Percent Applications: math lesson, Grade 7

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Math lessons Grade 7 : Percent Applications — Zyro the alien explorer of Planète Maths

Percents show up every time you shop, eat out, save money, or measure something. A store says “30% off,” a menu suggests an 18% tip, a bank pays 4% interest, and a scientist reports a 3% error. In this chapter you will use one simple idea, percent means “per 100,” to solve all of these real-world problems with confidence.

Sommaire

1. Finding a percent of a quantity

DefinitionA percent is a ratio that compares a number to 100. The symbol % means “out of 100,” so \( 35\% = \dfrac{35}{100} = 0.35 \).

To find a percent of a quantity, change the percent to a decimal (or fraction) and multiply. The word “of” in a percent problem usually signals multiplication.

Method: percent of a quantity

Write the percent as a decimal by moving the decimal point two places to the left.
Multiply the decimal by the quantity.
Write the answer with the right unit.

Example 1A school has 240 students and 35% of them ride the bus. How many ride the bus?
\( 35\% = 0.35 \) and \( 0.35 \times 240 = 84 \). So 84 students ride the bus.

A bar model makes this idea visible. Cut the whole into ten equal parts: each part is 10% of the whole. Below, the whole is a 250-mile road trip, so each part is 25 miles, and 4 parts (40%) make 100 miles.

Check: \( 0.40 \times 250 = 100 \) miles. The bar model and the multiplication agree.

2. Percent increase and decrease

When a quantity changes, we often describe the change as a percent of the original amount.

Percent change\[ \text{percent change} = \dfrac{\text{amount of change}}{\text{original amount}} \times 100\% \]
If the new value is larger, it is a percent increase. If it is smaller, it is a percent decrease. The amount of change is the difference between the new and original values, written as a positive number.

Example 2A video game costs 48 dollars in March and 60 dollars in April. What is the percent increase?
Change: \( 60 - 48 = 12 \). Percent change: \( \dfrac{12}{48} = 0.25 = 25\% \). The price went up by 25%.
Now suppose a game drops from 80 dollars to 68 dollars. Change: \( 80 - 68 = 12 \), and \( \dfrac{12}{80} = 0.15 \). The price fell by 15%.

You can also go the other way. To increase a number by \( r\% \), multiply by \( 1 + \dfrac{r}{100} \). To decrease it by \( r\% \), multiply by \( 1 - \dfrac{r}{100} \). For instance, an 8% increase means multiplying by 1.08, and a 12% decrease means multiplying by 0.88.

Watch outAlways divide by the original amount, not the new one. Also, a 50% increase followed by a 50% decrease does not bring you back to the start: 100 becomes 150, and 50% of 150 is 75, which leaves 75.

3. Markups and discounts

DefinitionsA markup is the amount a store adds to its cost to set a selling price. A discount is the amount subtracted from the original price during a sale. Both are usually given as a percent of a starting price: the markup is a percent of the store’s cost, and the discount is a percent of the regular price.

Example 3A jacket regularly costs 90 dollars and is 30% off.
Discount: \( 0.30 \times 90 = 27 \) dollars. Sale price: \( 90 - 27 = 63 \) dollars.
Shortcut: paying 70% of the price means \( 0.70 \times 90 = 63 \). The double number line shows both views.

Example 4A shop pays 32 dollars for a skateboard deck and uses a 50% markup.
Markup: \( 0.50 \times 32 = 16 \) dollars. Selling price: \( 32 + 16 = 48 \) dollars.

4. Sales tax and tips

Sales tax is a percent of the price that is added at the register. A tip (or gratuity) is a percent of the bill that you add to thank the person who served you. Both work like a markup: the total is the original amount plus the percent of it.

Method: tax or tip

Convert the rate to a decimal.
Multiply by the price (or bill) to get the tax (or tip).
Add it to the price to get the total, or multiply once by \( 1 + \text{rate} \).

Example 5Headphones cost 36.00 dollars and the sales tax rate is 7.5%.
Tax: \( 0.075 \times 36 = 2.70 \). Total: \( 36 + 2.70 = 38.70 \) dollars.
A diner’s bill is 42.50 dollars and the diner leaves an 18% tip: \( 0.18 \times 42.50 = 7.65 \) dollars.

Zyro’s trickOn my home planet we do tips in our heads! Move the decimal point one place left to get 10%. Double that for 20%, or halve it and add for 15%. On a 40-dollar bill, 10% is 4 dollars, 20% is 8 dollars, and 15% is 6 dollars.

5. Simple interest

When you deposit money in a savings account, the bank pays you interest. When you borrow money, you pay interest. With simple interest, the interest is always figured on the original amount.

Simple interest formula\[ I = P \times r \times t \]
Here \( I \) is the interest, \( P \) is the principal (the starting amount), \( r \) is the annual interest rate written as a decimal, and \( t \) is the time in years. The final amount is \( P + I \).

Example 6You deposit 1,500 dollars at 4% simple interest for 3 years.
\( I = 1500 \times 0.04 \times 3 = 180 \) dollars. The account holds \( 1500 + 180 = 1680 \) dollars.

Because the interest is the same every year, the graph of interest versus time is a straight line through the origin. This is a proportional relationship. The graph below shows 800 dollars at 5%: that is 40 dollars of interest each year.

Watch outUse the rate as a decimal (5% becomes 0.05) and the time in years. Eighteen months is 1.5 years, and 9 months is 0.75 year.

6. Commissions and fees

A commission is pay that equals a percent of the sales a person makes. A fee is a charge for a service, and it can be a flat amount, a percent, or both.

Example 7A furniture salesperson sells 12,500 dollars of furniture in a month and earns a 6% commission.
Commission: \( 0.06 \times 12500 = 750 \) dollars. If the agent also has a 1,800 dollar base salary, total pay is \( 1800 + 750 = 2550 \) dollars.
A payment app charges a 3% fee on an 80 dollar payment: \( 0.03 \times 80 = 2.40 \) dollars.

7. Percent error

No measurement is perfect. Percent error tells you how far a measured value is from the true (actual) value, compared with the true value.

Percent error\[ \text{percent error} = \dfrac{|\text{measured} - \text{actual}|}{\text{actual}} \times 100\% \]
The absolute value bars mean the error is always positive. A smaller percent error means a more accurate measurement.

Example 8A board is actually 50 centimeters long, but you measure 48.5 centimeters.
Error: \( |48.5 - 50| = 1.5 \). Percent error: \( \dfrac{1.5}{50} = 0.03 = 3\% \). The same idea works in U.S. units: measuring a 20-inch shelf as 19 inches gives \( \dfrac{1}{20} = 5\% \) error.

Key takeaways

Percent means per 100: \( 35\% = 0.35 \). To find a percent of a number, multiply.
Percent change \( = \dfrac{\text{change}}{\text{original}} \times 100\% \). Always divide by the original amount.
Discount: subtract the percent of the price. Markup, tax, and tip: add the percent of the amount.
Simple interest: \( I = Prt \), with the rate as a decimal and the time in years.
Commission and fees are percents of an amount of sales or money.
Percent error \( = \dfrac{|\text{measured} - \text{actual}|}{\text{actual}} \times 100\% \).