Knowsys SAT Blog: Equations

Algebra: Equations

Read the following SAT test question and then select the correct answer.

Always read the question carefully and identify the bottom line so that you do not waste time finding something unrelated to the question. Assess your options for solving the problem and choose the most efficient method to attack the problem. When you have an answer, take a second or two to loop back and make sure that your answer matches the bottom line.

If a, b, and c are numbers such that $\frac{a}{b}=3$ and $\frac{b}{c}=7$ , then $\frac{a+b}{b+c}$ is equal to which of the following?

Bottom line: $\frac{a+b}{b+c}$

Assess your Options: There are two ways that you can solve this equation, and both will arrive at the correct answer. You can solve it algebraically by substituting information into the equation, or you can pick your own numbers for the variables. Choose the method that is easier and faster for you.

Attack the problem: If you are going to solve a problem algebraically, always look for ways to simplify the problem that you are given. In this case, you will want to get rid of unnecessary fractions. Look at the first piece of information that you are given. If a divided by b is 3, you can get rid of the fraction by multiplying each side of the equation by b.

Now you have a = 3b.

Look at the numerator (the top part of the fraction) of your bottom line. You can now make sure that there is only one variable in this portion of the equation. Substitute 3b for a. Now you have 3b + b, which will simplify to 4b.

Here are the steps you just completed:

$\frac{a+b}{b+c}=\frac{3b+b}{b+c}=\frac{4b}{b+c}$

Look at the denominator of your equation. How can you simplify b + c? You might be tempted to substitute 7c for b, but remember your goal is to get to a number without a variable. If you have the same variable in the top and bottom, the two variables cancel. Therefore, you need to find what c is equal to in terms of b.

When you are given the information that b divided by c is 7, then you know that c divided by b is 1 over 7. You flip both equations. Solve for c by multiplying both sides of the equation by b.

$\frac{b}{c}=7$ so $\frac{c}{b}=\frac{1}{7}$ so $c =\frac{1}{7}b$

Plug this information into your bottom line equation and combine like terms.

$\frac{4b}{b+c}=\frac{4b}{b+\frac{1}{7}b}=\frac{4b}{\frac{8}{7}b}$

A fraction over a fraction is ugly, but remember that dividing by a fraction is the same thing as multiplying by the reciprocal of that fraction. In other words:

$\frac{4b}{\frac{8}{7}b}=4b(\frac{7}{8b})=4(\frac{7}{8})=\frac{28}{8}=\frac{7}{2}$

Notice that the variable b moves to the bottom of the second fraction and cancels out. You solved the equation!

Alternatively: If you dislike algebra, use the strategy of picking numbers to solve this problem. You want to get rid of ugly fractions, and the best way to do that is to put a number over 1. You cannot just put b = 1 because b affects two different equations and you might end up with numbers that are difficult to use in your other equation. However, c is on the bottom of a fraction in one equation. Pick c = 1. Plug 1 into the second piece of information with c and solve for b.