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.

Video Tips

Practice Exercises

Do the following exercises to practice applying percentages.