Original Amounts and Percent of Change Word Problems | Grade 7 Math | LevelKite

You already know how to go forward with a percent of change, or percent change for short: a percent increase or a percent decrease. You start with the original amount (the amount before the change), apply the percent, and get the new amount (the amount after the change).

In this lesson you go backward. You know the new amount and the percent change, and you find the original amount. Then you use both directions to solve word problems.

Everything rests on one question: what is the percent a percent of?

A percent is a rate per 100 of a stated positive whole. In a percent change, that whole is the original amount, which must be positive. It is never the new amount.

All of the original amount is 100% of it. A 20% decrease takes away 20 for every 100 of the original amount, so 80 for every 100 remain. The new amount is 80% of the original amount, or $0.80$ times it.

A 25% increase adds 25 for every 100 of the original amount, so the new amount is 125 for every 100 of the original amount. That is 125% of it, or $1.25$ times it.

This is the forward step you already know, written as one multiplication. The number you multiply by is the multiplier. Write the percent as $p$ %, where $p$ is greater than 0. For an increase, the multiplier is $1 + \frac{p}{100}$ . For a decrease, the multiplier is $1 - \frac{p}{100}$ . In this lesson every decrease is less than 100%, so that multiplier is greater than 0.

For any positive original amount, the forward step is:

\text{new amount} = \text{multiplier} \times \text{original amount}

Key concept

Divide by the multiplier to go back

In a percent change, the percent is a rate per 100 of the positive original amount, never of the new amount. Going forward, you multiply the original amount by the multiplier. Dividing by a number that is not zero undoes multiplying by that number, so going backward you divide the new amount by the same multiplier.

For every percent increase, and for every percent decrease of less than 100%, the multiplier is not zero, so this rule works. A decrease of exactly 100% leaves $0$ whatever the original amount was, so you cannot work back from it.

\text{original amount} = \text{new amount} \div \text{multiplier}

With a positive original amount, each of these multipliers works in both directions

Percent change	Multiplier	Forward: original to new	Backward: new to original
25% increase	$1 + 0.25 = 1.25$	multiply by $1.25$	divide by $1.25$
8% increase	$1 + 0.08 = 1.08$	multiply by $1.08$	divide by $1.08$
20% decrease	$1 - 0.20 = 0.80$	multiply by $0.80$	divide by $0.80$
35% decrease	$1 - 0.35 = 0.65$	multiply by $0.65$	divide by $0.65$

Worked example

Working back from a decrease

After a 20% decrease, an amount is $48$ . What was the original amount?

Let x be the original amount, a positive number. The 20% is a rate per 100 of x, never of the new amount, 48. The decrease takes away 20 for every 100 of x, so 80 for every 100 remain. Write that as the multiplier.
$1 - 0.20 = 0.80$
Write the forward relationship with the values you know: the multiplier times the original amount equals the new amount.
$0.80x = 48$
Divide both sides by 0.80 to undo the multiplication. Dividing both sides by the same number keeps an equation true when that number is not zero, and 0.80 is not zero.
$\frac{0.80x}{0.80} = \frac{48}{0.80} \quad \text{so} \quad x = 48 \div 0.80$
Work out the quotient. Multiplying both 48 and 0.80 by 10 does not change the quotient, and it lets you divide by a whole number. The result is positive, as the original amount must be.
$48 \div 0.80 = 480 \div 8 = 60$
Check by going forward from your answer. The 20% is taken of the positive original amount, 60, never of the new amount. Subtract that part from 60 and you should get the new amount, 48.
$0.20 \times 60 = 12, \quad 60 - 12 = 48$

Answer

The original amount was $60$ .

Worked example

Working back from an increase

A school club grew by 25% this year and now has $40$ members. How many members did it have last year?

Sort the amounts. Last year's count came before the change, so it is the original amount, and it is missing. Let m be the number of members last year, a positive whole number. This year's 40 members is the new amount.
The 25% is a rate per 100 of m, never of the new amount, 40. The increase adds 25 for every 100 of m, so the new amount is 125 for every 100 of m. That gives the multiplier and the forward relationship.
$1 + 0.25 = 1.25 \quad \text{so} \quad 1.25m = 40$
Divide both sides by 1.25, which is not zero, so the equation stays true. Multiply both 40 and 1.25 by 100 to divide by a whole number. Eight 125s make 1000, so thirty-two 125s make 4000. The result is a positive whole number, as a count of members must be.
$m = 40 \div 1.25 = 4000 \div 125 = 32$
Check by going forward. The 25% is taken of the positive original amount, last year's 32 members, never of the new amount. Add that part to 32 and you should get this year's count, 40.
$0.25 \times 32 = 8, \quad 32 + 8 = 40$

Answer

The club had $32$ members last year.

Now use both directions in word problems. A percent change word problem is built from three quantities: the original amount, the percent change and the new amount. Before you calculate, sort them. Which ones do you know, and which one is missing?

Identify the original amount: the positive amount before the change. Words such as was, before and regular price often point to it. The percent is a rate per 100 of this amount, never of the new amount.
Identify the new amount: the amount after the change. Words such as now, after and sale price often point to it.
Decide which of the three quantities is missing. Name it with a variable and state its unit.
Write the relationship, solve, and answer the question that was asked, with its unit.

One more term helps: the amount of change is how far apart the original amount and the new amount are. To find it, subtract the smaller one from the larger one. If the new amount is greater than the original amount, the change is an increase. If it is smaller, the change is a decrease.

Sort a problem by the quantity that is missing

Missing quantity	What you know	What you do
New amount	Original amount and percent change	Multiply the positive original amount by the multiplier.
Original amount	New amount and percent change	Divide the new amount by the multiplier, which must not be zero.
Percent change	Original amount and new amount	Divide the amount of change by the positive original amount, never by the new amount. Write it as a rate per 100.

Worked example

A two-part word problem

Two stores start with the same original price for a pair of headphones. Store A takes 35% off and charges $52$ dollars. Store B charges $60$ dollars. What percent discount does Store B give?

A discount is a decrease in price. Sort Store A: 52 dollars is the new amount, the change is a 35% decrease, and the original price is missing. Let h be the original price in dollars, a positive number. The 35% is a rate per 100 of h, never of 52, so 65 for every 100 of h remain. That gives the multiplier and the forward relationship.
$1 - 0.35 = 0.65 \quad \text{so} \quad 0.65h = 52$
Divide both sides by 0.65, which is not zero, so the equation stays true. Multiply both 52 and 0.65 by 100 to divide by a whole number. Eight 65s make 520, so eighty 65s make 5200. The result is positive, as the original price must be.
$h = 52 \div 0.65 = 5200 \div 65 = 80$
Sort Store B: the original price is 80 dollars, the new amount is 60 dollars, and the percent change is missing. Find the amount of change in dollars by subtracting the smaller price from the larger one.
$80 - 60 = 20$
A percent change is the amount of change written as a rate per 100 of the positive original amount. Divide by the original price, 80, never by the new price, 60.
$\frac{20}{80} = \frac{1}{4} = 0.25 = \frac{25}{100} = 25\%$
Check by going forward. The 25% is taken of the positive original price, 80 dollars, never of the new price. Subtract that part from 80 and you should get Store B's price, 60 dollars.
$0.25 \times 80 = 20, \quad 80 - 20 = 60$

Answer

Store B gives a 25% discount. Its price of $60$ dollars is $20$ dollars below the original price of $80$ dollars.

Takeaway. In a percent change, the percent is a rate per 100 of the positive original amount, never of the new amount. That one fact gives you the multiplier, and the multiplier works in both directions. Multiply by it to go from the original amount to the new amount. Divide by it to go from the new amount back to the original amount. You can do that for any increase and for any decrease of less than 100%, because then the multiplier is not zero. In a word problem, sort the original amount, the new amount and the percent change first, then find the one that is missing.

Next, you will learn about percent error, which compares a measured amount with the positive amount that is accepted as correct, and you will find and fix mistakes in percent of change work.