# GRE Algebra answer is wrong

GRE study guide has following

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 $(\frac{1}{9}$ of the lots are to have 80 feet of frontage each and the remaining 40 lots are to have 100 feet of frontage each, what is the value of x?

Choices

1. 400
2. 3,200
3. 3,700
4. 4,400
5. 4,760

I calculated x as

$(\frac{8}{9})x = 4000$

$x = 4000(\frac{9}{8})$

$x = 4500$

• For future reference, algebraic geometry typically references studying curves via commutative algebra or scheme theory, not to be confused with a tag such as "geometry" Commented Jan 19, 2016 at 0:23
• @Prospect What is the correct tag? I was unable to find geometry. Commented Jan 19, 2016 at 0:24
– JRN
Commented Jan 19, 2016 at 0:40
• @JoelReyesNoche Updated the OP. Commented Jan 19, 2016 at 0:45

Let $n$ be the number of lots, we have $\frac{8}{9}n=40$, or $n=45$. So we see that 40 lots have 100 ft and 5 lots have 80 feet, giving 4400 feet of lake front.

$$40 \cdot 100 + 5 \cdot 80 = 4400$$

In this equation, the first factor of each product is the number of lots, and the second factor of each product is the length of the lots.

If you assign units to your equation, you'll find that they are inconsistent.

Edit - because someone isn't a fan!

Writing $\dfrac{8}{9}x = 4000$ implies that $x$ is $8/9$ of the length, when it's actually $8/9$ of the [number of] lots.

• The original post said "8/9 %" which is NOT the same as the fraction 8/9. Commented Jan 19, 2016 at 0:49
• Compensatory upvote. Commented Jan 19, 2016 at 0:49
• @user247327: The OP has some ill-formed LaTeX. It's hard to say what the OP meant to say, but interpreting it as $8/9$ percent does not permit an integer number of lots. Commented Jan 19, 2016 at 0:51
• Surely the percent sign is a typo! Commented Jan 19, 2016 at 0:56
• @user247327: The original post said no such thing: math.stackexchange.com/revisions/1617611/1 Commented Jan 19, 2016 at 0:57

Since the lots are not equal in length 8/9 of the lots will not have 8/9ths of total length. These lots (100 feet) are larger than the remaining lots (80 feet) so they have more than 8/9 of the length. (Just like if 1% of population are very rich, they have more than 1% of the money because the money isn't distributed evenly.

So $\frac 8 9 x \ne 4000$.

If $n$ is the total number of lots $n*\frac 1 9*80 + 40*100 = n\frac 1 9*80 + n*\frac 8 9 *100 = x$

What is $n$ and what is $x$?

We are told $8/9 n = 40$ so $n = 40*(9/8) = 45$

So $5*80 + 40*100 = x$.