Percent Word Problems and Fixing Errors | Grade 8 Math | LevelKite

In this lesson you will solve percent word problems and then find and fix mistakes in percent work. Both skills rest on one question: a percent of what?

A percent means “per hundred,” so $p\%$ of an amount is $\frac{p}{100}$ of that amount. The amount a percent is taken of is the whole, also called the base. The amount you compare with the whole is the part. In “ $30\%$ of $50$ is $15$ ,” the whole is $50$ , the part is $15$ and the percent is $30\%$ , so $p = 30$ . A part can be larger than its whole when the percent is more than $100\%$ .

Key concept

Every percent has its own whole

A percent always describes a part of one stated whole, so name that whole before you calculate. Then decide which quantity is unknown: the part, the whole or the percent. The equation below connects the three. Here $p$ is the percent, the whole is greater than $0$ , and the part and the whole use the same unit: $30$ cm out of $2$ m is $30$ cm out of $200$ cm, or $\frac{30}{200} = 15\%$ . In a percent change, the whole is the positive original amount. When a second percent change is applied to the new amount, that new amount is the whole for the second change.

\text{part} = \frac{p}{100} \times \text{whole}

Name the unknown, then use the matching equation

Unknown	What you know	Equation	Why it is valid
The part	The percent $p$ and the whole	$\text{part} = \frac{p}{100} \times \text{whole}$	It is the key-concept equation itself.
The whole	The part and the percent $p$ , with $p \neq 0$	$\text{whole} = \text{part} \div \frac{p}{100}$	Divide both sides of the key-concept equation by the nonzero number $\frac{p}{100}$ ; the sides stay equal.
The percent	The part and the whole	$p = \frac{\text{part}}{\text{whole}} \times 100$	Divide both sides by the whole, which is not $0$ , then multiply both sides by $100$ ; the sides stay equal.

You already know percent change: $\frac{\text{new} - \text{original}}{\text{original}} \times 100\%$ , where the whole is the positive original amount. A positive result is an increase and a negative result is a decrease. When you give only the size of a decrease, say so with the word “decrease”: a change of $-25\%$ is a $25\%$ decrease.

A markup is an increase a store adds to its cost, given as a percent of that cost. A discount is a decrease given as a percent of the original price.

If the percent change is $c$ , the change is $c\%$ of the original, or $\frac{c}{100} \times \text{original}$ . Add it to the original and factor out the original (the distributive property) to get the equation below. The number $1 + \frac{c}{100}$ is the multiplier: $1.25$ for a $25\%$ markup and $0.75$ for a $25\%$ discount. That equation is the key-concept equation with the new amount as the part, the original as the whole and $p = 100 + c$ . For a fixed $c$ , the new amount is proportional to the original, and the multiplier is the constant of proportionality.

\text{new} = \left(1 + \frac{c}{100}\right) \times \text{original}, \quad \text{original} > 0

Worked example

A markup: finding the new price

A sports shop pays 64 dollars for a skateboard and marks it up $25\%$ . What is the selling price?

Name the whole and the unknown. A markup is a percent of the shop’s cost, so the whole is the original amount, 64 dollars, and the percent change is 25%. The selling price is the unknown, and it is the part: the new amount compared with the whole.
$\text{whole} = 64, \quad c = 25, \quad \text{part} = \text{selling price}$
Find the multiplier for a 25% increase. Multiplying the cost by it adds 25% of the cost to the cost.
$1 + \frac{25}{100} = 1 + 0.25 = 1.25$
Multiply the whole by the multiplier. Splitting 1.25 into 1 + 0.25 shows the cost plus a 16-dollar markup.
$1.25 \times 64 = 64 + 0.25 \times 64 = 64 + 16 = 80$
Check with the percent change formula, which divides by the positive original amount, 64 dollars. It gives back the stated 25%, so the price fits the markup.
$\frac{80 - 64}{64} \times 100\% = \frac{16}{64} \times 100\% = 0.25 \times 100\% = 25\%$

Answer

The shop sells the skateboard for 80 dollars.

Worked example

A discount: finding the original price

A jacket is on sale for $25\%$ off. Its sale price is 63 dollars. What was its original price?

Name the whole and the unknown. The 25% discount is a percent of the original price, so the original price is the whole, and it is the unknown. Let x be the original price in dollars, with x > 0. The sale price, 63 dollars, is the part.
$\text{part} = 63, \quad \text{whole} = x, \quad x > 0$
Find the multiplier. A 25% discount is a percent change of −25%, so the sale price is 75% of the original price.
$1 + \frac{-25}{100} = 1 - 0.25 = 0.75$
Write the multiplier equation, with the sale price as the new amount and x as the original amount.
$0.75x = 63$
Divide both sides by 0.75. The sides stay equal because both are divided by the same number, and 0.75 is not 0. Since 0.75 = 3/4, dividing by it is the same as multiplying by 4/3.
$x = 63 \div 0.75 = 63 \times \frac{4}{3} = \frac{252}{3} = 84$
Check by working forward from the whole: take 25% of 84 dollars away and compare the result with the sale price.
$84 - 0.25 \times 84 = 84 - 21 = 63$

Answer

The jacket’s original price was 84 dollars.

Now use the same question to find errors. For each percent in a solution, ask: a percent of what? Mark the work as an error when it does one of these three things:

It reads a percent against the wrong whole, such as finding a percent change by dividing by the new amount instead of the positive original.
It treats percents of different wholes as parts of one common whole, such as adding two percent changes made one after the other.
It concludes that one amount is larger only because its percent is larger, when the wholes differ.

Do not mark correct work as an error. Comparing two percents of different wholes is correct, and so is finding their difference in percentage points, which subtracts one percent from the other. Each percent is still read against its own whole, so these results describe relative size, not amounts.

Dividing by the wrong base (check 1). A hoodie’s price falls from 60 dollars to 45 dollars. A student writes $\frac{45 - 60}{45} = \frac{-15}{45} = -\frac{1}{3} \approx -33.3\%$ . Dividing by the new price reads the percent against the wrong whole, and the result fails a check: taking $\frac{1}{3}$ of 60 dollars away leaves $60 - 20 = 40$ dollars, not 45. Percent change measures the change against where the price started, so its whole is the positive original price: $\frac{45 - 60}{60} \times 100\% = \frac{-15}{60} \times 100\% = -0.25 \times 100\% = -25\%$ , a $25\%$ decrease. The corrected work is valid because $25\%$ of 60 dollars is 15 dollars, and $60 - 15 = 45$ .

Two price increases with different wholes

Item	Original price (dollars)	New price (dollars)	Increase (dollars)	Percent change
Notebook	$4$	$5$	$5 - 4 = 1$	$\frac{1}{4} \times 100\% = 25\%$
Headphones	$40$	$46$	$46 - 40 = 6$	$\frac{6}{40} \times 100\% = 15\%$

Here is how the checks apply to claims about this table.

Correct: $25\% > 15\%$ , so relative to its own original price, the notebook’s price rose more. The percents differ by $25 - 15 = 10$ percentage points. This $10$ compares the two relative changes; it is not a percent of either price.
Error (check 3): “The notebook’s price rose by more dollars, because $25\% > 15\%$ .” A larger percent alone means a larger amount only when the wholes are equal. To compare amounts, find each from its own whole: $25\%$ of 4 dollars is 1 dollar, and $15\%$ of 40 dollars is 6 dollars, so the headphones rose by more dollars.
Error (check 2): “Together, the prices rose $25\% + 15\% = 40\%$ .” These percents have different wholes, so adding them does not find the percent change of the combined price. Add the amounts instead: $1 + 6 = 7$ dollars on a combined original price of $4 + 40 = 44$ dollars, which is $\frac{7}{44} \times 100\% \approx 15.9\%$ .

Worked example

Correcting a sum of percent changes

A game costs 50 dollars. A store raises its price by $20\%$ , then lowers the new price by $20\%$ . Riley says the changes cancel, since $(+20\%) + (-20\%) = 0\%$ , so the game costs 50 dollars again. Find and correct Riley’s error.

Name the unknown. It is a percent: the overall percent change, whose whole is the original price, 50 dollars. The first 20% is a percent of 50 dollars, but the second 20% is a percent of the new price, which is a different whole.
Apply the increase to its own whole, 50 dollars. A 20% increase has multiplier 1 + 20/100 = 1.20.
$50 \times 1.20 = 50 + 0.20 \times 50 = 50 + 10 = 60$
Apply the decrease to its own whole, the new price of 60 dollars. A change of −20% has multiplier 1 − 20/100 = 0.80.
$60 \times 0.80 = 60 - 0.20 \times 60 = 60 - 12 = 48$
Find the overall percent change, dividing by the positive original price, 50 dollars.
$\frac{48 - 50}{50} \times 100\% = \frac{-2}{50} \times 100\% = -0.04 \times 100\% = -4\%$
Explain the error. Adding the percents treats both as parts of one common whole, 50 dollars. But the decrease is 20% of 60 dollars, or 12 dollars, which is more than the 10-dollar increase. The corrected work is valid because each multiplier acts on the amount it actually changes, so the two changes together act like the product of the multipliers.
$1.20 \times 0.80 = 0.96 = 1 + \frac{-4}{100}$

Answer

Riley added percents of two different wholes. The game ends at 48 dollars, a percent change of $-4\%$ , which is a $4\%$ decrease, not a $0\%$ change.

Takeaway: Every percent is a percent of its own whole. Name that whole first, then decide whether the part, the whole or the percent is unknown. Measure each percent change against its positive original amount, and when changes happen one after another, apply each one to the amount it actually changes. You can compare percents of different wholes as percents, but to compare or combine their amounts, first find each amount from its own whole.