Understanding Fractions for the 11-Plus Exam

Understanding Fractions

Fractions are a fundamental concept in mathematics, particularly important for the 11-plus exam. This section will cover various aspects of fractions, including equivalent fractions, simplifying fractions, comparing and ordering fractions, and performing operations with fractions.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value. For example, 1/2 is equivalent to 2/4 and 3/6. To find equivalent fractions, you can multiply or divide both the numerator and the denominator by the same number.

Worked Example

Problem: Find two equivalent fractions for 3/5.

Solution:

• Multiply the numerator and denominator by 2: 3 × 2 / 5 × 2 = 6/10
• Multiply the numerator and denominator by 3: 3 × 3 / 5 × 3 = 9/15

Thus, 6/10 and 9/15 are equivalent to 3/5.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest terms. This is done by finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by this number.

Worked Example

Problem: Simplify the fraction 8/12.

Solution:

• The GCD of 8 and 12 is 4.
• Divide both the numerator and denominator by 4: 8 ÷ 4 / 12 ÷ 4 = 2/3.

Therefore, 8/12 simplifies to 2/3.

Comparing and Ordering Fractions

To compare fractions, they must have a common denominator. If they do not, you can find a common denominator by determining the least common multiple (LCM) of the denominators.

Worked Example

Problem: Compare 1/4 and 1/3.

Solution:

• The LCM of 4 and 3 is 12.
• Convert both fractions: 1/4 = 3/12 and 1/3 = 4/12.
• Since 3/12 is less than 4/12, 1/4 is less than 1/3.

Adding and Subtracting Fractions

When adding or subtracting fractions with different denominators, convert them to have a common denominator first.

Worked Example

Problem: Add 1/4 and 1/3.

Solution:

• Find the LCM of 4 and 3, which is 12.
• Convert the fractions: 1/4 = 3/12 and 1/3 = 4/12.
• Add: 3/12 + 4/12 = 7/12.

Multiplying and Dividing Fractions

To multiply fractions, simply multiply the numerators and the denominators. For division, multiply by the reciprocal of the second fraction.

Worked Example

Problem: Multiply 2/3 by 3/4.

Solution:

• Multiply the numerators: 2 × 3 = 6.
• Multiply the denominators: 3 × 4 = 12.
• The result is 6/12, which simplifies to 1/2.

Converting Between Mixed Numbers and Improper Fractions

A mixed number consists of a whole number and a fraction, while an improper fraction has a numerator larger than its denominator. To convert between them, follow these steps:

Worked Example

Problem: Convert 2 1/2 to an improper fraction.

Solution:

• Multiply the whole number by the denominator: 2 × 2 = 4.
• Add the numerator: 4 + 1 = 5.
• Place over the original denominator: 5/2.

Finding Fractions of Amounts

To find a fraction of an amount, multiply the amount by the numerator and then divide by the denominator.

Worked Example

Problem: Find 2/5 of 20.

Solution:

• Multiply: 20 × 2 = 40.
• Divide by 5: 40 ÷ 5 = 8.

Thus, 2/5 of 20 is 8.

Understanding fractions is crucial for success in the 11-plus exam. Master these concepts to enhance your mathematical skills and confidence.