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

Subtracting Fractions (Same Denominator)

Problem: 4⁄7 - 1⁄7

  1. Keep the denominator the same.
  2. Subtract the numerators: 4 - 1 = 3.

Answer: 3⁄7

Subtracting Fractions (Different Denominators)

Problem: 5⁄6 - 1⁄4

  1. Find a common denominator (LCM) for 6 and 4.
    • Multiples of 6: 6, 12, 18, ...
    • Multiples of 4: 4, 8, 12, 16, ...
    • The LCM of 6 and 4 is 12. So, 12 will be our common denominator.
  2. Convert each fraction to an equivalent fraction with the common denominator (12):
    • For 5⁄6: To change the denominator from 6 to 12, we multiply by 2 (6 × 2 = 12). So, multiply the numerator by 2: 5 × 2 = 10. Thus, 5⁄6 = 10⁄12.
    • For 1⁄4: To change the denominator from 4 to 12, we multiply by 3 (4 × 3 = 12). So, multiply the numerator by 3: 1 × 3 = 3. Thus, 1⁄4 = 3⁄12.
  3. Now subtract the new numerators: 10 - 3 = 7. Keep the common denominator.

Answer: 7⁄12

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: 3⁄5 ÷ 2⁄3

  1. Keep the first fraction as it is: 3⁄5.
  2. Change the division sign to a multiplication sign.
  3. Flip the second fraction (find its reciprocal): 2⁄3 becomes 3⁄2.
  4. Now, multiply the fractions: 3⁄5 × 3⁄2.
    • Multiply numerators: 3 × 3 = 9.
    • Multiply denominators: 5 × 2 = 10.

Answer: 9⁄10

Converting Mixed Numbers to Improper Fractions

Problem: Convert 2 1⁄3 to an improper fraction.

  1. Multiply the whole number by the denominator: 2 × 3 = 6.
  2. Add the numerator to this product: 6 + 1 = 7.
  3. Place this sum over the original denominator.

Answer: 7⁄3

Converting Improper Fractions to Mixed Numbers

Problem: Convert 11⁄4 to a mixed number.

  1. Divide the numerator by the denominator: 11 ÷ 4.
  2. The whole number part of the answer is the quotient: 11 ÷ 4 = 2 with a remainder.
  3. The new numerator is the remainder: 11 - (4 × 2) = 11 - 8 = 3.
  4. The denominator remains the same (4).

Answer: 2 3⁄4

Fraction of an Amount

Problem: Find 2⁄5 of 30.

  1. Divide the amount by the denominator: 30 ÷ 5 = 6. (This tells you what 1⁄5 of 30 is).
  2. Multiply the result by the numerator: 6 × 2 = 12.

Answer: 12

Test Your Fraction Skills!

Test your understanding of fractions with these practice questions. Try to solve them on your own before revealing the answers!

1. What is 3⁄4 + 1⁄6?

Working Out:

1. Find a common denominator. The least common multiple (LCM) of 4 and 6 is 12.

  • Multiples of 4: 4, 8, 12, ...
  • Multiples of 6: 6, 12, 18, ...

2. Convert each fraction to an equivalent fraction with the common denominator (12):

  • 3⁄4 = 3 × 3⁄4 × 3 = 9⁄12
  • 1⁄6 = 1 × 2⁄6 × 2 = 2⁄12

3. Add the numerators:

9 + 2 = 11

Answer: 11⁄12

2. Calculate 7⁄8 - 1⁄3.

Working Out:

1. Find a common denominator. The LCM of 8 and 3 is 24.

  • Multiples of 8: 8, 16, 24, ...
  • Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, ...

2. Convert each fraction to an equivalent fraction with the common denominator (24):

  • 7⁄8 = 7 × 3⁄8 × 3 = 21⁄24
  • 1⁄3 = 1 × 8⁄3 × 8 = 8⁄24

3. Subtract the numerators:

21 - 8 = 13

Answer: 13⁄24

3. Evaluate 5⁄9 ÷ 2⁄3.

Working Out:

1. To divide by a fraction, we multiply by its reciprocal (flip the second fraction):

${displayFraction(5,9)} ÷ ${displayFraction(2,3)} = ${displayFraction(5,9)} × ${displayFraction(3,2)}

2. Multiply the numerators and the denominators:

${displayFraction(5*3,9*2)} = ${displayFraction(15,18)}

3. Simplify the resulting fraction by finding the HCF of the numerator (15) and denominator (18), which is 3:

${displayFraction(15,18)} = ${displayFraction(15/3,18/3)} = ${displayFraction(5,6)}

Answer: 5⁄6

4. Find the sum of 2⁄5 + 3⁄10.

Working Out:

1. Find a common denominator. The LCM of 5 and 10 is 10.

2. Convert the first fraction to an equivalent fraction with the common denominator (10):

  • 2⁄5 = 2 × 2⁄5 × 2 = 4⁄10

3. Add the numerators:

4 + 3 = 7

Answer: 7⁄10

5. What is 11⁄12 - 1⁄4?

Working Out:

1. Find a common denominator. The LCM of 12 and 4 is 12.

2. Convert the second fraction to an equivalent fraction with the common denominator (12):

  • 1⁄4 = 1 × 3⁄4 × 3 = 3⁄12

3. Subtract the numerators:

11 - 3 = 8

4. Simplify the resulting fraction by finding the HCF of 8 and 12, which is 4:

${displayFraction(8,12)} = ${displayFraction(8/4,12/4)} = ${displayFraction(2,3)}

Answer: 2⁄3

6. Calculate 4⁄7 ÷ 8⁄14.

Working Out:

1. Multiply by the reciprocal of the second fraction:

${displayFraction(4,7)} ÷ ${displayFraction(8,14)} = ${displayFraction(4,7)} × ${displayFraction(14,8)}

2. Multiply the numerators and the denominators:

${displayFraction(4*14,7*8)} = ${displayFraction(56,56)}

3. Simplify the fraction:

${displayFraction(56,56)} = 1

Answer: 1

7. Add 1⁄2 + 2⁄3.

Working Out:

1. Find a common denominator. The LCM of 2 and 3 is 6.

2. Convert each fraction to an equivalent fraction with the common denominator (6):

  • 1⁄2 = 1 × 3⁄2 × 3 = 3⁄6
  • 2⁄3 = 2 × 2⁄3 × 2 = 4⁄6

3. Add the numerators:

3 + 4 = 7

Answer: 7⁄6 or 1 1⁄6

8. Subtract 5⁄6 - 1⁄9.

Working Out:

1. Find a common denominator. The LCM of 6 and 9 is 18.

  • Multiples of 6: 6, 12, 18, ...
  • Multiples of 9: 9, 18, 27, ...

2. Convert each fraction to an equivalent fraction with the common denominator (18):

  • 5⁄6 = 5 × 3⁄6 × 3 = 15⁄18
  • 1⁄9 = 1 × 2⁄9 × 2 = 2⁄18

3. Subtract the numerators:

15 - 2 = 13

Answer: 13⁄18

9. Divide 9⁄10 by 3⁄5.

Working Out:

1. Multiply by the reciprocal of the second fraction:

${displayFraction(9,10)} ÷ ${displayFraction(3,5)} = ${displayFraction(9,10)} × ${displayFraction(5,3)}

2. Multiply the numerators and the denominators:

${displayFraction(9*5,10*3)} = ${displayFraction(45,30)}

3. Simplify the resulting fraction by finding the HCF of 45 and 30, which is 15:

${displayFraction(45,30)} = ${displayFraction(45/15,30/15)} = ${displayFraction(3,2)}

Answer: 3⁄2 or 1 1⁄2

10. What is 3⁄7 + 1⁄2?

Working Out:

1. Find a common denominator. The LCM of 7 and 2 is 14.

2. Convert each fraction to an equivalent fraction with the common denominator (14):

  • 3⁄7 = 3 × 2⁄7 × 2 = 6⁄14
  • 1⁄2 = 1 × 7⁄2 × 7 = 7⁄14

3. Add the numerators:

6 + 7 = 13

Answer: 13⁄14

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Improper Fraction:

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