Last year Kate spent between 1/4 and 1/3 of her gross income on

Last year Kate spent between $1/4$ and $1/3$ of her gross income on her mortgage payments. If Kate spent $13{,}470$ on her mortgage payments last year, which of the following could have been her gross income last year?

Indicate all such gross incomes.

  1. $\$40{,}200$
  2. $\$43{,}350$
  3. $\$47{,}256$
  4. $\$51{,}996$
  5. $\$53{,}808$

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 19 of Section 4 of Practice Test 1. Those questions testing our knowledge of Fractions 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 your paper.

We can see two fractions in this question, so it likely tests our knowledge of Fractions. 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. Kate spent between $1/4$ and $1/3$ of her gross income on her mortgage last year
  2. Kate spent $\$13{,}470$ on her mortgage last year
  3. We want to know possible values for her gross income

 

Develop a Plan

Let’s start with a top-down approach, where we will begin with what we’re looking for and work down to the details of what we’re given in this question. We want to know Kate’s gross income. We see that we’re given her mortgage and what fraction that is of her gross income. We know that the general form of an equation when we have a fraction of something is:

$$\Amount \of \Something= (\Fraction \of \Total) · \Total \Amount$$

Now let’s write this specifically for our problem. We know that Kate’s mortgage is a certain fraction of her gross income:

$$\Mortgage = \Fraction \of \Gross \Income · \Gross \Income$$

We’re looking for her gross income, so let’s solve this equation for it:

$$\Gross \Income = {\Mortgage}/{\Fraction \of \Gross \Income}$$

We know her mortgage is $\$13{,}470$ and we have values for the fraction of gross income, so let’s use these numbers to solve for the gross income.

Solve the Question

We have two extreme values for the fraction of gross income, since we know it’s somewhere between $1/4$ and $1/3$ of the gross income. It makes sense that we should get the extreme possible values for gross income if we use the extreme values for the fraction of gross income. Let’s solve for these two extreme values for the gross income, first starting with the fraction of gross income being $1/4$:

$\Gross \Income$ $=$ ${\$13{,}470}/{\1\/4}$
$\Gross \Income$ $=$ ${\$13{,}470}·4$
$\Gross \Income $ $=$ $\$53{,}880$

We can see that whenever we divide by a fraction, we can just flip the top and bottom and multiply by that number. So dividing by $1/4$ is the same as multiplying by $4/1$. So at one extreme value, Kate’s gross income was $\$53{,}880$. Let’s solve for the other extreme value of gross income, using the fraction $1/3$:

$\Gross \Income$ $=$ ${\$13{,}470}/{\1\/3}$
$\Gross \Income$ $=$ ${\$13{,}470}·3$
$\Gross \Income $ $=$ $\$40{,}410$

So we now know that the gross income is somewhere between $\40{,}410$ minutes and $\$53{,}880$ minutes. We can see that answer choices B, C, D, and E are between these two values.

The correct answers are B, C, D, and E.

What Did We Learn

When we are given extreme values (the greatest and least possible values) for one variable, we can often use those extreme values to determine the extreme values for another variable. This can be very helpful for finding the greatest and least possible values for a certain variable.

 

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