Conversion from Percentage to Decimal and Decimal to Fraction – JSS1 Mathematics Lesson Note (First Term)

Conversion is the process of changing a number from one form (fraction, decimal or percentage) to another without changing its value.

• Fraction to decimal: divide the numerator by the denominator.
• Decimal to fraction: write the digits over 10, 100 or 1000, then simplify.
• Percentage to fraction: write the number over 100, then simplify.
• Percentage to decimal: divide the percentage by 100.

Behavioural objectives

By the end of the lesson, students should be able to:

1. change a fraction into a terminating decimal;
2. change a decimal into a fraction in its lowest terms;
3. express a percentage, including one with a mixed number, as a fraction in its simplest form;
4. change a percentage into a decimal;
5. change a decimal into a percentage.

What is a percentage?

A percentage is a fraction with 100 as its denominator, written with the symbol %.

• Meaning of %: “per cent” means “out of 100”.
• Example: 45% means 45 out of 100, which is 45/100.
• Example: if a student scores 70% in a test, the student got 70 marks out of every 100.

What is a terminating decimal?

A terminating decimal is a decimal number that ends after a fixed number of digits, such as 0.75 or 0.625.

• Tenths: the first digit after the decimal point, for example 0.3 = 3/10.
• Hundredths: the second digit after the point, for example 0.07 = 7/100.
• Thousandths: the third digit after the point, for example 0.009 = 9/1000.

How do we convert a fraction to a decimal?

To convert a fraction into a decimal, divide the numerator by the denominator.

1. Write the numerator as a decimal number, such as 3.0, 3.00 or 3.000.
2. Divide it by the denominator, using long division.
3. Add a zero each time a remainder is left, and keep dividing.
4. Stop when the remainder is zero.

Example 1

(a) Change 3/4 into a terminating decimal number. (b) Change 5/8 to a decimal fraction.

Solution

(a) 3/4:

1. The numerator 3 can be written as 3.0, 3.00 or 3.000.
2. 4 divides 3 zero times, so add a zero.
3. 4 goes into 30 seven times and 2 remains.
4. Add 0 to make 20. 4 goes into 20 five times.

Therefore, 3/4 = 0.75.

(b) 5/8:

1. 8 divides 5 zero times, so add a zero to make 50.
2. 8 goes into 50 six times and 2 remains.
3. Add a zero to make 20. 8 goes into 20 two times and 4 remains.
4. Add a zero to make 40. 8 goes into 40 five times.

Therefore, 5/8 = 0.625.

How do we convert a decimal to a fraction?

To change a decimal into a fraction, write the digits over a power of 10 and simplify.

1. Count the digits after the decimal point.
2. Write the digits as the numerator, without the point.
3. Write the denominator as 1 followed by as many zeros as there are digits: 10, 100 or 1000.
4. Simplify by dividing the numerator and denominator by their highest common factor.

Example 2

Convert each of the following decimal numbers into a fraction in the lowest terms: (a) 0.067 (b) 0.64

Solution

(a) 0.067 = 67/1000.

• There are 3 digits after the decimal point, so the denominator is 1000.
• 67 is a prime number, so the fraction cannot be simplified.

(b) 0.64 = 64/100.

• There are 2 digits after the decimal point, so the denominator is 100.
• Divide the numerator and the denominator by 4 to get 16/25.

How do we convert a percentage to a fraction?

To change a percentage to a fraction, write it over 100 and simplify.

1. Write the percentage number as the numerator and 100 as the denominator.
2. Divide the numerator and denominator by their highest common factor.
3. If the percentage has a mixed number, change it to an improper fraction first, then multiply by 1/100.

Example 3

Express each of the following as a fraction in its simplest form: (a) 30% (b) 45% (c) 16½%

Solution

(a) 30% = 30/100.

• The zeros cancel each other.
• The answer is 3/10.

(b) 45% = 45/100.

• Divide both numerator and denominator by 5.
• The answer is 9/20.

(c) 16½% = 33/2 × 1/100.

• Change the mixed number 16½ to the improper fraction 33/2.
• Multiply it by 1/100.
• The answer is 33/200.

How do we convert a percentage to a decimal?

To change a percentage to a decimal, divide it by 100, which moves the decimal point two places to the left.

1. Write the percentage as a number over 100.
2. Divide by 100 by moving the decimal point two places to the left.
3. Add a zero in front where there is no digit to move past.
4. Drop the % sign.

Example 4

Change each of the following to a decimal: (a) 45% (b) 7% (c) 16½% (d) 120%

Solution

• (a) 45% = 45/100 = 0.45.
• (b) 7% = 7/100 = 0.07. A zero is added before the 7 to fill the tenths place.
• (c) 16½% = 16.5% = 16.5/100 = 0.165.
• (d) 120% = 120/100 = 1.2. A percentage above 100 gives a decimal above 1.

How do we convert a decimal to a percentage?

To change a decimal to a percentage, multiply it by 100 and write the % sign.

• Method: move the decimal point two places to the right.
• Example: 0.35 × 100 = 35%.
• Example: 0.6 × 100 = 60%.

What are some common equivalent values?

The table shows fractions that students meet often, with their decimal and percentage forms.

Evaluation

1. Explain how to change a fraction into a decimal.
2. What do you count when changing a decimal into a fraction?
3. Change 0.75 into a fraction in its lowest terms.
4. Express 60% as a fraction in its simplest form.
5. Change 8% into a decimal.

Assignment

Do all of the following, showing your working.

1. Convert 0.125 into a fraction in its lowest terms.
2. Express 25% as a fraction in its simplest form.
3. Express 12% as a fraction in its simplest form.
4. Change 35% into a decimal.