In the sunshine, an upright pole 12 feet tall is casting a shadow 8

In the sunshine, an upright pole $12$ feet tall is casting a shadow $8$ feet long. At the same time, a nearby upright pole is casting a shadow $10$ feet long. If the lengths of the shadows are proportional to the heights of the poles, what is the height, in feet, of the taller pole?

  1. $10$
  2. $12$
  3. $14$
  4. $15$
  5. $18$

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 11 of Section 6 of Practice Test 1. Those questions testing our knowledge of Ratios and Proportions 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.

The question tells us that the lengths of the shadows are proportional to the heights of the poles, so it likely tests how well we know Ratios and Proportions. 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 have one pole that is $12$ feet tall casting a shadow $8$ feet long
  2. A nearby pole casts a shadow that is $10$ feet long
  3. The lengths of the shadow are proportional to the height of their corresponding poles

 

Develop a Plan

The question tells us that the pole height and shadow lengths are proportional. When two quantities are proportional, their ratio will remain constant. In other words, we can set the ratio of the pole height to the shadow length for the first pole equal to the ratio of the pole height to the shadow length for the second pole.

$${\Height_1}/{\Shadow_1}={\Height_2}/{\Shadow_2}$$

It doesn’t matter which pole we label $1$ or $2$. Let’s go ahead and label the pole we have all the information about as pole $2$. So for pole $2$, we know that its height is $12$ feet and its shadow length is $8$ feet. We also know that the shadow length of the other pole, pole $1$, is $10$ feet.

$${\Height_1}/{10 \feet}={12 \feet}/{8 \feet}$$

Let’s solve this equation for $\Height_1$ to finish this question.

Solve the Question

Let’s isolate $\Height_1$ by multiplying both sides of our last equation by $10$ feet:

$$\Height_1={12\feet·10\feet}/{8 \feet}$$

Instead of using a calculator to finish this arithmetic, a useful technique for solving this without a calculator is to rewrite the numbers as prime factors, then cancel out like terms in the numerator and denominator. Prime factors are the result of prime factorization where we write each number as the product of prime numbers, which are numbers that are only divisible by $1$ and itself, and doesn’t have any other positive divisors. As an example, the prime factorization for $12$ is $2·3·2$. Let’s go ahead and rewrite out numbers as prime factors now.

$$\Height_1={(2·3·2)·(5·2)}/{2·2·2} \feet$$

Nicely done! And it didn’t take that long either. Now let’s cancel out any terms repeated in the numerator and denominator. We can see three $2\s$ in the numerator and denominator, so let’s cancel those out.

$\Height_1$ $=$ ${(2·3·2)·(5·2)}/{2·2·2} \feet$
$ $ $ $
$\Height_1$ $=$ $15$

So the height of the taller pole is $15$ feet. The correct answer is D, $15$.

What Did We Learn

Whenever we see the word proportional we should think about finding a ratio and setting equal two of these ratios.

 

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