# The quantities S and T are positive and are related by the equation

The quantities \$S\$ and \$T\$ are positive and are related by the equation \$S=k/T\$, where \$k\$ is a constant. If the value of \$S\$ increases by \$50\$ percent, then the value of \$T\$ decreases by what percent?

1. \$25%\$
2. \$33 1/3%\$
3. \$50%\$
4. \$66 2/3%\$
5. \$75%\$

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 20 of Section 4 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 your paper.

We see the word “percent” mentioned a couple of times. So we should expect to use our knowledge of Fractions, Decimals, and Percents here.

## What Do We Know?

Let’s carefully read through the question and make a list of the things that we know.

1. \$S\$ and \$T\$ are related by the equation \$S=k/T\$
2. \$k\$ is a constant that doesn’t change
3. We will increase the value of \$S\$ by \$50\$ percent
4. We want to know by what percent \$T\$ decreases

## Develop a Plan

We know that we have two situations for \$S\$ and \$T\$: 1) Before increasing \$S\$ by \$50\$ percent and 2) After increasing \$S\$ by \$50\$ percent. Our equation with \$S\$ and \$T\$ in it has a constant, \$k\$. We know that if we solve this equation for \$k\$, then the product between \$S\$ and \$T\$ must be constant since \$k\$ is constant.

\$\$S·T = k\$\$

We know that the product between the original values of \$S\$ and \$T\$ must be \$k\$:

\$\$S_{original}·T_{original} = k\$\$

We also know that the product between the new values of \$S\$ and \$T\$ must also be \$k\$:

\$\$S_{new}·T_{new} = k\$\$

Since both equations are equal to \$k\$, we can set them equal to each other.

\$\$S_{new}·T_{new} = S_{original}·T_{original}\$\$

We are told \$S\$ increased by \$50\$ percent. We remember that the equation for percent increase is:

\$\$\New \Value = (1 + {\Percent \Change}/100) * \Old \Value\$\$

Now let’s rewrite this equation for the \$S\$ value being increased by \$50\$ percent:

\$\$S_{new} = 1.50*S_{original}\$\$

We know if we can plug this equation into our equation with four variables, then we can solve for the new value of \$T\$.

## Solve the Question

Solving the system of equations we get:

 \$S_{new}·T_{new}\$ \$=\$ \$S_{original}·T_{original}\$ \$1.5*S_{original}*T_{new}\$ \$=\$ \$S_{original}*T_{original}\$ \$T_{new}\$ \$=\$ \$T_{original}/1.5\$ \$T_{new}\$ \$=\$ \$0.6667*T_{original}\$

We know we want an answer as a percent decrease, so let’s go ahead and change this decimal value into a percentage.

\$\$0.6667 = 66.67%\$\$

Great, so we’ve solved this equation and found out that the new \$T\$ is \$66.67\$ percent of the original value of \$T\$. We know that we can subtract this number from \$100\$ percent to get the percent decrease in \$T\$:

\$\$100 – 66.67% = 33.33%\$\$

So our final correct answer will be B, \$33 1/3%\$.

Math equations rock! While it might have been tempting to try to figure out this answer without writing out these equations with the original and new values of \$S\$ and \$T\$, the problem was definitely very manageable once we decided to rely on these math equations. Translating words into math equations is a valuable skill that we should continue to improve.

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