047. Percentages and Decimals

Learning Intentions

• To understand the meaning of ‘per cent’ (%).
• convert between percentages and decimals

Pre-requisite Summary

• Understand that a fraction can represent part of a whole
• Know that decimal numbers can represent tenths, hundredths and thousandths
• Understand that the decimal point separates whole numbers from parts of a whole
• Know that (\text{per cent}) means “for every (100)” or “out of (100)”
• Be able to multiply and divide by (100)
• Recognise equivalent forms of the same quantity, such as fraction, decimal and percentage

Worked Examples

Worked Example 1

a) Explain what (35%) means.
b) Write (35%) as “out of (100)”.
c) Write (35%) as a decimal.

Worked Example 2

Convert each percentage to a decimal:
a) (8%)
b) (45%)
c) (120%)

Worked Example 3

Convert each decimal to a percentage:
a) (0.6)
b) (0.27)
c) (1.4)

Worked Example 4

Convert between percentages and decimals:
a) (0.05) to a percentage
b) (75%) to a decimal
c) (0.325) to a percentage

Worked Example 5

A test score is (0.84) of the total.
a) Write this as a percentage.
b) Explain what the percentage means.

Worked Example 6

A battery is charged to (125%).
a) Write this as a decimal.
b) Explain why a percentage can be greater than (100%).

Problems

Problem 1

a) Explain what (62%) means.
b) Write (62%) as “out of (100)”.
c) Write (62%) as a decimal.

Problem 2

Convert each percentage to a decimal:
a) (9%)
b) (54%)
c) (135%)

Problem 3

Convert each decimal to a percentage:
a) (0.7)
b) (0.43)
c) (1.25)

Problem 4

Convert between percentages and decimals:
a) (0.08) to a percentage
b) (65%) to a decimal
c) (0.415) to a percentage

Problem 5

A test score is (0.91) of the total.
a) Write this as a percentage.
b) Explain what the percentage means.

Problem 6

A machine is operating at (150%) of its usual output.
a) Write this as a decimal.
b) Explain why a percentage can be greater than (100%).

Exercises

Understanding and Fluency

1. State the meaning of each percentage:
   a) (10%)
   b) (25%)
   c) (80%)

2. Write each percentage as “out of (100)”:
   a) (6%)
   b) (48%)
   c) (125%)

3. Convert each percentage to a decimal:
   a) (3%)
   b) (20%)
   c) (57%)

4. Convert each percentage to a decimal:
   a) (75%)
   b) (140%)
   c) (8.5%)

5. Convert each decimal to a percentage:
   a) (0.2)
   b) (0.45)
   c) (0.9)

6. Convert each decimal to a percentage:
   a) (1.2)
   b) (0.07)
   c) (0.375)

7. Convert in either direction as needed:
   a) (34%)
   b) (0.68)
   c) (2.05)

8. Convert in either direction as needed:
   a) (150%)
   b) (0.04)
   c) (99%)

Reasoning

9. Explain why (45% = 0.45).

10. A student says (7% = 0.7). Explain the mistake.

11. Explain why converting a decimal to a percentage involves multiplying by (100).

12. A student says (1.3 = 13%). Explain why this is incorrect.

Problem-solving

13. A student scored (0.86) of the total marks on a test. Write this as a percentage.

14. A phone battery is at (35%) charge. Write this as a decimal.

15. A tank is (0.72) full. Write this as a percentage.

16. A shop advertises a discount of (15%). Write this percentage as a decimal.

17. A machine is running at (1.1) of its normal speed. Write this as a percentage.

18. A class attendance rate is (98%). Write this as a decimal.

Potential Misunderstandings

• Students may think (\text{per cent}) means “out of (10)” instead of “out of (100)”
• Students may move the decimal point in the wrong direction when converting between decimals and percentages
• Students may think a percentage must always be less than (100%)
• Students may confuse (0.5) with (5%) instead of (50%)
• Students may write (25%) as (25.0) instead of (0.25)
• Students may not recognise that decimals greater than (1) correspond to percentages greater than (100%)

Next: 048. Comparing Proportions