6 < x < 7 y = 8

$$6<x<7$$
$$y=8$$

Quantity A $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ Quantity B
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
$x/y$ $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ $0.85$
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given

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 5 of Section 4 of Practice Test 1. Those Fraction questions 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 that Quantity A has a fraction, so this question should test what we learned in arithmetic about Fractions and Decimals. Also, we see an inequality symbol ($<$), which might test what we learned in algebra about Solving Linear Inequalities knowledge. Let’s keep what we’ve learned about these skills 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. We want to compare the fraction $x/y$ to a certain value
  2. We know the value of $y$
  3. We have an inequality for the value of $x$

 

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 know that if we calculate Quantity A ($x/y$), then we can just compare it to the value in Quantity B ($0.85$). Quantity A has two variables in it, so let’s try to find out their values.

Well, the question tells us that $y=8$, so we now know half of Quantity A. Yay, we’re halfway through! Hmmm… $x$ looks a tiny bit trickier though.

Let’s think about this logically. We know that Quantity A is $x/y$. We have an inequality for $x$, so let’s divide this inequality by $y$ so that it looks more like Quantity A, then compare it to Quantity B.

Solve the Question

First, let’s write the inequality with $x$ on our paper:

$$6<x<7$$

Now let’s divide by $y$ and plug in $y=8$.

$$6/y<x/y<7/y$$
$$6/8<x/y<7/8$$

We can see that Quantity A is somewhere between $6/8$ and $7/8$. Quantity B is written as a decimal though. It’ll be easier to compare the two quantities if they’re both written similarly, so let’s convert our inequality for $x/y$ into decimals. We’ll use our knowledge from what we know about Fractions and Decimals to convert them. This gives us to get $6/8=0.75$ and $7/8=0.875$. Of course, if we ever forget the decimal form of these common fractions, we can always use a calculator to divide $6$ by $8$ to get $0.75$, then do the same to get $7/8=0.875$.

$$6/8<x/y<7/8$$
$$\;\;0.75<x/y<0.875$$

Okay, so now we can see that Quantity A $(x/y)$ is somewhere between $0.75$ and $0.875$. Since Quantity B is $0.85$, it’s possible for Quantity A to be less than, equal to, or even greater than Quantity B, so we don’t have enough information to figure out which quantity is bigger.

The correct answer is D, the relationship cannot be determined from the information given..

What Did We Learn

We were given an inequality in our question that looked somewhat similar to Quantity A. We solved this question by forcing the inequality to look EXACTLY like Quantity A. We did this by asking ourselves how to change $x$ to be $x/y\;$, which is why we divided by $y$. Let’s remember that rewriting equations and inequalities so that they look more like the quantities in Quantitative Comparison questions can be very helpful. And we can divide inequalities by any positive number without worrying about the need to reverse the sign direction of inequalities.

 

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