Lesson 42: Solving Percent Problems

New Concept

Algebra is a powerful mathematical language that uses symbols and rules to represent numbers, explore relationships, and solve for unknown values in any situation.

What’s next

To start, we'll apply these algebraic ideas by solving percent problems. You'll master using equations and proportions to tackle real-world financial calculations.

Property

A percent is a ratio that compares a number to 100. The whole is the total amount, the percent is a rate that quantifies an amount with respect to the whole, and the percentage is the number representing that percent. This can be shown as a proportion:

Examples

What is 20% of 80? Answer: $c = 0.20 \cdot 80 = 16$
15 is what percent of 60? Answer: $\frac{15}{60} = \frac{x}{100} \rightarrow x = 25$, so it is 25%.
10 is 50% of what number? Answer: $\frac{10}{x} = \frac{50}{100} \rightarrow x = 20$

Explanation

Think of percents as a slice of a 100-piece pizza! The 'whole' is the entire pizza, the 'percent' is your share rate, and the 'percentage' is how many slices you actually get. It’s a simple way to compare a part to the total amount.

Property

To find a percentage, change the percent to its decimal form and then multiply the decimal by the whole number.

Examples

What number is 30% of 60? Answer: $c = 0.30 \cdot 60 = 18$
What number is 150% of 80? Answer: $n = 1.50 \cdot 80 = 120$
What is 7.5% of 200? Answer: $c = 0.075 \cdot 200 = 15$

Explanation

To find a piece of a number, just turn the percent into a decimal—scoot the decimal point two spots left! Then, multiply. It's a lightning-fast calculation to find the exact slice of the whole you're looking for. Zap! Instant answer.

Property

Set up a proportion where one ratio compares the part to the whole, and the other ratio is the percent rate written in the form of a ratio. Then, solve for the unknown.

Examples

What number is 120% of 50? Answer: $\frac{c}{50} = \frac{120}{100} \rightarrow 100c = 6000 \rightarrow c = 60$
20 is what percent of 80? Answer: $\frac{20}{80} = \frac{x}{100} \rightarrow 80x = 2000 \rightarrow x = 25$
30 is 60% of what number? Answer: $\frac{30}{x} = \frac{60}{100} \rightarrow 60x = 3000 \rightarrow x = 50$

Explanation

Set up a super-fair trade! One fraction is the part over the whole, and the other is the percent over 100. Cross-multiply to find the missing piece. It's like balancing scales to find the unknown value, solving any percent puzzle perfectly.