Rewrite Percent Change Expressions to Solve Problems | Grade 7 Math | LevelKite

Prices rise, sales cut them, and clubs grow. People often describe changes like these with a percent.

A percent is a rate per 100 of a stated whole. The whole is the amount you take the percent of, and it must be positive. So 5% of a price means 5 for every 100 of that price, which is $\frac{5}{100}$ , or $0.05$ , times the price.

Suppose a ticket costs $p$ dollars, where $p > 0$ , and then the price goes up by 5% of that original price. The whole is the original price $p$ , so the increase is $0.05p$ dollars. You already know how to write an expression for a situation like this. The new price in dollars is the original price plus the increase: $p + 0.05p$ .

Call this the two-part form. It tells the story in order: start with $p$ , then add 5% of $p$ . In this lesson you will rewrite it in a shorter form, and then use the shorter form to solve problems.

You can join the two parts into one. First, $p$ is the same as $1p$ , because multiplying by 1 does not change a number. Now each part is a number times $p$ .

The distributive property says that $a \cdot p + b \cdot p = (a + b) \cdot p$ for all numbers $a$ , $b$ and $p$ . Here $a = 1$ and $b = 0.05$ , so you add the numbers in front and keep the $p$ . The steps look like this:

p + 0.05p = 1p + 0.05p = (1 + 0.05)p = 1.05p

The short expression $1.05p$ is the one-multiplier form. Two expressions are equivalent only when they have the same value for every value of the variable. The distributive property holds for every number, so $p + 0.05p$ and $1.05p$ are equivalent. Trying one or two values of $p$ can catch a slip, but it does not prove that two expressions are equivalent.

The number in front of the variable is the multiplier. It is the number you multiply the original amount by to get the new amount. So increasing a positive amount by 5% of itself means the same as multiplying it by $1.05$ . The $1$ keeps all of the original price, and the $0.05$ adds 5% more. The new price is 105% of the original price.

Worked example

Rewrite a percent increase

A gym charges $m$ dollars per month, where $m > 0$ . Next year the monthly price will go up by 8% of this price. Write the new monthly price in two-part form. Then rewrite it as one multiplier times $m$ .

Find the whole and the rate. The 8% is a rate per 100 of the original price m, which is positive. Write 8 per 100 as a decimal.
$\frac{8}{100} = 0.08$
Write the increase in dollars. It is 0.08 of the original price, so multiply m by 0.08.
$0.08m$
Write the two-part form. The new price is the original price plus the increase.
$m + 0.08m$
Rewrite m as 1m, because multiplying by 1 does not change its value. Each part is now a number times m, so the distributive property lets you add 1 and 0.08. This works for every value of m, so the two forms are equivalent.
$m + 0.08m = 1m + 0.08m = (1 + 0.08)m = 1.08m$
Check with one value, m = 100. The two-part form gives 100 + 8 = 108 dollars, and the one-multiplier form also gives 108 dollars. One value cannot prove that the forms are equivalent, but it can catch a slip such as writing 1.8 in place of 1.08.
$100 + 0.08(100) = 100 + 8 = 108 \qquad 1.08(100) = 108$

Answer

The new monthly price is $m + 0.08m$ dollars, which is equivalent to $1.08m$ dollars. An 8% increase means the new price is $1.08$ times the original price, or 108% of it.

A decrease works the same way, with subtraction. Suppose the ticket price of $p$ dollars drops by 20% of the original price. The decrease is $0.20p$ dollars, so the new price in dollars is $p - 0.20p$ . The distributive property works here too, because subtracting $0.20p$ is the same as adding $-0.20p$ :

$p - 0.20p = 1p - 0.20p = (1 - 0.20)p = 0.80p$

So decreasing a positive amount by 20% of itself means the same as multiplying it by $0.80$ . You are left with 80% of the original price.

You can also go the other way and read a multiplier by comparing it with 1. Because the original amount is positive, a multiplier greater than 1 means an increase, and a multiplier between 0 and 1 means a decrease. For example, $1.30p$ is $(1 + 0.30)p$ , a 30% increase. And $0.96p$ is $(1 - 0.04)p$ , a 4% decrease, because 4 per 100 is $0.04$ .

Key concept

A percent change is one multiplication

Let $x$ be the original amount, with $x > 0$ . Let $r$ be the percent of the change written as a decimal, with $r > 0$ . For 5%, $r = 0.05$ . The percent is always a rate per 100 of the original amount $x$ .

Increase: $x + rx = (1 + r)x$ . You keep all of $x$ and add the part $rx$ .
Decrease: $x - rx = (1 - r)x$ . You keep all of $x$ and take away the part $rx$ . In this lesson $r$ is less than 1 for a decrease, so the new amount stays positive.

In each line the two expressions are equivalent, because the distributive property makes them equal for every value of $x$ .

x + rx = (1 + r)x \qquad x - rx = (1 - r)x

Each percent is a rate per 100 of x, the positive original amount.

Change	Two-part form	One-multiplier form	New amount as a percent of the original
Increase by 5%	$x + 0.05x$	$1.05x$	105%
Increase by 12%	$x + 0.12x$	$1.12x$	112%
Increase by 40%	$x + 0.40x$	$1.40x$	140%
Decrease by 20%	$x - 0.20x$	$0.80x$	80%
Decrease by 35%	$x - 0.35x$	$0.65x$	65%
Decrease by 7%	$x - 0.07x$	$0.93x$	93%

Worked example

Rewrite a percent decrease, then use it

A jacket has a regular price of $j$ dollars, where $j > 0$ . It is on sale for 25% off the regular price. Write the sale price in two-part form and rewrite it with one multiplier. Then find the sale price when $j = 64$ .

The 25% is a rate per 100 of the regular price j, the positive original amount. As a decimal, 25 per 100 is 0.25, so the discount is 0.25j dollars. Subtract the discount from the regular price to get the two-part form.
$j - 0.25j$
Rewrite j as 1j. Then use the distributive property to subtract the numbers in front of j. The two forms have the same value for every value of j, so they are equivalent.
$j - 0.25j = 1j - 0.25j = (1 - 0.25)j = 0.75j$
Read the multiplier. 0.75 is 75 per 100, so the sale price is 75% of the regular price. Taking away 25% leaves 75%.
Use the one-multiplier form to find the sale price. Substitute 64 for j. Since 0.75 is three fourths, find one fourth of 64, which is 16, and multiply it by 3.
$0.75(64) = 3 \times 16 = 48$
Check with the two-part form. The discount is 0.25 times 64, which is 16 dollars. Equivalent forms must give the same value, and they do.
$64 - 0.25(64) = 64 - 16 = 48$

Answer

The sale price is $j - 0.25j$ dollars, which is equivalent to $0.75j$ dollars. When the regular price is 64 dollars, the sale price is 48 dollars.

The one-multiplier form helps most when you know the new amount and need the original amount. In the two-part form the unknown appears twice, so it is hard to undo. In the equivalent one-multiplier form the unknown appears once, so one division finds it.

To work backward, you solve an equation. A solution is a value of the variable that makes the equation true. Keep three rules in mind:

Do the same operation to both sides, so the two sides stay equal.
Never divide by zero. Before you divide by a multiplier, confirm that it is not zero.
A solution must make the original equation true, so check your answer there.

Worked example

Work backward to the original amount

This year a robotics club grew by 20% of its starting size, and it now has 60 members. Let $n$ be the number of members at the start of the year, where $n > 0$ . How many members did the club have at the start of the year?

Write an equation in two-part form. The 20% is a rate per 100 of the starting number n, the positive original amount, so the growth is 0.20n members. The starting number plus the growth equals 60.
$n + 0.20n = 60$
Rewrite the left side in its one-multiplier form. Since n is 1n, the distributive property lets you add 1 and 0.20. The two forms are equal for every value of n, so they are equivalent.
$n + 0.20n = 1n + 0.20n = (1 + 0.20)n = 1.20n$
Put the equivalent form into the equation. The left side has the same value as before, so the new equation has the same solution as the original one.
$1.20n = 60$
Divide both sides by 1.20. Doing the same operation to both sides keeps them equal, and 1.20 is not zero, so you may divide by it. To make the division easier, multiply the top and bottom of the fraction by 10, which does not change its value.
$\frac{1.20n}{1.20} = \frac{60}{1.20} \qquad n = \frac{600}{12} = 50$
Check in the original equation. The growth is 0.20 times 50, which is 10 members. The left side equals 60, so the original equation is true and 50 is the solution.
$50 + 0.20(50) = 50 + 10 = 60$

Answer

The club had 50 members at the start of the year.

Here is the idea to keep. A percent change of a positive original amount $x$ has two equivalent forms. The two-part form, such as $x + 0.05x$ or $x - 0.20x$ , shows the original amount and the change. The one-multiplier form, such as $1.05x$ or $0.80x$ , shows the number you multiply the original amount by. The forms are equivalent because the distributive property makes them equal for every value of $x$ .

Pick the form that fits the question. To find an original amount, set the one-multiplier form equal to the new amount, confirm the multiplier is not zero, divide both sides by it, and check your answer in the original equation.

Next, you will decide whether different expressions for the same situation are equivalent.