Saturday, February 27, 2021

ACT math practice: business sales algebra


The following question, released into the public domain by ACT Inc., is reproduced here in order to help high school students and their parents become more familiar with the types of math problems students will be expected to solve on the ACT college entrance exam.

Sales for a business were 3 million dollars more the second year than the first, and sales for the third year were double the sales for the second year. If sales for the third year were 38 million dollars, what were sales, in millions of dollars, for the first year?

Correct answer and Common Core standards

Sales in year 1 were $16 million.

In the seventh-grade Common Core math standards, students are required to

Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

The standard appears under the expressions and equations section … “Solve real-life and mathematical problems using numerical and algebraic expressions and equations” … “Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.”

Solution strategy (there are others)

Describe the situation with an equation and solve that equation.

Let x = sales for the first year in millions of dollars. Then,

\textrm{Sales in second year} = x+3
\textrm{Sales in third year} = 2(x+3)

We are given the sales for the third year, $38 million, so we set the last expression in our work above equal to that and find x:

2(x+3) = 38
x+3 = 19
x = 16

Checking our work, we can see that sales in the second year were $3 million more than that, or $19 million. Then, sales in the third year were double what they were in the second year, so $38 million, which matches what we were given as the sales in the third year.

Paul Katula
Paul Katula is the executive editor of the Voxitatis Research Foundation, which publishes this blog. For more information, see the About page.

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