Understanding Fractions

What are Fractions?

A fraction represents a part of a whole or, more generally, any number of equal parts. It's a way of showing a value that is not a whole number.

A fraction has two main parts:

Numerator Denominator
  • The Numerator (top number) tells us how many parts we have.
  • The Denominator (bottom number) tells us how many equal parts the whole is divided into.

Types of Fractions:

Understanding HCF and LCM

When working with fractions, two important concepts are the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM). They help us simplify fractions and perform operations like addition and subtraction.

Highest Common Factor (HCF)

The HCF (also known as Greatest Common Divisor or GCD) of two or more numbers is the largest number that divides into all of them without leaving a remainder.

Method 1: Listing Factors

One common method is to list all the factors (numbers that divide exactly into it) of each number and then find the largest factor that appears in all lists.

Example: Find the HCF of 12 and 18.

  • Factors of 12 are: 1, 2, 3, 4, 6, 12.
  • Factors of 18 are: 1, 2, 3, 6, 9, 18.

The common factors are 1, 2, 3, and 6. The largest of these is 6.

So, the HCF of 12 and 18 is 6.

Method 2: Euclidean Algorithm

The Euclidean Algorithm is an efficient method for finding the HCF of two numbers, especially larger numbers. The steps are:

  1. Divide the larger number by the smaller number and find the remainder.
  2. If the remainder is 0, the smaller number is the HCF.
  3. If the remainder is not 0, replace the larger number with the smaller number, and the smaller number with the remainder.
  4. Repeat steps 1-3 until the remainder is 0. The HCF is the last non-zero remainder (which will be the divisor at that stage).

Example 1: Find the HCF of 48 and 18 using the Euclidean Algorithm.

  • Step 1: Divide 48 by 18: 48 = 2 × 18 + 12 (Remainder is 12)
  • Step 2: Replace larger (48) with smaller (18), and smaller (18) with remainder (12). Now find HCF of 18 and 12.
  • Step 3: Divide 18 by 12: 18 = 1 × 12 + 6 (Remainder is 6)
  • Step 4: Replace larger (12) with smaller (6), and smaller (6) with remainder (6). Now find HCF of 12 and 6.
  • Step 5: Divide 12 by 6: 12 = 2 × 6 + 0 (Remainder is 0)

Since the remainder is 0, the HCF is the last divisor, which is 6.

So, the HCF of 48 and 18 is 6.

Example 2: Find the HCF of 105 and 70.

  • Step 1: 105 = 1 × 70 + 35 (Remainder is 35)
  • Step 2: Now find HCF of 70 and 35.
  • Step 3: 70 = 2 × 35 + 0 (Remainder is 0)

The remainder is 0. The HCF is the last divisor, 35.

So, the HCF of 105 and 70 is 35.

When is HCF used with fractions?

The HCF is primarily used for simplifying fractions. To simplify a fraction, you divide both the numerator and the denominator by their HCF. This reduces the fraction to its lowest terms without changing its value.

E.g., To simplify 12⁄18, divide both by HCF(12,18) = 6. So, 12÷6⁄18÷6 = 2⁄3.

Lowest Common Multiple (LCM)

The LCM of two or more numbers is the smallest positive number that is a multiple of all the numbers.

How to find the LCM:

One common method is to list the multiples of each number until you find the smallest multiple that appears in all lists.

Example: Find the LCM of 8 and 12.

  • Multiples of 8 are: 8, 16, 24, 32, 40, 48...
  • Multiples of 12 are: 12, 24, 36, 48, ...

The first common multiple appearing in both lists is 24.

So, the LCM of 8 and 12 is 24.

When is LCM used with fractions?

The LCM is crucial when you need to:

  • Add or subtract fractions with different denominators. The LCM of the denominators is used as the common denominator.
  • Compare fractions with different denominators. By converting them to equivalent fractions with a common denominator (the LCM), you can easily see which is larger.

E.g., To add 1⁄8 + 5⁄12, the LCM of 8 and 12 is 24. So, 3⁄24 + 10⁄24 = 13⁄24.

Visualizing Common Fractions

Seeing fractions visually can help understand what they represent.

1⁄2 (One Half)

1⁄3 (One Third)

1⁄4 (One Quarter)

3⁄4 (Three Quarters)

1⁄1 (One Whole)

2⁄3 (Two Thirds)

Working with Fractions: Examples

Simplifying Fractions

Problem: Simplify 4⁄8

  1. Find the largest number that divides both the numerator (4) and the denominator (8). This is called the Highest Common Factor (HCF). To find the HCF of 4 and 8:
    • Factors of 4 are: 1, 2, 4.
    • Factors of 8 are: 1, 2, 4, 8.
    • The common factors are 1, 2, and 4. The Highest Common Factor is 4.
  2. Divide both the numerator and the denominator by the HCF (4):
    • 4 ÷ 4 = 1
    • 8 ÷ 4 = 2

Answer: 1⁄2

Adding Fractions (Same Denominator)

Problem: 1⁄5 + 2⁄5

  1. Keep the denominator the same.
  2. Add the numerators: 1 + 2 = 3.

Answer: 3⁄5

Adding Fractions (Different Denominators)

Problem: 1⁄3 + 1⁄2

  1. Find a common denominator. This is a number that both 3 and 2 divide into. The easiest way is to find the Lowest Common Multiple (LCM).
    • Multiples of 3 are: 3, 6, 9, 12, ...
    • Multiples of 2 are: 2, 4, 6, 8, 10, ...
    • The Lowest Common Multiple (LCM) of 3 and 2 is 6. So, 6 will be our common denominator.
  2. Convert each fraction to an equivalent fraction with the common denominator (6):
    • For 1⁄3: To change the denominator from 3 to 6, we multiply by 2 (since 3 × 2 = 6). So, we must also multiply the numerator by 2: 1 × 2 = 2. Thus, 1⁄3 = 2⁄6.
    • For 1⁄2: To change the denominator from 2 to 6, we multiply by 3 (since 2 × 3 = 6). So, we must also multiply the numerator by 3: 1 × 3 = 3. Thus, 1⁄2 = 3⁄6.
  3. Now that the denominators are the same, add the new numerators: 2 + 3 = 5. Keep the common denominator.

Answer: 5⁄6

Multiplying Fractions

Problem: 2⁄3 × 1⁄4

  1. Multiply the numerators together: 2 × 1 = 2.
  2. Multiply the denominators together: 3 × 4 = 12.
  3. The result is 2⁄12. Simplify if possible (HCF of 2 and 12 is 2).
  4. 2 ÷ 2 = 1, 12 ÷ 2 = 6.

Answer: 1⁄6

Dividing Fractions (Keep, Change, Flip)

Problem: 1⁄2 ÷ 1⁄4

The "Keep, Change, Flip" method is used for dividing fractions:

  1. KEEP the first fraction the same: 1⁄2.
  2. CHANGE the division sign (÷) to a multiplication sign (×).
  3. FLIP the second fraction (find its reciprocal): 1⁄4 becomes 4⁄1.
  4. Now multiply the fractions: 1⁄2 × 4⁄1 = (1×4)⁄(2×1) = 4⁄2.
  5. Simplify the result: 4⁄2 = 2.

Answer: 2

Finding a Fraction of an Amount

Problem: Find 2⁄3 of £18.

  1. Divide the whole amount by the denominator: £18 ÷ 3 = £6. (This finds 1⁄3).
  2. Multiply this result by the numerator: £6 × 2 = £12.

Answer: £12

Test Your Fraction Skills!

1. Simplify 6⁄9.

Answer: 2⁄3

Explanation: HCF of 6 and 9 is 3. (6÷3=2), (9÷3=3).

2. Calculate 3⁄7 + 2⁄7.

Answer: 5⁄7

Explanation: Add numerators (3+2=5), keep denominator.

3. Calculate 1⁄4 × 2⁄5.

Answer: 1⁄10

Explanation: (1×2)/(4×5) = 2/20. Simplified = 1/10.

4. Find 3⁄4 of 20.

Answer: 15

Explanation: (20 ÷ 4) = 5. Then 5 × 3 = 15.

Exam-Style Fraction Problems

1. Sarah is baking a cake and a batch of cookies. The cake recipe requires 3⁄4 kg of flour. The cookie recipe requires 2⁄5 kg of flour.
a) How much flour does Sarah need in total for both recipes (give your answer as a mixed number)?
b) If Sarah starts with a 2 kg bag of flour, how much flour will she have left after baking both?

Answers:

a) Total flour needed: 1 3⁄20 kg

b) Flour left: 17⁄20 kg

Explanation:

a) To find the total flour, add the fractions: 3⁄4 + 2⁄5.
Find a common denominator. Multiples of 4: 4, 8, 12, 16, 20. Multiples of 5: 5, 10, 15, 20. LCM is 20.
3⁄4 = (3×5)⁄(4×5) = 15⁄20.
2⁄5 = (2×4)⁄(5×4) = 8⁄20.
Total = 15⁄20 + 8⁄20 = 23⁄20 kg.
As a mixed number, 23⁄20 = 1 3⁄20 kg.

b) To find the flour left, subtract the total used from the initial amount:
Initial flour = 2 kg = 40⁄20 kg (since 1 kg = 20⁄20 kg).
Flour left = 40⁄20 - 23⁄20 = (40-23)⁄20 = 17⁄20 kg.

2. John painted 1⁄3 of a fence on Monday. On Tuesday, he painted 1⁄2 of the remaining part of the fence. What fraction of the whole fence is still unpainted after Tuesday?

Answer: 1⁄3 of the fence is still unpainted.

Explanation:

1. Fraction painted on Monday: 1⁄3.

2. Fraction remaining after Monday: 1 - 1⁄3 = 3⁄3 - 1⁄3 = 2⁄3.

3. Fraction painted on Tuesday (of the whole fence): 1⁄2 of the remaining 2⁄3 = 1⁄2 × 2⁄3 = (1×2)⁄(2×3) = 2⁄6 = 1⁄3.

4. Total fraction painted: 1⁄3 (Monday) + 1⁄3 (Tuesday) = 2⁄3.

5. Fraction still unpainted: 1 - 2⁄3 = 1⁄3.

3. A school has 120 students. 2⁄5 of the students play football. Of the students who play football, 1⁄4 also play cricket. How many students play both football and cricket?

Answer: 12 students play both football and cricket.

Explanation:

1. Number of students who play football: 2⁄5 of 120.
First, find 1⁄5 of 120: 120 ÷ 5 = 24 students.
Then, 2⁄5 of 120 = 24 × 2 = 48 students play football.

2. Number of students who play both football and cricket: 1⁄4 of those who play football (which is 48 students).
1⁄4 of 48 = 48 ÷ 4 = 12 students.

4. A baker has 3⁄4 of a kilogram of sugar. He wants to make small bags of sugar, each weighing 1⁄8 of a kilogram. How many small bags of sugar can he make?

Answer: 6 small bags

Explanation: This is a division problem: How many 1⁄8 kg are in 3⁄4 kg? So, we calculate 3⁄4 ÷ 1⁄8.

  1. KEEP the first fraction: 3⁄4.
  2. CHANGE the division (÷) to multiplication (×).
  3. FLIP the second fraction (1⁄8 becomes 8⁄1).
  4. The problem becomes: 3⁄4 × 8⁄1.
  5. Multiply the numerators: 3 × 8 = 24.
  6. Multiply the denominators: 4 × 1 = 4.
  7. The result is 24⁄4.
  8. Simplify: 24 ÷ 4 = 6.

So, the baker can make 6 small bags of sugar.

5. A piece of ribbon is 2 1⁄2 metres long. How many pieces of ribbon, each 1⁄4 metre long, can be cut from it?

Answer: 10 pieces

Explanation: We need to calculate 2 1⁄2 ÷ 1⁄4.

  1. Convert the mixed number to an improper fraction: 2 1⁄2 = (2×2)+1⁄2 = 5⁄2.
  2. The problem is now: 5⁄2 ÷ 1⁄4.
  3. KEEP the first fraction: 5⁄2.
  4. CHANGE the division (÷) to multiplication (×).
  5. FLIP the second fraction (1⁄4 becomes 4⁄1).
  6. The problem becomes: 5⁄2 × 4⁄1.
  7. Multiply the numerators: 5 × 4 = 20.
  8. Multiply the denominators: 2 × 1 = 2.
  9. The result is 20⁄2.
  10. Simplify: 20 ÷ 2 = 10.

So, 10 pieces of ribbon can be cut.

Which Fraction is Bigger?

To compare two fractions and find out which one is larger (or if they are equal), we need to make sure they are talking about the same size of parts. The easiest way to do this is to give them a common denominator.

Steps to Compare Fractions:

  1. Find a Common Denominator:
    • The best common denominator to use is the Lowest Common Multiple (LCM) of the two denominators.
    • The LCM is the smallest number that both original denominators can divide into evenly.
    • For example, to compare 2⁄3 and 3⁄5, the denominators are 3 and 5. The LCM of 3 and 5 is 15.
  2. Convert to Equivalent Fractions:
    • Change each fraction into an equivalent fraction that has the common denominator.
    • To do this, figure out what you multiplied the original denominator by to get the common denominator. Then, multiply the numerator by the same number.
    • Using our example:
      • For 2⁄3: To get from denominator 3 to 15, you multiply by 5 (since 3 × 5 = 15). So, multiply the numerator by 5: 2 × 5 = 10. The equivalent fraction is 10⁄15.
      • For 3⁄5: To get from denominator 5 to 15, you multiply by 3 (since 5 × 3 = 15). So, multiply the numerator by 3: 3 × 3 = 9. The equivalent fraction is 9⁄15.
  3. Compare the Numerators:
    • Now that both fractions have the same denominator, you can compare their numerators.
    • The fraction with the larger numerator is the larger fraction.
    • In our example, we compare 10⁄15 and 9⁄15. Since 10 is greater than 9, 10⁄15 is greater than 9⁄15.
    • Therefore, 2⁄3 is greater than 3⁄5.

Example: Comparing 5⁄6 and 7⁄9

  1. Find LCM of 6 and 9:
    • Multiples of 6: 6, 12, 18, 24...
    • Multiples of 9: 9, 18, 27...
    • LCM is 18.
  2. Convert fractions:
    • 5⁄6: 18 ÷ 6 = 3. So, (5×3)⁄(6×3) = 15⁄18.
    • 7⁄9: 18 ÷ 9 = 2. So, (7×2)⁄(9×2) = 14⁄18.
  3. Compare numerators:
    • Compare 15⁄18 and 14⁄18.
    • Since 15 > 14, then 15⁄18 > 14⁄18.

Answer: 5⁄6 is greater than 7⁄9.

Interactive Comparison Tool

Enter two fractions below to see which one is larger and the steps involved. (Assumes positive fractions for comparison).

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Result: ?

Interactive Fraction Visualizer

Enter a fraction below to see a visual representation using shaded blocks. This can help you understand the part-to-whole relationship, including whole numbers (e.g., 3/1) and improper fractions (e.g., 7/4).

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Interactive Fraction Simplifier

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Interactive: Mixed Number to Improper Fraction

Enter a whole number, numerator, and denominator for a mixed number to convert it to an improper fraction. Detailed workings will be shown.

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Interactive: Fraction of an Amount

Enter a fraction and a whole amount to find the fractional part. Detailed workings will be shown.

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Interactive Fraction Calculator

Enter two fractions and select an operation to calculate the result. Detailed workings will be shown, including conversion to a mixed number if applicable.

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Changing Fractions to Decimals

A fraction represents a division. To change a fraction to a decimal, you simply divide the numerator by the denominator.

Examples:

Interactive Fraction to Decimal Converter

Enter a fraction below to convert it to a decimal and see the division working.

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Decimal Value: ?

Key Fraction Facts