Linear equations in one variable
Solving a Linear Equation With a Constant
If the equation above is true, and is a constant, which of the following is the value of ?
Solution:
Add to both sides.
Divide both sides by .
Free SAT® math practice questions
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Algebra
Practice linear equations, systems of equations and inequalities, and word problems with worked solutions for every question.
Linear word problems
Solving a Linear System Word Problem
Three friends, Adam, Benjamin and Claire, are playing a game. Adam gets three more points than Claire. Benjamin’s points, on the other hand, are two times the difference of Claire’s points and five. Adam and Benjamin get an equal number of points. How many points does Adam get?
Correct answer: 16
Solution:
Let’s denote Claire’s points by . According to the problem, Adam gets three more points than Claire. Hence, Adam gets points. Similarly, Benjamin gets two times the difference between Claire’s points and , which is equal to points. According to the problem, Adam and Benjamin get an equal number of points. Thus,
Subtract from both sides.
Add to both sides.
Hence, Adam gets points.
Advanced Math
Practice exponential growth, polynomials, radical equations, function composition and quadratics with worked solutions for every question.
Exponential growth
Exponential Growth of an Investment
The estimated annual value increase of an investment is percent. The value of the initial investment was , and after years the value is , where is a constant. Which of the following is closest to the value of ?
Solution:
According to the problem, the initial value is , and it increases by percent each year. Hence, after one year, the value is
Similarly, after two years, the value is
Furthermore, after years, the value is
According to the problem, the value of the investment is after years. Let’s solve for .
We can see that
Polynomial factors from a table
Finding a Polynomial Factor From a Table
Function is a polynomial. The table above shows selected values of and the corresponding values of . Based on the table, which of the following must be a factor of ?
Solution:
We know that if is a polynomial function and , then must be a factor of . We can see from the table that . Hence, must be a factor of .
Problem-Solving and Data Analysis
Practice reading tables and dot plots, probability, and statistics with worked solutions for every question.
Range and standard deviation
Comparing Range and Standard Deviation
Which of the following statements correctly compares the ranges and standard deviations of data set A and data set B, that are represented by the dot plots above?
Solution:
In data set A, the greatest value is and the smallest value is . Hence, the range of data set A is . Similarly, in data set B, the greatest value is and the smallest value is . Hence, the range of data set B is . Thus, the range of data set A is greater than the range of data set B.
We can see that the values of data set B are more concentrated near the mean, and the values of data set A are more spread out. Hence, the standard deviation of data set A is greater than the standard deviation of data set B.
The range of data set A is greater than the range of data set B, and the standard deviation of data set A is greater than the standard deviation of data set B.
Two-way tables and probability
Probability From a Two-Way Table
| Red | Blue | Green | Total | |
|---|---|---|---|---|
| Small | 2 | 4 | 3 | 9 |
| Big | 4 | 1 | 3 | 8 |
| Total | 6 | 5 | 6 | 17 |
The table above summarizes the sizes and colors of building blocks in a toy set. If a big block is selected at random, what is the probability that it is red?
Correct answer: 1/2
Solution:
We can see from the table that there are a total of big blocks, and of them are red. Hence, if a big block is selected at random, the probability of it being red is
Geometry and Trigonometry
Practice right triangles, similar triangles, circles and volume with worked solutions for every question.
Right triangles / Pythagoras
Finding a Leg Length in a Right Triangle
Which of the following is equal to the value of in the right triangle above?
Solution:
The length of the hypotenuse is , and the lengths of the legs are and . According to the Pythagorean Theorem:
Subtract from both sides.
As a length, must be positive. Hence,
Radians and degrees
Converting Radians to Degrees
The measure of angle in the picture above is , in radians. What is the measure of angle , in degrees?
Correct answer: 120
Solution:
We know that radians is equal to . Hence,
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