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
Exercise 1.
State the meaning of each percentage:
a) (10%)
b) (25%)
c) (80%)
Exercise 2.
Write each percentage as “out of (100)”:
a) (6%)
b) (48%)
c) (125%)
Exercise 3.
Convert each percentage to a decimal:
a) (3%)
b) (20%)
c) (57%)
Exercise 4.
Convert each percentage to a decimal:
a) (75%)
b) (140%)
c) (8.5%)
Exercise 5.
Convert each decimal to a percentage:
a) (0.2)
b) (0.45)
c) (0.9)
Exercise 6.
Convert each decimal to a percentage:
a) (1.2)
b) (0.07)
c) (0.375)
Exercise 7.
Convert in either direction as needed:
a) (34%)
b) (0.68)
c) (2.05)
Exercise 8.
Convert in either direction as needed:
a) (150%)
b) (0.04)
c) (99%)
Reasoning
Exercise 9.
Explain why (45% = 0.45).
Exercise 10.
A student says (7% = 0.7). Explain the mistake.
Exercise 11.
Explain why converting a decimal to a percentage involves multiplying by (100).
Exercise 12.
A student says (1.3 = 13%). Explain why this is incorrect.
Problem-solving
Exercise 13.
A student scored (0.86) of the total marks on a test. Write this as a percentage.
Exercise 14.
A phone battery is at (35%) charge. Write this as a decimal.
Exercise 15.
A tank is (0.72) full. Write this as a percentage.
Exercise 16.
A shop advertises a discount of (15%). Write this percentage as a decimal.
Exercise 17.
A machine is running at (1.1) of its normal speed. Write this as a percentage.
Exercise 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%)