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May 2013

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The ideas in this lesson are taken from Math Mammoth Percent book.

The concepts and ideas of this lesson are also explained in this video:

Percentage of a Number: Using Decimals

You have learned that to find 1% of a number means finding 1/100 of it. Similarly, finding 60% of a number means finding 60/100 (or 6/10) of it.

In these expressions, the word “of” translates into multiplication:

1% of 90 → 1% × 90 OR 60% of $700 → 60% × $700.

We can also write those percentages as decimals:

1% of 90 → 0.01 × 90 OR 60% of $700 → 0.6 × $700.

This gives us another way to calculate the percentage of a number (or percentage of some quantity):

To calculate a percentage of some number, change the percentage into a decimal,
and the word "of" into multiplication.

Example 1. Find 70% of 80.

Following the shortcut, we write this as 0.7 × 80.

Remember that in decimal multiplication, you multiply as if there were no decimal points, and the answer will have as many “decimal digits” to the right of the decimal point as the total number of decimal digits of all of the factors. So when you multiply 0.7 × 80, think of multiplying 7 × 80 = 560. Since 0.7 has one decimal digit, and 80 has none, the answer has one decimal digit: 56.0 Thus, 0.7 × 80 = 56.

You can also use “common sense” to reason it through logically: 0.7 × 80 must be less than 80, yet more than 1/2 of 80, which is 40. Since 7 × 8 = 56, you know that the answer must be 56—not 5.6 or 560.

Example 2. Find 3% of $4,000.

First write it as 0.03 × $4,000. Then multiply 3 × $4,000 = $12,000. Lastly put the decimal point where it gives the answer two decimal digits: $120.00.

Example 3. Find 23% of 5,500 km.

Write the expression as 0.23 × 5,500 km and use a calculator to calculate the product. The answer is 1,265 km. This answer makes sense because 10% of 5,500 km is 550 km, so 20% is 1,100 km. Thus 1,265 km as 23% of 5,500 km is a reasonable answer.

1. "Translate" the expressions into multiplication by a decimal. Calculate.

a. 20% of 70 ______ × ______ = ______	b. 90% of 50 ______ × ______ = ______	c. [available in the book]
d. [available in the book]	e. 9% of 3,000 ______ × ______ = ______	c. [available in the book]

2. "Translate" the other way: write the multiplications as expressions of "percentage of the number".

a. 0.6 × 50 _____% of ______ = ______	b. 0.03 × $400 _____% of ______ = ______	c. [available in the book]
d. 0.08 × 6 _____% of ______ = ______	e. [available in the book]	f. [available in the book]

3. Use a calculator to find percentages of these quantities.

a. 17% of $4500

b. 67% of 27 m

c. [available in the book]

4. Use mental math to find percentages of these quantities.

a. 25% of 240 mi

b. [available in the book]

c. [available in the book]

5. a. A lake has a 30-km long shoreline. 6% of it is sandy beach.
What percentage of the shoreline is not sandy beach?

b. [available in the book]

6. Twenty percent of a university’s 4,000 students have a scholarship.

a. What percent of the students do not have a scholarship?

b. How many students have a scholarship?

c. [available in the book]

7. [available in the book]

8. Identify the errors that these children made. Then find the correct answers.

a. Find 80% of 50.

Gladys’s solution:
80 × 50 = 4,000

b. Find 75% of 84,000.

Glenn’s solution:
[available in the book]

9. Find the expressions with the same value as 20% of $620.

0.02 × $620

$620 ÷ 5

$620 ÷ 10 × 2

2 × $62

× $620

0.2 × $620

20 × $620

$620 ÷ 4

10. [available in the book]

11. The table below shows Andy’s usage of time in one day.

a. Calculate the time he spent in each activity.
Round the minutes to the nearest minute.

b. Label the sections in the circle graph with the name of each activity.

Activity	Percentage	Minutes	Hours/minutes
Sleep	38%
School	21%
Soccer	10%	144	2 h 24 min
Play	11%
Eating	9%
Chores	9%
Hygiene	2%
TOTAL	100%	1440	24 hours

Andy’s Usage of Time

The ideas in this lesson are taken from Math Mammoth Percent book.