A developer has land that has x feet of lake frontage. The land is to

A developer has land that has $x$ feet of lake frontage. The land is to be subdivided into lots, each of which is to have either $80$ feet or $100$ feet of lake frontage. If $1/9$ of the lots are to have $80$ feet of frontage and the remaining $40$ lots are to have $100$ feet of frontage each, what is the value of $x$ ?

  1. $400$
  2. $3{,}200$
  3. $3{,}700$
  4. $4{,}400$
  5. $4{,}760$

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 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 definitely see a fraction in this question, so it likely tests what we know about 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. We want to solve for $x$, the total feet of lake frontage on some land
  2. The land is split into lots that have either $80$ feet or $100$ feet of lake frontage
  3. $1/9$ of the lots each have $80$ feet of lake frontage
  4. The remaining $40$ lots each have $100$ feet of lake frontage

 

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 the total number of lake frontage feet, $x$. We can see that the lake frontage is split into lots with either $80$ feet or $100$ feet of lake frontage. We know that if we add together the lake frontage for both types of lots, we’ll have the total feet of lake frontage.

We know that the feet of lake frontage in the $80$-foot lots would be the number of $80$-lot lots multiplied by $80$. Similarly, the feet of lake frontage in the $100$-foot lots would be the number of $100$-foot lots multiplied by $100$. Writing this as an equation gives us:

$$x = 80·\Number \of 80 \Foot \Lots + 100·\Number \of 100 \Foot \Lots$$

So now we just need to figure out how many of each of these lots there are and we can answer the question. The question does tell us that we have $40$ $100$-foot lots:

$$\Number \of 100 \Foot \Lots= 40$$

The question is a bit more cryptic about the number of $80$-foot lots though. Almost makes us wonder what exactly are they hiding in these lots. Anyway, it tells us that $1/9$ of the lots are $80$-foot lots and the remaining lots are $100$-foot lots. Since the fraction of lots that have $80$-feet of lake frontage and the fraction of lots that have $100$-feet of lake frontage must add up to $1$, then since $1-1/9=8/9$, $8/9$ of the arrangements have $80$ feet of lake frontage. Let’s write this on our paper now to keep track:

$$\;\,\Number \of 80 \Foot \Lots = 8/9 \of \Total \Number \of \Lots$$
$$\Number \of 100 \Foot \Lots = 1/9 \of \Total \Number \of \Lots$$

Since $8/9$ is $8$ times more than $1/9$, we can get the number of $80$-foot lots if we divide the number of $100$-foot lots by $8$, so let’s do that now.

$\Number \of 80 \Foot \Lots$ $=$ ${\Number \of 100 \Foot \Lots} / 8$
$\Number \of 80 \Foot \Lots$ $=$ $40 / 8$
$\Number \of 80 \Foot \Lots$ $=$ $5$

Now that we have the number of $80$- and $100$-foot lots, we can finally finish this question!

Solve the Question

Let’s use our equation for $x$, plugging in $5$ for the number of $80$-foot lots and $40$ for the number of $100$-foot lots.

$x$ $=$ $80·\Number \of 80\Foot \Lots + 100·\Number \of 100 \Foot \Lots$
$x$ $=$ $80·5 + 100·40$
$x$ $=$ $400+4{,}000$
$x$ $=$ $4{,}400$

The correct answer is D, $4{,}400$.

What Did We Learn

This problem had a lot going on. Using a top-down approach and gradually expanding our original equation was very helpful for keeping all of the bits of info organized. Also, this top-down approach forced us to develop a plan in steps, making such a long word problem much more manageable.

 

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