SparkNotes

Money Problems

The SAT makers love writing math questions that involve money. This isn't so surprising: We all buy things and know how money works. But sometimes things can get a little complicated. For example, you might come across a question that asks you to calculate someone’s salary or how much someone has to pay for a gift, and multiple parts of these questions will be in terms of variables. We'll call these questions variable money problems.

The key to solving variable money problems is to not freak out about the variables. Most of the questions use fairly simple concepts; the test makers are just trying to throw you off by using two, three, or even more variables. Here's an example:

Samuel earns m dollars a week working at the car dealer, plus an additional k dollars for every car that he sells. If Samuel sold 9 cars in the past week, in terms of m and k, how much did he earn for that week?

(A)   m + k/9
(B)   9m + k
(C)   m + k + 9
(D)   m + 9k
(E)    (m + k)/9

This question is asking us to find Samuel’s salary for one particular week. The first step is to recognize that there are two portions to his salary: a base portion (money he earns every week, regardless of how many cars he sells) and a commission (money he earns for each car that he sells). Now, we need to find the value of each:

Base: Samuel earns m dollars a week.

Commission: Samuel earns k dollars for every car that he sells. In the past week, he sold 9 cars.

Samuel’s base salary, m, is not affected by the number of cars that he sold. He also earns k dollars for each of the 9 cars that he sold, so 9k dollars. His total salary is m + 9k; your answer is (D).

That question was pretty straightforward. Let’s look at a trickier one:

For Lisa’s birthday, n of her friends decided to contribute equal shares to buy her a new camera that costs f dollars. If b of her friends forget to contribute money, which of the following represents the additional amount, in dollars, that each of the remaining friends must contribute to pay for the camera?

(A)   fb / [n(n-b)]
(B)   f / [n-b]
(C)   [f / n] – b
(D)   f / [n+b]
(E)    [n – b] / [f / n]

Now we have three variables, n, b, and f) and our answer choices look a lot more complicated. But don’t psych yourself out just yet. First, let’s determine how many people are contributing to the gift. Originally all n of Lisa’s friends intended to throw in some dough. However, b friends "forget" (yeah, right) to pay up, so only (n – b) friends actually put in money.

Now, the question asks how much additional money each friend will have to contribute to the gift, after b of them forget to pay. In order to calculate this value, we need to set up an equation to determine how much money each of the original n friends intended to contribute and how much each of the (n – b) friends actually had to contribute. The difference between these two values represents the additional money.

Additional money contributed per person =

Actual amount contributed by (n – b) friends – Amount n friends intended to contribute

If all n friends had contributed, how much would each person have had to pay? The camera costs f dollars and n friends contribute equally:

f dollars / n friends = Amount n friends intended to contribute

But only (n – b) friends contribute to pay for the f dollar camera:

f dollars / (n-b) friends = Actual amount contributed by (n - b) friends

Plug those values into our overall equation:

Additional money contributed per person = f / (n-b) - f / n

This doesn’t look like any of the answer choices yet, so we probably have to simplify our equation. First, we need to find a common denominator in order the subtract the two fractions. The easiest common denominator will be n(n-b). Then, we need to multiply the numerator by n in the first fraction, and (n-b) in the second fraction.

f / (n - b) - f / n = fn / [n(n-b)] - f(n-b) / [n(n-b)]

Now that both fractions have the same denominator, we can subtract the first fraction from the second:

[fn – f(n-b)] / [n(n-b)]

Multiply out the numerator and combine like terms:

[fn – (fn – fb)] / [n(n-b)] = [fn – fn + fb] / [n(n-b)]

= fb / [n(n-b)]

Our answer is (A).

This question is definitely one of the most difficult you'll find on the SAT. If you have time, you might want to double-check your answer by plugging in values for the variables and seeing if the answer you get makes sense.

Pick values that will be easy to work with. Let’s say 5 friends had intended to purchase the camera for Lisa, and 3 forgot to contribute money. The camera costs $100. This means n = 5, b = 3, and f = 100. Originally, the 5 friends meant to split the $100 camera, or to chip in $20 each. However, 3 forgot to pay, so only 2 are splitting the $100 price. They each have to pay $50. What is the additional amount that each of the 2 friends pay? $50 - $20 = $30.

Does this match answer (A)?

fb / [n(n-b)] = [100 * 3] / [5*(5-3)]

= 300 / [5*2] = 300 / 10 = 30

The math works; your answer is correct.

Here are some keys for variable money problems:

• Understand the question and what the variables mean. When the question includes variables, you need to understand what they mean in order to work toward the answer.
• Break down the question as much as possible. In the first question, we broke down Samuel’s salary into two parts. In the second question, we also broke down the “additional money contributed” into two parts. Working on each part separately helps to simplify the question.
• Double-check. If the question is fairly complicated and you're not confident in your answer, go back and work out the problem with real numbers. Plug in actual values for the variables and do the math. Remember to choose values that make your calculation easy.

For more practice, take this quiz.

Got money problems? Ones that have to do with SAT math? Let us know about them in the comments, or email testpreptutor@sparknotes.com.

Related Post: Finding Patterns in SAT Math, Part II

By: Adam

Topics: SAT, act, math, word problems, variables

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