pattern-based reminders.
Simple
Simple remainder questions will provide some kind of division problem, which will leave some remainder. The student will then generally have to find whatever whole number is left over when the nearest factor of the denominator is taken away from our original number.
For example: what is the remainder when 2387 is divided by 38?
$$\frac = 62.816$$ This means that the nearest factor of 38 to the original number is 38 * 62 = 2356
The remainder is therefore: 2387 − 2356 = 21
Pattern-Based
For pattern-based remainder questions, the question will provide some kind of repeating decimal, and will then ask for a large-numbered digit of that repeating decimal: for example, the 4567th digit after the decimal point. A repeating decimal means that there is an infinite string of digits after the decimal, which repeat in a set pattern.
The repetition often starts next to the decimal, or after the first two digits of the decimal. For example, 1/12 = .08, so the first two numbers after the decimal point are not a part of the repeating pattern.
- Key takeaway: not all numbers after the decimal point have to repeat for there to be a pattern.
Just like with patten-based remainder questions, factors are still key. However, instead of being interested in the factors of the denominator in a fraction/ratio, pattern-based remainder questions are more about how many digits repeat in the decimal pattern.
Common Repeating Decimals
Repeating-decimal questions can create or provide any decimal or pattern, but there are some common repeating decimals that might show up. The most common repeating decimals to show up on these questions (or even other questions) are:
1/9 = .111 For any fraction with 9 as the denominator, the repeating decimal is simply equal to the numerator repeating after the decimal point (2/9 = .222)
1/11 = .090909 For any fraction with 11 as the denominator, the repeating decimal is simply equal to the numerator multiplied by 9 (expressed in two digits) after the decimal point (2/11 = .181818)
31. Weird shape area
Some ACT Math area questions will involve shapes that are irregular, meaning that they are not the standard triangle, circle, rectangle, or square. Instead, they might simply be polygons that combine portions of these shapes together.
- For weird shaped area problems, break down the large, weird shape into smaller, more familiar shapes. Just like with equations, it’s about making the unfamiliar seem more familiar.
Sample Problem
Here is an example problem from the July 2019 ACT:
There are two different ways to approach this problem. Either, the area of the original rectangle can be calculated, as well as the squares that were removed, and then the two numbers can be subtracted. Or the remaining shape can be broken down into a group of five rectangles, and their areas can be summed up together.
Now, since the answers to the question all involve subtraction, it would make sense that the subtraction method might work best in this case.
The original area of the rectangle was 18 • 8 = 144. The area of each square removed was equal to x 2 . Four of these squares were removed, and so the area of the remaining rectangle is 144 − 4x 2 or G.
Key Takeaways
With weird shape areas it’s often helpful to remember that weird shapes are made from cutting out familiar shapes from familiar shapes, or by adding familiar shapes to familiar shapes.
Also remember that especially when it comes to shapes made from cutting things out. There are some quick rules to remember about how this affects the areas and perimeters of the shapes.
- Area: Just like the question above, remember that the area of the larger shape can still be calculated to find the area of the new shape. It’s just about remembering to take away the areas of the shapes that were cut out.
- Perimeter: A slight more fun trick. If the shapes that are cut out from the original shape are rectangles (are made up of only perpendicular lines), then the perimeter of the new shape is the same as the perimeter of the old shape. So in the above example, the original rectangle and the new shape would have the same perimeter.
The perimeter trick is particularly useful for saving time on questions. Perimeter doesn’t change by taking rectangles out of shapes.
32. Periodic function graph
Sine and Cosine graphs appear fairly frequently on the ACT math section. It is less likely that you’ll be asked to calculate specific values on the graphs (though this does happen occasionally).
Instead, it’s much more likely you’ll be asked to:
- Look at a graph and identify a possible equation for that graph
- Be given an equation and asked to identify an accurate graph of that equation
Either of these types of questions can be quite easy, as long as a student remembers some key facts about the sine and cosine graphs.
Basic Principles
The Sine and Cosine functions are wave functions. They repeat forever making the same pattern. The values for the sine and cosine graphs are the same values that one could find by looking at the unit circle.
Sine and Cosine graphs look almost identical, but they start at slightly different places. The Sine graph starts at the origin (0,0), while the Cosine graph starts at (0,1). Therefore, the two graphs are basically the same, but shifted by $$\frac<\pi>$$in the x-direction from each other.
Amplitude
The Sine and Cosine functions have the same amplitude, which is the distance from the center line (the x-axis for the default graph) to the maximum point of the graph.
- The default amplitude of the Sine and Cosine graphs = 1
Period (Frequency)
The period of a repeating graph like sine and cosine refers to how long the graph needs to go along the x-axis until it begins to repeat. The period of the sine and cosine graphs is also the same.
- The default period of the Sine and Cosine graphs = 2π
Shifts
The Sine and Cosine graphs are affected by x and y shifts just like any other functions. Y shifts will affect the amplitude of the graphs, while x shifts will affect the frequency.
Shifts are accomplished either through the addition of leading coefficients in front of the variables or by adding constant terms next to the variables.
Y-Shifts
Y-Shifts are more intuitive. When a constant is added to the sine or cosine function, it shifts the graph up or down based on the number and sign of the constant.
If the y constant is positive, the graph shifts up, and if the y constant is negative, the graphs shifts down.
The same is true for coefficients in front of cos(x) or sin(x). Coefficients greater than 1 will stretch the graph up/down, while coefficients smaller than 1 will compress the graph.
X-Shifts
Just like with other functions, X-shifts are a bit more counterintuitive. For example, the graph of sin(x + 1) shifts left instead of to the right. Also, sin(4x) compresses the graph by a factor of 4, rather than expanding it by that amount.
The easiest way to think about this is that by affecting the x value, the amount of work being asked of the x variable is changing. If each x variable is multiplied by 4, then that means that it only takes a number ¼ as big to get the same output, and so everything compresses to account for that.
Similarly, with (x + 1), the number -1 will produce the same output that x = 0 used to create, and so the graph has to shift one unit to the left to account for this.
- Sine and cosine are repeating wave functions when graphed. They have an amplitude of 1 and a period of2
- Y-constants and coefficients in front of sin/cos will stretch or move the graph proportionally with these added terms
- Constants and coefficients inside of the sin and cos with the x variable will stretch or move the graph in the inverse direction of that coefficient/constants sign or magnitude
Sample Problems
This can be one of the trickier concepts at first, but with a little practice these questions can become quick and easy. Here are two sample problems to give some practice:
The question does not identify which of the two graphs is sine or cosine, but since the dotted line graph starts at the origin and the solid line graph starts above the origin, it’s clear the solid line is the cosine graph and the dotted line is the sine graph.
The sine graph has a higher amplitude than the cosine graph, which means that its coefficient must be larger than that of the cosine graph (a1 > a2). Since the sin graph has been stretched out, it’s also clear that a1 > 0. As a result, the answer must be B.
This question is a more direct example of quantitative shifts. The Cosine function normally starts at (0,1), but in this graph that is now (0,3), meaning that the equation has been stretched out by a factor of 3. This eliminates answers F and G.
Looking next at the period, the graph appears to repeat after x = π . This means that the graph has been compressed along the x-axis, which means the x value is multiplied by a coefficient greater than 1. Only K does both things correctly and so this is the answer.
33. Shaded area
Shaded area questions will generally involve one of two things: a geometric shape (or possibly one inside of the other) or an inequality graph.
Geometric Shading Problems
Geometric shading problems will focus on calculating area. Students will be asked to calculate a shaded area from a larger whole area. This will often involve subtracting one area from another.
The most common version of this is to have a square inside of a circle or vice versa. It may also involve a triangle inside a sector of a circle. The question would then ask for the area of the circle sector minus the area of the triangle.
Inequality Graphs
Inequality shading problems will focus on identifying areas on a graph. Students will need to identify which area above or below a line should be shaded, or which portions of a shape should be shaded to satisfy an inequality.
Like with normal inequality problems, the easiest way to approach these questions is to replace the inequality sign with an equal sign. Normal algebra rules then apply for the equation: isolate the y on the left hand side to make it look like the usual y = mx + b form.
Here is an example problem from the June 2017 ACT:
ax + by = c can be rewritten as $$y = \ \frac + \frac$$. Remembering that 0 < a < b < c. The slope of the line will be negative, so the line should be downward sloping. This eliminates F. $$\frac$$is positive, so the y-intercept should be positive: this eliminates J and H.
The only difference between G and K is that one has shading below the line, while the other has shading above the line.
- Shading above a line means that y should be greater than the other side of the equation
- Shading below a line means that y should be less than the other side of the equation
The inequality version of the equation is: $$y \leq \ \frac + \frac$$. Y is less than the other side of the equation, meaning that the shading should be below the line.
This means the answer is K.
34. Counting principle
Counting principle questions are similar to probability questions, but instead of probabilities they are asking about possibilities. Often in ways such as: how many different possible combinations are there of several different groups of items or options, such as letters and numbers for a license plate or dice rolls from multiple pairs of dice (possibly with different sets of numbers).
The counting principle simply states that if there are x possible ways to do one thing and y possible ways to do another thing, then there are x • y possible ways of doing both things. So count and then multiply.
This can be thought of in a similar way to multiplying probabilities from above, but where the tree branch of options is more about the possibilities created by each branch rather than what proportion of them is successful. Basically only the denominators calculated for that tree matter, rather than the whole fractions.
Here is an example question from the December 2017 ACT:
Single digits 0 through 9 = 10 possible numbers that can be chosen
Letters of the alphabet = 26 possible numbers that can be chosen.
Single digits must be chosen for each of the first three slots and they can repeat. Therefore, the are 10 ways to fill slot #1, 10 ways to fill slot #2, and 10 ways to fill slot #3 = 10 • 10 • 10 = 1, 000 possibilities. This is how many possible ways the first three slots can be filled.
The last three slots require letters. There are 26 ways to fill each slot, and so there are 26 • 26 • 26 = 17, 576 possibilities. This is how many ways the last three slots can be filled.
How many ways can all 6 slots be filled= first three possibilities x last three possibilities = 1, 000 • 17, 576 = 17, 576, 000. Our answer therefore is K.
35. Logarithms
A logarithm is the inverse of taking something to an exponential power. When something is expressed as log5125, in a sense it’s asking what exponent would 5 need to be taken to to get 125. This is why log5125 = 3.That’s why the first core skill of logarithms is to be able to rewrite logs in exponential form.
Rewrite Exponentially
If given logax = b, be able to rewrite this formula as a b = x. This will make any question asking you to solve for a variable in the original equation much more intuitive and easy to answer.
Log Operations
In the same way that exponents have certain rules for fundamental math operators, logarithms have their own rules for adding, subtracting, multiplying, and dividing. Just like with exponents, these rules generally rely on having the same base number.
Add
Loga(X•Y) = LogaX + LogaY The log of any given number is equal to the sum of the logs of its factors. Or read the other way, two logs added together is equivalent to the log of their product
Subtract
Special Skills and Facts
Moving Exponents
Since logs are related to exponents, exponents that are a part of a log can be moved around in a convenient way! Specifically, exponents on numbers inside of the log function can be moved in front of the log and turned into a linear coefficient.
For example loga x 2 , can be rewritten as 2 loga x.
Natural Log
A normal logarithm uses the shorthand log, but there are other types of logarithms that are commonly used in math and other fields. THe most common one is the natural log, which is written as ln x. This still follows the same rule as any other log, but instead of being the log of an integer, natural log is log base e, where e is the special number known as
Log Base 10
In the same way that without a specific number on the radical is understood to mean the square root. A log without a specified base is understood to mean log10.
Log 1
Another great example of how log rules relate to exponent rules. Any number taken to the power of 0 = 1. So, Log 1 = 0, no matter what the Log base is.
36. Imaginary numbers
Imaginary numbers refer to irrational numbers that cannot exist in the real word, but which do exist in math. The most common imaginary number to see on the ACT is the imaginary number i, or the square root of -1.
When in Doubt, Try to Square It
The best piece of advice for problems involving i, especially those that involve calculating a real number or simplifying a fraction such that it only contains real numbers is:
- See if there is a way to square i. Since it is equal to -1, squaring i can take an imaginary number and make it a real number
- When i is part of a binomial in the denominator, it may require using a conjugate to simplify out i. See #55 on this list for a longer explanation of conjugates.
Powers of i
On the ACT, it’s worth knowing not only that i = , but also that there is an easy way to know what i raised to any power is equal to. i goes through a pattern of four different values, depending on what power it is raised to. These values always go in the same order, and as long as a student knows how the pattern starts, they can always figure out the value of i to any exponent.
Any number taken to the 0 power is = 1, and so i 0 = 1.
i = , and so i 1 =
Any square root squared is simply equal to the value underneath the square root, so i 2 = () 2 = − 1
And finally, i 2 = − 1 and so i 4 = i 2 • i 2 = − 1 • − 1= 1
And so the pattern of ilooks like: 1, i, − 1, − i.
To figure out the value of i at any given exponent, simply take the exponent and find it’s nearest factor of 4. The conversion chart then looks like:
| Remainder After Factor | Value |
|---|
| 0 | 1 |
|---|
| 1 | i |
|---|
| 2 | − 1 |
|---|
| 3 | − i |
|---|
The Complex Plane
Occasionally ACT problems involving i will include the concept of the “complex plane”.
This means that the xy plane is replaced with a plane that includes:
37. Least Common Multiple (LCM)
The Least Common Multiple refers to the smallest possible number that is evenly divisible by a given pair or set of numbers. For example, the least common multiple of the numbers 2 and 3 is 6.
The easiest trick for finding the least common multiple of any series of numbers is to find the multiples of the largest number in the set, and find the first of those numbers that is evenly divisible by both of the other two numbers.
For the set 4, 10, and 25. Go through the multiples of 25 until one is evenly divisible by both 4 and 10. The first number that satisfies this is 100, so this is the least common multiple of this set of numbers.
38. Scientific notation
Scientific Notation is a way to express very large or very small numbers by the number’s most significant digits combined with an exponential expression of base 10. This notation usually includes a whole number with one or two decimals after it: 2.38. This will then be followed by an exponential expression of: × 10 a , where a is going to either be a positive or negative number, generally with a magnitude greater than 3.
The exponent on the 10 shows if the number is a very large number or a very small number. If the exponent is positive, then the number being expressed is quite large. If the exponent is negative, then the number being expressed is very small.
The magnitude of the exponent then tells you how many places the original digits were moved by to get to the new expression. In other words, how many places were originally between the decimal point and the new currently visible digits of the number.
For example, 6.52 × 10 8 is another way of writing 652,000,000. The scientific notation version is raised to the 8th power because in our original number there were 8 places between the leading 6 and the decimal point. It’s important to note that this would be the same even if the number were 6.2 × 10 8 , but there would be 7 0s instead of 6, as now there is one fewer significant digit before the decimal point.
The inverse is true for when there is a negative exponent instead of a positive one. It’s still the number of places between the original decimal point and the first significant digit, but now it’s to the right of the decimal point instead of the left.
6.52 × 10 −8 is another way of writing .0000000652. The simple conversion is the whole number will move to the right of the decimal point, and then zeroes are inserted. The number of zeros = |Exponent|− 1. In other words since the exponent here is -8, there are 7 zeroes added rather than 9 (8-1 rather than − 8 − 1).
The last thing to note is that with positive scientific notation, sometimes there are two or three digits after the decimal point that “absorb” some of the zeroes places that are being multiplied by. With the negative exponents, there is generally just the single digit whole number to the right of the decimal, and so the number of zeroes will always be equal to that exponent minus one rule.
39. Ellipses
Ellipses are a rare but important shape to know about for the ACT math section. Here are the main properties of an ellipse that students will be expected to know:
Key Skills for Ellipses
What is an Ellipse?
An ellipse is an oval shape. It is similar to a circle, but where one of its axises is stretched. As a result, an ellipse has both a major axis and a minor axis.
- The major axis is the axis in between the longer ends of the oval
- The minor axis is the axis in between the shorter ends of the oval
Reading and Graphing Ellipse Equations
Since the shape of an ellipse is similar to a circle, the equation is similar too. An ellipse equation will look like this:
- Unlike the circle equation, there are now denominators under the squared binomials. This is to differentiate the lengths of the major and minor axes.
- The ellipse equation will always be = 1
Like with the circle equation, the constants inside of the binomial indicate where the center of the ellipse is.
- (x-1)2 means that the center point lies along the vertical line x=1
- (y+2)2means that the center point lies along the horizontal line y= -2
- Combining these two things means the center is at (1,-2)
The denominator term is then equal to half the length of the major/minor axis squared.
- The denominator under the x term is 9, meaning that the each half of the major axis has a length of 3
- The x-axis is the major axis because it has a larger denominator, and therefore its axis is longer
- The denominator under the y term is 1, meaning that each half of the minor axis has a length of 1
ACT questions about ellipses may give an ellipse equation and require students to graph it to find a point on the ellipse or the center point of the ellipse. A question could also provide a graph of an ellipse and then provide several equations, from which the student must then select the equation that correctly matches the graph.
40. Vectors
Vectors are a way of expressing magnitude (size) and direction. The magnitude of a vector will often be described using variables, most commonly the variables i and j. The direction of the vector will be indicated by an arrowhead at the end of the vector line.
Vector Skills
- There are three main operations students might be asked to perform on vectors: addition, subtraction, and multiplication.
Adding
To add together any two vectors, take each vector and attach it to the endpoint of the vector that it is being added to. This will create a diamond shape. The diamond starts where the two vectors start, and it ends where they meet (after being added on to the other vector).
The sum of the two vectors is then equal to a new vector created by going from one end of the diamond to the other. The direction/arrow of this new vector will move in the same direction as the two added vectors.
Subtracting
Subtracting vectors is similar to addition. However, instead of simply attaching both vectors to their ends of their added pairs, first one of the vectors has to be flipped.
Specifically, the vector that is being subtracted needs to have it’s direction flipped (though it’s magnitude remains the same).
This flipped vector is then added to the other, original vector using the same diamond pattern from above. The product of the subtraction is once again equal to the vector created between the two ends of the diamond.
Scaling Vectors (Scalar Multiplication)
On the ACT, the only kind of vector multiplication that a student might be asked to know is scalar multiplication. Like the name suggests, this is simply scaling the vector up or down (either growing or shrinking it). There are ways to multiply actual vectors together, but this should not be covered on the ACT Math section.
Scalar Multiplication follows some quick and simple rules:
- If a vector is multiplied by a number whose magnitude is greater than 1, then the vector will increase in magnitude (length) by that number
- If a vector is multiplied by a number whose magnitude is less than 1, then the vector will shrink in magnitude (length) by that number
- If a vector is multiplied by a positive number, then the direction of vector will not change
- If a vector is multiplied by a negative number, then the direction of vector will be reversed
i and j notation
Vector problems sometimes use the variables iand j. These are known as the unit vectors, and each variable is associated with a specific axis. i is the unit vector for the x-axis. j is the unit vector for the y-axis.
When these show up on an ACT math question, they will often be used to refer to direction. For example, here is a sample question on vectors from an ACT Diagnostic test:
Since vector j represents 1 mile per hour north, that means that − j would represent 1 mile per hour south. Maria is jogging south at 12 miles per hour, meaning that − j would need to be scaled up by 12 to properly represent her velocity.
Therefore, the answer is 12 • − j = − 12j or B.
Worth Knowing (>25% chance):
41. Permutation
Permutation is similar to the counting principles discussed earlier. However, permutation questions involve options which are mutually exclusive.
Here is a sample ACT question involving Permutation:
Unlike with the security lock question covered above, once a plant is used in one display spot, it cannot be used in a different display spot. So instead of simply multiplying all of the possibilities together 6 * 6 * 6, now there is one fewer possibility each step of the way (since regardless of which plant is chosen, there will only be 5 plants to choose from for the second spot and 4 for the third).
The answers is therefore 6 * 5 * 4 = 120 or D.
Counting principle questions and permutation can intersect when there are limits introduced to the counting principle setup. For example, if the license plate example from above included the stipulation that no number or letter be repeated on a license plate, then the question would be similar to permutation. The number of options would reduce by 1 for each digit series (numbers or letters) that was continuously drawn from.
42. Volume of a prism
Simple Shapes
The volume of a simple 3d object is equal to the area of the base object multiplied by the height of the object
Complex Shapes
The shapes above all have right vertices, meaning that their heights are consistently perpendicular. Not all 3D shapes have perpendicular vertices, which means that their volume equations are more complex.
The most likely weird volume equation to be asked about is:
Other strange area volumes that can be worth knowing:
- Volume of a Right Cone: $$\pi r^\frac$$
Inscribed Volume
One of the more complex volume problems to see on the ACT involves inscribed objects, meaning where one 3D object is inside of another. On the ACT this usually involves a sphere and a circle, either of which can be the interior or exterior object.
The first key to these questions is to draw it out. It will seem complex when it’s first described, but the calculation should become more intuitive once it is drawn out. The objects will have to be described as touching each other at at least one point, but most likely all points. This will then give key information about the relationship between things like the length of the sides of the cube and the radius of the sphere.
43. c = product of roots, b = sum of roots, a = scaling
When factoring an exponential equation (ax2+bx+c into (x-d)(x-e)), the signs and magnitudesof b and c can provide a lot of information that is useful for factoring.
Example Equation: x 2 + 3x − 10
C
C should always be the starting place any time a student factors. It gives direct and specific information about the constant terms of the pair of binomials that would be created by factoring,
C is the equal to the product of the two constants. This means that:
- Magnitude: The magnitude of C is equal to the product of the magnitude of the two constants in the binomials
- Sign: If the sign of C is positive, then the constants in the binomial have the same sign (two negatives or two positives). If the sign of C is negative, then the constants in the binomial have opposite signs (one positive and one negative).
In the example equation above, the magnitude of C is 10, meaning that the constants would be factors of 10. The sign of C is negative, which means that the constants have opposite signs.
B
B is equal to the sum of the two constants. Combining this with what is known from C:
- If the signs of the constants are the same, then the sign of B is the same as the signs of both constants
- If the signs of the constants are opposites, then the sign of B is the same as the constant with the larger magnitude (the bigger number regardless of sign)
A
A is the product of the coefficients in front of the variable terms in the binomials. Most ACT quadratic equations will not have a coefficient in from of the variable term (the coefficient will just be one), so B and C are the main things students should be comfortable with translating.
However, if the A term is something other than 1, it’s worth knowing the following things:
- Normally A will be a perfect square: 1, 4, 9, 16. Meaning that the signs in front of the variable term will be the same in each binomial.
- If A is not a perfect square: It will usually be an odd, prime number such as 3, 5, or 7, as then the coefficients in front of the variables have to be that number and 1.
- A does not affect C, but it will scale the terms that add up to B: since C is just the product of the two variable terms, it is not affected by coefficients on the x term. However, it will affect the variable terms that sum to b.
- If A is a perfect square, then the terms that sum to B will be scaled evenly
- If A is not a perfect square, then one term will be scaled by a larger factor than the other.
44. Circle equations
The default equation of a circle looks like this: (x−h) 2 + (y−k) 2 = r 2 where r is the radius of the circle, and the x and y binomials indicate where the center of the circle is.
- The sign inside of the equations will be the opposite from how the circle moves from the origin. (x-3)means that the center of the circle is located at x = 3, rather than x = -3.
- This is true for both the x and y portions of the equation, rather than just the x term
Circle questions on the ACT math section will normally provide either a graph or an equation. Whichever one is provided must be used to find the one that isn’t provided.
45. Difference of two squares
Difference of two squares is a pattern that can be used to factor certain quadratic equations on the ACT math section. The default formula is that:
a 2 − b 2 = (a+b)(a−b). The terms in this formula can be either variables or constants. There is no center term because the two binomials are conjugates of each other.
When the difference of squares appears on the ACT, the a term is normally a variable and the b term is a constant. This means the left side of identity normally looks like: x 2 − 16.
46. Arithmetic sequence
Arithmetic sequences are similar to function, in that they indicate a way to work with numbers. However, instead of working with numbers continuously (including terms with decimals), sequences work only with whole numbers.
Questions will normally provide a sequence, which students will then have to determine a specific term in (for example, the 27th term in the sequence). However, students can also be given numbers from the sequence and asked to identify a formula that could be the sequence those numbers follow.
Here is an example question from the 2017 April ACT:
Here the question is looking for the 50th term in the sequence. The 3rd and 4ths terms of the sequence are given. Since there is no other information, the sequence likely has to be something quite simple.
There is an increase of 5 in between the 3rd and 4th terms, so the sequence might be: nx = 5 + nx − 1, where xn is the current number of the sequence and xn − 1is the previous number in the sequence.
Since the sequence increases by 5 every term, and there it will take 46 more terms to get from the 4th term to the 50th, that means the 50th term will be 46 x 5 = 230 greater than the 4th term.
Therefore the 50th term is 4th term + 230 = 18 + 230 = 248. So the answer is A.
47. Laws of Sines and Cosines
The Laws of Sines and Cosines are useful for geometry questions involving non-right triangles.
Law of Sines
Questions involving the Law of Sines or Law of Cosines will not normally involve calculating a specific number. Instead, it will often be about plugging numbers into the equation, and then isolating a variable in that equation.
Here is a sample question using the equation:
This is a perimeter question. Since there needs to be a firefighter every 4 meters, the perimeter will need to be divided by 4 to calculate the number of firefighters needed. This eliminates answer C, since it is the only answer not divided by 4.
Answers D and E can also be eliminated, since they utilize an area equation rather than calculating a perimeter.
So the answer must be either A or B. This also makes sense, since the perimeter sum should include 130 meters, as well as the lengths of the other two sides (which will be calculated using the law of sines, so having the two angle ratios for the other two sides makes sense).
For this triangle, the Law of Sines would look like: $$\frac\ = \frac = \frac\ \frac<><>$$.
So the lengths of b or c are equal to $$\frac$$multiplied by the sin of their opposite angles. So the correct equation should have sin(91) in both denominators and then sin(42) and sin(47) multiplied by 130 in each numerator. So the correct answer is A.
Law of Cosines
The Law of Cosines is similar to the Law of Sines, but rather than comparing individual ratios of each side/angle pair it includes all:
48. Triangle opposite side rule
The sides of triangles and the angles opposite them are related and this relationship may be useful on certain triangle questions.
The most important thing to remember is that the length of the side of a triangle is proportional to the angle opposite that angle. So the largest side will also have the largest angle opposite it.
This is why the hypotenuse of a triangle always has the 90 degree angle opposite it in a right triangle.
49. Change the base – If 9^x = 27^5, what is x?
Some questions will require changing the base of a number with an exponent. For example,
16 x = 256 4 , then what is x?
256 = 16 2 , meaning that the original equation can be rewritten as 16 x = (16 2 ) 4 = 16 8
This will usually involve numbers whose square or cube roots are simple integers.
So it’s useful to get familiar recognizing the first few numbers in these exponent sequences:
| Number | x 2 | x 3 | x 4 |
|---|
| 2 | 4 | 8 | 16 |
|---|
| 3 | 9 | 27 | 81 |
|---|
| 4 | 16 | 64 | 256 |
|---|
50. Similar triangles
Similar triangles are triangles that have the same angles, but different side lengths. The easiest way to think of this is to imagine a triangle being scaled up or scaled down.
- With similar triangles: angles are the same but side lengths are different
- To maintain the same internal angles, the ratio of the sides of similar triangles has to remain consistent.
If specific numbers aren’t given for a similar triangles problem, then it’s best to look at the shape of the triangle.
If the original triangle has relatively equilateral sides (is roughly symmetric) then a similar triangle will have to be similarly symmetrical. If the original triangle looks stretched to one side or very flat (long base and long sides), then the similar triangle has to be the same as well.
51. Probability with “not”
Probability questions with not are similar to normal probability questions. The main difference is that the odds that need to be calculated are the odds of failure rather than success.
In the earlier examples on probability, the focus was on drawing a specific color of starburst out of a bag. The odds there were simply: $$\frac$$
If the question were changed to “what are the odds of drawing a color that is not red out of a bag of starbursts”, then the odds are simply equal to: $$\frac$$. So instead of adding up the red starbursts, add up all of the other colors.
Another way to think about it is that the probability of something not happening is equal to: 1 − Ps where Psis defined as the odds of that same thing happening. Since probability is just a likelihood out of 1 (or 100%), the odds of success and failure always add up to 1. So a student can also simply calculate the odds of success, and then subtract that from 1.
52. Factors
Factors refer to numbers that can be multiplied together to reach a specific product. For example, 2 and 3 are factors of 6 because their product is 6. Quadratic factors were discussed above, while this section will focus on whole number/constant factors. These types of factors usually come up on the ACT in the following ways:
Greatest Common Factor
Similar to the Least Common Multiple expressed above, the Greatest Common Factor is the largest factor that is shared between a set of numbers.
So instead of looking for the smallest number that every number in the set factors into. Now, the goal is to find the largest number of the factors into the numbers in the set. Put another way:
- Least Common Multiple looks at the set of numbers as factors
- Greatest Common Factor looks for factors of the set
For example, the Greatest Common Factor of the set: 12, 24, 42 is 6, because it is the last whole number that divides evenly into all numbers of the set.
Prime Factorization
Some Greatest Common Factor questions will ask about the concept of Prime Factorization.
- Prime Factorization means factoring a number using only prime numbers. In other words, figuring out what prime numbers need to be multiplied together to reach a given number.
For example, the prime factorization of 60 would look like this: 5 •3 •2 •2
To get the prime factorization of any number, factor it normally, and then keep factoring until all of the numbers in the sequence are prime. Regardless of which factors you start with, the end product will be the same.
60 = 20 • 3 = 5 • 4 • 3= 5 • 3 • 2 • 2
60 = 15 • 4= 5 • 3 • 4= 5 • 3 • 2 • 2
53. Trapezoid
Trapezoid questions on the ACT usually relate to the area of the shape or the internal angles of the shape. As a result, it’s worth knowing that the area formula for a trapezoid is:
And that a trapezoid has 360 degrees of internal angles inside of it.
54. Domain and Range
Domain and Range are two common terms used in reference to functions and data sets, respectively.
- Domain refers to the set of all possible values of x for a given function
- Range refers to the distance between the highest and lowest numbers in a data set, OR it can also refer to all of the possible output (or y) values for a given function
- When referring to a data set, the range is one, discrete number: “The range of the data set is 36”
- When referring to a function, the range will be a set of all possible outputs of the function
55. Conjugates
Conjugates are binomial pairs that have the same numbers/variables within them, but opposite signs. So for example (x-1) and (x+1) are conjugates. Conjugates have special properties that can be useful to know when they appear in questions or answers on the ACT.
Simplifying and Rationalizing Denominators
This particularly true when the original denominator contains either a radical (square root) or imaginary numbers (i = sqrt(-1)).
For instance, if the fraction $$\frac$$is given, then this can be multiplied by $$\frac$$(standard algebra rule: multiplying it by one does not change the value of the fraction).
This leads to the fraction: $$\frac>$$. i 2 = − 1, so − 3i 2 = + 3. So the denominator of the fraction can be simplified to 103, giving the final fraction $$\frac$$.
Imaginary roots of polynomials always come in conjugate pairs. So whenever you see a binomial that includes i, think conjugate.
Conjugates and Quadratics
When conjugates are factors of quadratic equations they cancel out the middle terms of the equation and leave only a 2 + c. Whenever this equation appears in a problem, it means that the equation can be factored into a conjugate pair.
56. Exponential Growth/Decay
The Exponential Growth/Decay formula is one of the most common equation forms to see on the ACT. It can be used to express a multitude of different situations, which means it can show up theoretically as a variable function or practically within a word problem. The formula looks like this:
Key Takeaways
- If it is a growth function, then the number inside of the parenthesis should be greater than 1
- If it is a decay function, then the number inside of the parenthesis should be less than 1
- Final is the output of the equation. It is like the y-value in a linear equation. It is the thing that is being calculated
- Initial is the starting point of the equation. It is like the intercept in a linear equation. It says what the starting point is for the equation.
- T is usually the main variable in the equation. It says how long the equation needs to be run for (how many cycles it goes through), as well as how often it will happen.
How It Shows Up In Problems
Exponential growth and decay problems are often word problems. The problem will describe an example of a real-life situation that utilizes this formula. The problem will then either:
- Present the correct version of the formula, and then ask students to interpret one element of the formula
- Give several different possible iterations of the formula, from which the student must choose the correct version of the formula as described in the problem
Common Exponential Growth/Decay Formulas or Questions
This formula can be applied to a wide variety of situations, but there are a few variations of it that have shown up on multiple exams and are therefore more likely to show up on any given exam.
Financial/Compound Interest
The compound interest formula shows how an investment growths with time. The exponent is normally equal to the number of years that the investment grows for or the number of months divided by 12.
Half-Life
The “half-life” says that after X number of years, the amount of a given element remaining will be half of what was initially there. This is why the rate inside of the parenthesis is $$\frac$$and the exponent will be equal to $$\frac>$$.
Doubling
The doubling formula is the same as the half-life formula, but with a two inside of the parenthesis and where the denominator of the exponent is the time it takes to double rather than the time it takes to halve. This can be expressed in years, but is often expressed in smaller units like hours.
57. Weighted average
Weighted averages are a way to combine the averages of two data sets of different sizes. This is similar to calculating sum totals on a histogram.
For example, two classes are described. Class A has 10 students who had an average test score of 75. Class B has 15 students with an average test score of 90. A question might then ask what the combined average of the two classes is.
The most basic way is to simply calculate the total scores of both groups: number of students • average score, and then add those two totals together and divide by the total number of students.
(75 •10) + (90•15) = 750 + 1350 = 2100 $$\frac = 84$$
The other way to do this is to take the ratio of the two numbers of students, and then weight each of the averages according to that ratio.
$$\frac = \frac = \frac$$ There are 2 students in group A for every 3 students in group B. Multiply both averages by their weights in the ratio, and then divide by the total number of students within that ratio.
(75•2) + (90•3) = 150 + 270 = 420 $$\frac = 84$$
The math is the same as above, but scaled down by a factor of 5, which makes the numbers a bit easier to work with.
Finally, one can also take the ratio numbers, and use them to calculate which portion of each average should be used for the weight average.
Group A weight = $$\frac$$ Group A Weight * Group A Average $$\frac \bullet 75 = 30$$
Group B weight = $$\frac$$ Group B Weight * Group B Average $$\frac \bullet 90 = 54$$
Any of these methods yields the same number, so students should choose whichever seems most easy or intuitive to them.
58. Reciprocal vs. Inverse trig
Reciprocal Trig Functions
The normal trigonometric functions are sine, cosine, and tangent. However, there are also expressions for the reciprocals of these functions.
These functions are secant, cosecant, and cotangent. They are equal to the reciprocals of the ratios from SOHCAHTOA:
The easiest way to remember this is that for sine and cosine, each pair should have one S and one C. So Secant and Cosine are paired and Cosecant and Sine are paired. Cotangent and tangent are generally much easier to remember, as they are the only identities that have a T in them.
How It Appears on the Exam
The reciprocal trig functions are generally used in right triangle problems. This means they are used just like in a normal SOHCAHTOA problem, but where the ratios are inverted.
More rare problems will involve looking at graphs of these functions. The important thing to remember is that just like their functions, their graphs will also be the reciprocals of the sine and cosine graphs (cotangent graphs are not likely to come up).
Sine there are points where sine and cosine are = 0, these graphs have vertical asymptotes at those points. This is why the graphs of these functions look like this:
Inverse Trig Functions
Inverse trig functions are how to calculate angles from the SOHCAHTOA ratios
- They are written using the normal trig functions, but with negative exponents: sin −1 , cos −1 , tan −1
Since $$sin(\theta) = \frac>>$$, $$\textn^(\frac>>) = \theta$$, where θis the angle of the triangle with those opposite and hypotenuse sides.
Problems using the inverse trig functions generally will not ask a student to actually calculate a numerical value for the angle. Instead, students will simply be asked to understand how the inverse trig functions are applied to find an angle measure.
59. Parallelogram
Parallelograms are an important thing to review for the ACT, because they tend to be something students are less familiar or comfortable with. Problems involving parallelograms can therefore become an easy place for students to lose points. Some key things to remember about parallelograms are as follows.
Adjacent Angles
- Adjacent angles in a parallelogram are supplementary, meaning that they add up to 180
Area Formula
The area of a parallelogram has the same formula as a rectangle: A = bh, but because it’s vertical sides are not perpendicular to the base, the height is not equal to the length of the side of the shape.
- The area of a parallelogram is equal to the shape’s base multiplied by the height of a vertical line drawn from the shape’s top left corner down to the base
This is similar to finding the height of a non-right triangle, where the height has to be created by dropping a vertex down to the base.
60. Use the radius
Any time a circle appears in a problem, the first thing to think about is the radius of the circle. It is the core piece of information about the circle, and it is the thing that is most commonly used to calculate or infer information about things that appear in the rest of the problem.
The radius might be given numerically, but sometimes it can be given in a problem without an actual numerical value attached to it. For example, this is a problem from the December 2020 ACT.
Here it is not necessary to calculate the value of r, but instead simply to understand the relationship between each circle’s radii. The first circle has a radius of r, the second circle has radius of 2r, and the third circle has a radius of 3r.
The perimeter function for a circle is 2πr. Therefore, the sum of the circumferences of the circles is 2πr + 2π(2r) + 2π(3r). This can be rewritten as 6(2πr). The question said that the circumference of the smallest circle is 10, and since the circumference of the first circle = 2πr, then 6(2πr) = 6 • 10 = 60. So the answer is G.
61. Value/frequency charts
Frequency charts summarize data sets by showing how often each data point appears within a set. This can be displayed as either a table or a graph. Questions about these charts will generally focus on the mean/median of the data set.
Students should be comfortable calculating the mean/median from either of these two types of charts. Like other mean and median questions, ACT questions will often ask about what would happen to the mean/median of the data set if a new number was added or if a previous number was changed.
62. Pythagorean Triples
There are a few common triangles that appear on the ACT math section that are called “Pythagorean Triples”. These are right triangles where if a student knows two of the sides, they also can know the third side automatically without having to use the pythagorean theorem. These triangles are known as 3,4,5 and 5-12-13 triangles.
The key for these triangles is to remember that the largest number has to be the hypotenuse for these to be true. So if a triangle has side lengths 5 and 13, but they are the two shorter sides, then it is not a 5-12-13 triangle.
These pythagorean triples remain true even when the numbers are scaled up by any given integer. So, when 3,4,5 becomes 6,8,10 (each original number multiplied by 2), it’s still a pythagorean triple. As long as the proportions remain consistent, the relationship remains true for these types of triangles.
63. Lowest Common Denominator
The Lowest Common Denominator is the smallest number that can be a denominator for two separate fractions. Another way to think of it is that it’s the smallest number that both original denominators can factor into. This is why the most common way to find the LCD is simply to multiply the two denominators together. For example:
$$\frac + \frac$$ 2 • 5 = 10 That makes 10 the lowest common denominator.
To get both fractions to that form, multiply them by the opposite denominator divided by itself. In the above example, that means multiply $$\frac$$by $$\frac$$. Since anything divided by itself = 1, that means the fraction is not being changed, only scaled.
The above equation thus becomes: $$\frac + \frac = \frac$$
64. System of equations with three equations
Most systems of equations problems only involve two equations, but occasionally they’ll include three equations, often in the form of a word problem. For these problems, students need to isolate two different variables in two of the equations. These then need to be substituted in for each other in the 3rd equation.
The main key is that at some point all of the variables in the equation will need to be substituted or eliminated until only one variable and a constant remain.
65. Compare numbers
Some questions on the ACT math section might require comparing the size of numbers expressed in different forms, or where the expressions contain only variables.
For these questions, it’s important to remember simple things about different expressions:
Radicals/Square Roots: The equivalent/value of a square root gets large as the number under the root/radical gets larger, but it does so at a decreasing rate. So going from 1 to 4 (an increase of 3) leads to a doubling of the equivalent value: = 2 vs. = 1.
However, going from 49 to 52 (another increase of 3) leads to a much smaller increase in equivalent value: = 7 vs. = 7.211 .
x a Exponents: Larger exponents make larger numbers. Each increase in a positive exponent leads to an even larger growth in the number affected (increases at an increasing rate).
Negative exponents do the opposite. Negative exponents lead to smaller numbers and the numbers will shrink at an increasing rate.
.abcde Decimals: The leading digits of decimals determine their relative value. If the first digit of a decimal has a greater magnitude than the first digit of another decimal, then the first decimal has a larger overall value, regardless of the following digits. If the first digit or first several digits are the same, then the relative size of the decimals is determined by the first non-shared digit in the decimals.
|a| Absolute Value: Absolute value is about magnitude, so the further away the digit inside the square root is from 0 (regardless of its positive or negative), the larger the equivalent value will be.
Sample Question
P is a positive number. n is a negative number, and p has a greater magnitude than n. The question wants the biggest number possible. All of the answers are inside of an absolute value, meaning that the largest number will be the fraction that results in the furthest magnitude away from 0.
Looking at the answers:
F and G: These answers have the same numerators, but different denominators. P > n, so the numerator is positive. P > n, so the denominator is greater in F. A larger denominator means a smaller magnitude, so G has a greater value than F.
H, J, and K: These answers all have the same numerator, but different denominators. K’s denominator (n) will be the smallest, since n is a negative number smaller than p, which means that p-n > p and the magnitude of p is greater than the magnitude of n (again, absolute value means that magnitude is the important measure).
G and K are the last two options left.
They are the same, except for their numerators. The question wants the largest value. Bigger denominator = larger value. Since n < 0 and p >0, (p − n) > (p+n). Therefore, G has the largest denominator and thus the larger value.
66. Translations
Translation questions will involve moving a shape or the graph of a function in one of three ways: translation, rotation, or reflection.
Translation
- Translation: the orientation of the shape or graph will not change, but it will move to a new space in the xy-plane.
Rotation
- Rotation: the shape or graph will be rotated about the origin. This means that the origin will be the center of rotation, so whichever point of the graph or shape is closest to the origin will remain closest to the origin, but it will likely change to a new quadrant of the xy-plane.
Reflection
- Reflection: the shape or graph will be “flipped” along the y-axis or x-axis. Another way to phrase this is that a reflection creates the “mirror” image of the shape/graph on the other side of the axis that it is being reflected along.
67. Stem and Leaf Plots
Stem and leaf plot questions are a way of displaying a set of data in a concise manner. The name refers to the fact that the leading digit (usually the tens digit) is removed from each data point, and instead is used as a label for a row that includes all data points with the same tens digit.
Stem and Leaf Plot questions will look like this:
The important thing to remember is that each “leaf” represents a different data point, while the stem only refers to their tens digit.
The question asks about students who jumped at least 60 inches, so for data points ≥ 60. There are 5 students in the 6-stem row and 3 students in the 7-stem row. This means that 8 students had a score of at least 60, out of a total of 19 students.
The odds are therefore $$\frac$$, so the answer is K.
68. Converting Angles to Radians or Vice Versa
Angles within a triangle can be expressed one of two ways. This can either be in degrees or radians.
- Each circle contains 360 degrees or 2πradians
Therefore to convert between the two:
- Converting from degrees to radians: multiply byπand divide by 180
- Converting from radians to degrees: multiply by 180 and divide byπ
69. Factorials
Factorials express a product of multiple consecutive integers starting at 1. Factorials look like an exclamation point, which is placed next to either a variable or a whole number. However, even when placed next to a variable it still indicates a whole number reached by multiplying that variable by all of the integers before that variable.
The main trick with factorials is to remember that when they are divided by each other, the smaller factorial will divide out and the remainder will be the product of all remaining integers from the larger factorial.
For example, $$\frac$$=$$\frac$$= 17 • 16 • 15. So even though the fraction is being “simplified”, it is the larger numbers that will remain rather than the smaller numbers.
Since factorials are products, the above rule is true regardless of whether the larger number is in the numerator or denominator (it just determines where the remainder goes).
70. Faces on a 3D Figure
The ACT sometimes likes to include questions that involve 3D objects that have a number of faces on the object. The faces on the object will be common geometric shapes such as triangles, squares, or occasionally more complex shapes such as hexagons.
Key things to remember for faces on 3D figure questions are:
- Shared Edges: Faces on the shape will have shared edges. Each edge will be shared by two faces, and each face will have the same numbers of edges as it has sides.
- Area: the area of the shape is equal to the summed area of all of the individual faces
Here is a sample question from the April 2017 ACT:
There are 20 sides to the die, each of which has 3 edges. Each edge is shared by two faces. Therefore the number of sides = $$\frac<(20 \bullet 3)> = 30$$. The answer is D.
71. Step Function
A step function is a function that includes a series of horizontal lines, which jump from one to the other at regular intervals.
- A step function graph will include open and closed dots. These open and closed dots pairs are where the function “jumps”
- The jumps will usually occur at whole number values of the x-axis
Here is an example question from the 2017 ACT:
For ABC cabs, the price of a taxi fare jumps $2 each time the fare reaches a new whole number value. This means for any distance of less than 1 whole mile, the fare is $2. As soon as 1 whole mile is reached, the price then jumps to $4. It stays at that price until 2 full miles are reached, and so on.
The prices on the Y-axis are flat fares (they are not per mile). So to compare the price of a 5-mile fare for each company, a student must simply compare the y-values at x = 5 for both companies. For a 5-mile trip, ABC will charge $12 and Tary will charge $9. Therefore, the cheaper fare is $9.
Bonus: General Math Problem Advice
Things to Do With the Questions
Simplify
Sometimes an equation will be given in an unsimplified form. If a formula is given unsimplified, the first thing to do is to simplify it. The question is not written that way by accident, it’s done to make the problem seem more challenging or harder to solve. Once it’s been simplified, there is a good chance it will look more familiar or more manageable than it did in its original form.
Pull Out the Numbers
This is especially a good idea for word problems that seem confusing on first read. Remember that some words in word problems aren’t useful mathematically, and others are only useful when translated into something actually mathematical. Pull out numbers, variables, and any mathematical statements of fact (relative sizes, positive/negatives, sample sizes, etc).
Draw it Out
Especially true for graphing questions or word problems (especially if they seem to involve some kind of geometry).
Things to Do With the Answers
Look at the Answers
If all of the answers have a certain denominator or include a special symbol (like a square root or negative), then that says the question revolves around that concept. Or that one needs to take the number/formula that is originally given and convert it to look more like the answer choices.
Plug in Answers
This should be done when a student doesn’t know how to approach a problem conceptually. It’s important to remember that it can take a fair amount of time, depending on how complex the problem/equation given is.
Eliminate Answers
It’s never optimal to have to guess, but if one has to guess, then it’s best to give oneself the greatest odds possible of getting things right. When there are five possible answers, even eliminating two wrong answers can improve the odds of guessing correctly from ⅕ to ⅓.
It’s easiest to eliminate answers in batches. For example, if a student knows that the number has to be positive to be correct, then all of the negative answers can be eliminated. Same is true for if the answer needs to be even or to have a certain denominator.
Boost Your ACT Math Score With SoFlo Tutors
Getting comfortable with these 71 concepts is a great start towards being successful on the ACT Math section. But there is always more to be done to be prepared, both for the math section and the English, Reading, and Science sections as well.
This can seem intimidating to do alone, and that’s why SoFlo Tutoring offers smart, reliable, and helpful tutoring services that are personalized for each individual student. High-scoring, expert tutors can help anyone to set goals, practice problems and concepts, and put themselves in the best possible position to be successful. Schedule a free consultation with SoFlo today for help on the SAT or ACT!
