Percentages

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1. Percent of a quantity; finding the percent and the whole

Warm-up

Three quick questions from earlier lessons.

Discover

A class of 40 students: 25% walk to school. How many walk?

  1. 10
  2. 25

Key idea

Percent means per hundred.

p%p\% of a whole is p100×whole\tfrac{p}{100} \times \text{whole}. What percent? part ÷ whole × 100. Find the whole: part ÷ p100\tfrac{p}{100}.

Worked example

30% of a number is 12.

whole=12÷0.3\text{whole} = 12 \div 0.3
0.3 × whole = 12.
4040
Divide.

Your turn, guided

Your turn: what percent of 80 is 20?

2525
20 ÷ 80 × 100.

Practice ★

  1. What is 50% of 340?
  2. What is 5% of 300?
  1. What percent of 140 is 7?

Watch out!

Practice ★★

  1. 60% of a number is 228. What is the number?
  2. 45% of a number is 162. What is the number?

Exit ticket

Three questions to check you have got it.

  1. What is 50% of 340?
  2. What is 5% of 300?
  3. What is 40% of 400?

Summary card

Percent of a quantity

p%=p/100p\% = p/100; percent = part ÷ whole × 100.

Watch out: divide the part by the whole, not the reverse.

2. Percent increase and decrease: tax, tips, discounts

Warm-up

Three quick questions from earlier lessons.

Discover

A 60-dollar coat is 10% off. What do you pay?

  1. 54 dollars
  2. 6 dollars

Key idea

Increase: add; decrease: subtract, always a percent of the original.

An increase of p%p\% gives (100+p)%(100 + p)\% of the original; a decrease gives (100−p)%(100 - p)\%. Tax and tips are increases, discounts are decreases. Percent change = change ÷ original × 100.

Worked example

Price from 40 to 46 dollars.

66
Change.
0.150.15, a 15% increase
6 ÷ 40.

Your turn, guided

Your turn: 80 dollars plus 5% tax. The total.

8484
80 · 1.05.

Practice ★

  1. A 160-dollar jacket's price goes up by 40%. What is the new price (dollars)?
  2. A 300-dollar jacket's price goes up by 20%. What is the new price (dollars)?
  1. A 320-dollar game is 35% off. What is the sale price (dollars)?

Watch out!

Practice ★★

  1. A town grew from 280 to 518 people. By what percent did it grow?
  2. A town grew from 260 to 468 people. By what percent did it grow?
  1. Boss problem. A meal costs 120 dollars. The sales tax is 10% and you leave a 20% tip on the price before tax.

Exit ticket

Three questions to check you have got it.

  1. A 160-dollar jacket's price goes up by 40%. What is the new price (dollars)?
  2. A 300-dollar jacket's price goes up by 20%. What is the new price (dollars)?
  3. A 60-dollar jacket's price goes up by 60%. What is the new price (dollars)?

Summary card

Percent increase and decrease

New = (100±p)%(100 \pm p)\% of the original; change ÷ original for the percent.

Watch out: the base is the original amount.

3. Percent error

Warm-up

Three quick questions from earlier lessons.

Discover

A 50 cm board is measured as 52 cm. Which describes the error?

  1. A 4% error
  2. A 2% error

Key idea

Percent error compares the error with the actual value.

Percent error =∣measured−actual∣actual×100= \dfrac{|\text{measured} - \text{actual}|}{\text{actual}} \times 100. A tolerance of p%p\% allows values within p%p\% of the target, above or below.

Worked example

A 400 g bag with at most 5% error.

2020 g
5% of 400.
From 380 g to 420 g
Range.

Your turn, guided

Your turn: actual 40, measured 38. The percent error.

55
2 ÷ 40 × 100.

Practice ★

  1. A board is really 36 cm long, but Ana measures 40 cm. What is her percent error?
  2. A board is really 73 cm long, but Ana measures 76 cm. What is her percent error?
  1. A box really holds 400 cups, but a label says 520. What is the percent error?

Watch out!

Practice ★★

  1. A bag should hold 600 g of rice, with at most 7% error. What is the smallest allowed mass (g)?
  2. A bag should hold 400 g of rice, with at most 8% error. What is the smallest allowed mass (g)?

Exit ticket

Three questions to check you have got it.

  1. A board is really 36 cm long, but Ana measures 40 cm. What is her percent error?
  2. A board is really 73 cm long, but Ana measures 76 cm. What is her percent error?
  3. A board is really 52 cm long, but Ana measures 53 cm. What is her percent error?

Summary card

Percent error

Error ÷ actual × 100.

Watch out: divide by the actual value, not the measured one.

Answers: Percentages

1. Percent of a quantity; finding the percent and the whole

A

  1. 170170
  2. 1515

B

  1. 55

C

  1. 380380
  2. 360360

D

  1. 170170
  2. 1515
  3. 160160

2. Percent increase and decrease: tax, tips, discounts

A

  1. 224224
  2. 360360

B

  1. 208208

C

  1. 8585
  2. 8080

D

  1. 156156

E

  1. 224224
  2. 360360
  3. 9696

3. Percent error

A

  1. 1009\frac{100}{9}
  2. 30073\frac{300}{73}

B

  1. 3030

C

  1. 558558
  2. 368368

D

  1. 1009\frac{100}{9}
  2. 30073\frac{300}{73}
  3. 2513\frac{25}{13}