Part 5: Proportional Reasoning

# 5.1: Applying percentages

Notes

A percentage is simply a fraction with a denominator of 100 and a numerator that need not be a whole number. For example, 25% (twenty five percent) is the fraction 25/100, which can also be written as the decimal number 0.25. Word problems involving percentages comes in various flavours:

##### Apply a percentage to a number
• Word problem: Natsuki’s dinner bill comes to \$35.00. How much tip should Natsuki leave if she wants to tip 15%?
• Percentage problem: Find 15% of 35.
• Solve using equivalent fractions:
• tip/35 = 15/100
• 35 x tip/35 = 35 x 15/100    [multiply both sides of the equation by 35]
• tip = 525/100 = 5.25    [calculate and simplify]
• Solve using a percent table:
• 100% → \$35.00
• 1% → \$35.00 ÷ 100 = \$0.35    [divide both sides by 100]
• 15% → 15 x \$0.35 = \$5.25    [multiply both sides by 15]
• Natsuki should leave a tip of \$5.25.
##### Find the percentage of a number
• Word problem: Ted paid a cleaning fee of \$18 on his room charge of \$200. What is Ted’s cleaning fee as a percentage?
• Percentage problem: What percent of 200 is 18?
• Solve using equivalent fractions:
• fee/100 = 18/200
• 100 x fee/100 = 100 x 18/200    [multiply both sides of the equation by 100]
• fee = 1800/200 = 9    [calculate and simplify]
• Solve using a percent table:
• 100% → \$200
• 100% ÷ 200 = 0.5% → \$1    [divide both sides by 100]
• 0.5% x 18 = 9% → \$18    [multiply both sides by 18]
• Ted’s cleaning fee is 9%.
##### Find the whole amount corresponding to a percentage
• Word problem: Béatrice earned \$60 in interest last year on an investment that yields a 4% annual interest rate. How much does Béatrice have invested?
• Percentage problem: 60 is 4% of what number?
• Solve using equivalent fractions:
• 4/100 = 60/investment
• investment x 4/100 = investment x 60/investment    [multiply both sides of the equation by investment]
• investment x 4/100 = 60    [calculate and simplify]
• investment x 4/100 x 100/4 = 60 x 100/4    [multiply both sides of the equation by 100/4]
• investment = 6000/4 = 1500    [calculate and simplify]
• Solve using a percent table:
• 4% → \$60
• 1% → \$60 ÷ 4 = \$15    [divide both sides by 4]
• 100% → 100 x \$15 = \$1500    [multiply both sides by 100]
• Béatrice has \$1500 invested.
##### Find the total after applying a percentage increase/decrease
• Word problem: Veronica bought a t-shirt priced at \$25.00. How much did she have to pay overall if the sales tax is 12%?
• Percentage problem: Calculate \$25.00 plus 12% of \$25.00. In other words, find 112% of \$25.00.
• Solve using equivalent fractions:
• total/25 = 112/100
• 25 x total/25 = 25 x 112/100    [multiply both sides of the equation by 25]
• total = 2800/100 = 28.00    [calculate and simplify]
• Solve using a percent table:
• 100% → \$25.00
• 1% → \$25.00 ÷ 100 = \$0.25    [divide both sides by 100]
• 112% → 112 x \$0.25 = \$28.00    [multiply both sides by 112]
• Veronica has to pay \$28.00 overall.
##### Applying multiple percentage changes

Be careful when applying multiple percentage changes together. Think in terms of multiplication, not addition. For example, suppose the \$25.00 t-shirt is on sale for a 12% discount. Since the sales tax is 12%, does that mean Veronica has to pay \$25.00 overall? Let’s see:

• Pre-tax sale price = \$25.00 – (\$25.00 x 12/100) = \$25.00 – \$3.00 = \$22.00.
• After-tax sale price = \$22.00 + (\$22.00 x 12/100) = \$22.00 + \$2.64 = \$24.64.

The reason that the after-tax sale price is \$24.64, not \$25.00, is because the tax applies to the sale price (\$22.00), not the original price (\$25.00). In other words, we can’t simply add –12% (to represent the sale) and 12% (the sales tax) to get 0% overall.

Another way to approach this problem is as follows.

• First, find the pre-tax sale price by finding \$25.00 minus 12% of \$25.00. In other words, find 88% of \$25.00.
• pre-tax sale price/25 = 88/100
• 25 x pre-tax sale price/25 = 25 x 88/100
• pre-tax sale price = 25 x 88/100
• Next, find the after-tax sale price by finding the pre-tax sale price plus 12% of the pre-tax sale price. In other words, find 112% of the pre-tax sale price.
• after-tax sale price/pre-tax sale price = 112/100
• pre-tax sale price x after-tax sale price/pre-tax sale price = pre-tax sale price x 112/100
• after-tax sale price = 25 x 88/100 x 112/100 = \$24.64

This is why we have to think in terms of multiplication rather than addition when applying multiple percentage changes. Here, we’re multiplying the original price by 88/100 to apply the 12% sale discount and by 112/100 to apply the 12% tax.

For another example, suppose Annie has a large dog, Nora has a medium dog, and Félicité has a small dog. Annie’s dog weighs 30% more than Nora’s dog and Nora’s dog weighs 30% more than Félicité’s. Does that mean Annie’s dog weighs 60% more than Félicité’s dog? No, because 130/100 x 130/100 = 169/100, so Annie’s dog weighs 69% more than Félicité’s dog.

The video below works through some examples of applying percentages.

Practice Exercises

Do the following exercises to practice applying percentages.