Fractions, Decimals, and Percentages

Fractions, decimals, and percentages are three different ways of expressing parts of a whole. The FSCE 11+ exam tests your ability to convert between them, calculate with them, and apply them to real-world problems. This lesson covers everything you need to know, with six fully worked examples and a conversion reference table you should memorise.

Understanding Fractions

A fraction has two parts:

• The numerator (top number) tells you how many parts you have.
• The denominator (bottom number) tells you how many equal parts the whole is divided into.

So 3/8 means "3 parts out of 8 equal parts."

Types of Fractions

• Proper fraction: numerator is less than the denominator (e.g. 3/4). Value is less than 1.
• Improper fraction: numerator is greater than or equal to the denominator (e.g. 7/4). Value is 1 or more.
• Mixed number: a whole number and a fraction together (e.g. 1 3/4).

Converting Between Mixed Numbers and Improper Fractions

Mixed to improper: Multiply the whole number by the denominator, add the numerator. Keep the same denominator.

• 2 3/5 = (2 x 5 + 3)/5 = 13/5

Improper to mixed: Divide the numerator by the denominator. The quotient is the whole number, the remainder is the new numerator.

• 17/4 = 4 remainder 1, so 17/4 = 4 1/4

Equivalent Fractions and Simplifying

Two fractions are equivalent if they represent the same value: 2/4 = 1/2 = 3/6 = 50/100.

To simplify a fraction, divide both numerator and denominator by their highest common factor (HCF).

• 12/18: HCF of 12 and 18 is 6. So 12/18 = 2/3.

Adding and Subtracting Fractions

Rule: You can only add or subtract fractions when they have the same denominator (common denominator).

Steps:

1. Find a common denominator (the lowest common multiple of both denominators).
2. Convert both fractions to equivalent fractions with this denominator.
3. Add or subtract the numerators. Keep the denominator the same.
4. Simplify if possible.

Example: 2/3 + 1/4

1. Common denominator of 3 and 4 = 12.
2. 2/3 = 8/12. 1/4 = 3/12.
3. 8/12 + 3/12 = 11/12.
4. 11/12 is already in its simplest form.

Multiplying Fractions

Rule: Multiply the numerators together and multiply the denominators together.

3/4 x 2/5 = (3 x 2)/(4 x 5) = 6/20 = 3/10

To multiply by a whole number, write the whole number as a fraction over 1:

• 3/5 x 4 = 3/5 x 4/1 = 12/5 = 2 2/5

Dividing Fractions

Rule: To divide by a fraction, flip the second fraction (find its reciprocal) and multiply.

3/4 ÷ 2/3 = 3/4 x 3/2 = 9/8 = 1 1/8

Memory aid: "Keep, Change, Flip" — keep the first fraction, change ÷ to x, flip the second fraction.

Understanding Decimals

Decimals use a decimal point to represent parts of a whole:

Ones	.	Tenths	Hundredths	Thousandths
3	.	4	5	2

This represents 3.452 = 3 + 4/10 + 5/100 + 2/1000.

Understanding Percentages

"Per cent" means "out of 100." So 45% means 45 out of 100, or 45/100, or 0.45.

Finding a Percentage of an Amount

To find 15% of 240:

• Method 1: Convert to a decimal and multiply. 15% = 0.15. 0.15 x 240 = 36.
• Method 2: Find 10% and 5%. 10% of 240 = 24. 5% of 240 = 12. 15% = 24 + 12 = 36.

Conversion Reference Table

This table is essential. Memorise these conversions:

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
2/5	0.4	40%
3/5	0.6	60%
1/10	0.1	10%
1/3	0.333...	33.3...%
2/3	0.666...	66.6...%
1/8	0.125	12.5%
3/8	0.375	37.5%
1/100	0.01	1%

How to Convert

Fraction to decimal: Divide the numerator by the denominator. 3/8 = 3 ÷ 8 = 0.375.

Decimal to percentage: Multiply by 100. 0.375 x 100 = 37.5%.

Percentage to fraction: Write over 100 and simplify. 45% = 45/100 = 9/20.

Percentage to decimal: Divide by 100. 37.5% ÷ 100 = 0.375.

Decimal to fraction: Use the place value. 0.375 = 375/1000 = 3/8.

Worked Examples

Worked Example 1: Converting a Fraction to a Percentage

Question: Convert 7/20 to a percentage.

Step-by-step solution:

1. Divide: 7 ÷ 20 = 0.35.
2. Multiply by 100: 0.35 x 100 = 35%.
3. Alternative method: 7/20 = ?/100. Since 20 x 5 = 100, multiply numerator by 5 too: 7 x 5 = 35. So 7/20 = 35/100 = 35%.

Answer: 35%

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Worked Example 2: Adding Fractions with Different Denominators

Question: Calculate 3/4 + 5/6. Give your answer as a mixed number in its simplest form.