A third-grade teacher has n boxes, each containing 12 pencils

A third-grade teacher has $n$ boxes, each containing $12$ pencils. After the teacher gives $p$ pencils to each student in the class, the teacher has $t$ pencils left over. Which of the following represents the number of students in the class?

  1. ${12n-t}/p$
  2. ${12n+t}/p$
  3. ${12n}/p-t$
  4. ${12p-t}/n$
  5. ${12p+t}/n$

So, you were trying to be a good test taker and practice for the GRE with PowerPrep online. Buuuut then you had some questions about the quant section—specifically question 17 of Section 6 of Practice Test 1. Those questions testing our knowledge of Operations with Algebraic Expressions can be kind of tricky, but never fear, PrepScholar has got your back!

Survey the Question

Let’s search the problem for clues as to what it will be testing, as this will help shift our minds to think about what type of math knowledge we’ll use to solve this question. Pay attention to any words that sound math-specific and anything special about what the numbers look like, and mark them on our paper.

Boxes of pencils…passing out some of the pencils…counting the remaining pencils…this definitely sounds like we’ll be deeply involved in translating words into a math equation which sounds like it’ll focus on Operations with Algebraic Expressions. Let’s keep what we’ve learned about this skill at the tip of our minds as we approach this question.

What Do We Know?

Let’s carefully read through the question and make a list of the things that we know.

 

  1. A teacher has $n$ boxes of pencils
  2. Each box of pencils has $12$ pencils
  3. Each student gets $p$ pencils
  4. There are $t$ pencils leftover
  5. We want to know how many students there are in this class

 

Develop a Plan

Definitely a lot of information to process here. First, we should realize that if we can come up with an equation to describe this question, then we can just solve that equation for the variable we’re interested in: the number of students. It definitely seems more logical to write an equation for the pencils than for the students, so let’s do that.

Solve the Question

Logically, the number of pencils leftover should be the difference between the number of pencils at the start and the number of pencils given to the students:

$$\Leftover \Pencils = \Pencils \at \Start – \Pencils \Given \to \Students$$

The question tells us that the leftover pencils is the variable $t$. Let’s put that into our equation.

$$t = \Pencils \at \Start – \Pencils \Given \to \Students$$

Okay, now we need to figure out how many pencils the teacher started with. She started with $n$ boxes of pencils, and each box had $12$ pencils. So for example, if she started with $1$ box of pencils, then she would have started with $12$ pencils. Or if she started with $2$ boxes of pencils, then she would have started with $24$ pencils. If we’re ever confused,
we can always plus in numbers like these, logically think about the answer, and then write an expression based upon our findings. Definitely seems like we need to multiply $12$ by the number of boxes $(n)$ to get the starting number of pencils.
Let’s put this into our grand equation now.

$$t = 12n – \Pencils \Given \to \Students$$

Now…for the pencils given to the students. Since each student got $p$ pencils, the total number of pencils given to the students should just be the number of students multiplied by $p$. Let’s use the variable $s$ to stand for the number of students, and then put $s·t$ into our equation for the pencils given to the students.

$$t = 12n – s·p$$

Expertly done! Now let’s solve this equation for $s$, then we’ll have our answer! We want to isolate the variable $s$, so we’ll use our order of operations (PEMDAS) backwards to accomplish this (PEMDAS = Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). Let’s first start by subtracting $12n$ from both sides of the equation, then dividing by $p$.

$12n-s·p$ $=$ $t$
$ $ $ $
$-s·p$ $=$ $t-12n$
$ $ $ $
$-s$ $=$ ${t-12n}/p$

Almost done! Now if we multiply both sides of our equation by $-1$, we’ll have solved for $s$ :

$$s={12n-t}/p$$

Looking at our answer choices, we can see that the correct answer is A, ${12n-t}/p$.

What Did We Learn

Trying to figure out this question without an equation would have been insanely difficult. Let’s always remember that math is a language, and we translate languages one sentence, one phrase at a time. Doing so definitely made this question much more manageable than it otherwise would have been.

 

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