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Fractions - Addition and Subtraction of Fractions

Grade 5ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Like Fractions: These are fractions that have the same denominator, such as 27\frac{2}{7} and 47\frac{4}{7}. Visually, imagine two identical circles both divided into 7 equal slices; because the slices are the same size, you can easily compare or combine them.

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Unlike Fractions: These are fractions with different denominators, such as 12\frac{1}{2} and 13\frac{1}{3}. Visually, a circle cut into 2 large pieces looks different from one cut into 3 smaller pieces. To add or subtract these, they must first be converted into fractions with a common denominator.

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Equivalent Fractions and LCM: To add unlike fractions, we find the Least Common Multiple (LCM) of the denominators to create equivalent fractions. Visually, this is like taking a large 12\frac{1}{2} slice of cake and cutting it into 3 smaller pieces so it becomes 36\frac{3}{6}, which can then be added to 16\frac{1}{6} slices.

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Addition of Fractions: When adding like fractions, we add only the numerators and keep the denominator the same. For example, 15+25=35\frac{1}{5} + \frac{2}{5} = \frac{3}{5}. Visually, if you have 1 slice of a 5-slice pizza and add 2 more slices, you have 3 slices of that same 5-slice pizza.

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Subtraction of Fractions: Similar to addition, when subtracting like fractions, we subtract the numerators and keep the denominator the same. If the fractions are unlike, we must find a common denominator before subtracting the numerators.

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Mixed Fractions: A mixed fraction consists of a whole number and a proper fraction, like 2142 \frac{1}{4}. Visually, this represents 2 complete shapes (like squares) and one-quarter of a third shape. Before adding or subtracting, it is often easier to convert these into improper fractions.

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Simplifying Fractions: After adding or subtracting, the result should be reduced to its simplest form by dividing both the numerator and the denominator by their Highest Common Factor (HCF). For example, 48\frac{4}{8} simplifies to 12\frac{1}{2} because both numbers can be divided by 4.

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Improper Fractions: These are fractions where the numerator is greater than or equal to the denominator, such as 74\frac{7}{4}. Visually, this represents more than one whole unit. We often convert the final answer of an addition problem from an improper fraction back into a mixed number.

📐Formulae

Addition of Like Fractions: ac+bc=a+bc\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}

Subtraction of Like Fractions: ac−bc=a−bc\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}

Converting Mixed Fraction to Improper Fraction: WND=(W×D)+NDW \frac{N}{D} = \frac{(W \times D) + N}{D}

Finding Common Denominator for Unlike Fractions: ab±cd=(a×k1)±(c×k2)LCM(b,d)\frac{a}{b} \pm \frac{c}{d} = \frac{(a \times k_1) \pm (c \times k_2)}{LCM(b, d)}

Simplifying a Fraction: p÷HCF(p,q)q÷HCF(p,q)\frac{p \div HCF(p, q)}{q \div HCF(p, q)}

💡Examples

Problem 1:

Add the unlike fractions: 310+25\frac{3}{10} + \frac{2}{5}

Solution:

  1. Find the LCM of the denominators 10 and 5. The LCM is 10.
  2. Convert 25\frac{2}{5} to an equivalent fraction with denominator 10: 2×25×2=410\frac{2 \times 2}{5 \times 2} = \frac{4}{10}.
  3. Add the numerators of the like fractions: 310+410=3+410=710\frac{3}{10} + \frac{4}{10} = \frac{3+4}{10} = \frac{7}{10}.
  4. Since 7 and 10 have no common factors other than 1, the fraction is already in its simplest form.

Explanation:

To add fractions with different denominators, we first find a common denominator (LCM), convert the fractions so they are 'like', and then add the numerators.

Problem 2:

Subtract the mixed numbers: 312−1343 \frac{1}{2} - 1 \frac{3}{4}

Solution:

  1. Convert the mixed numbers to improper fractions: 312=(3×2)+12=723 \frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{7}{2} 134=(1×4)+34=741 \frac{3}{4} = \frac{(1 \times 4) + 3}{4} = \frac{7}{4}
  2. Find the LCM of 2 and 4, which is 4.
  3. Convert 72\frac{7}{2} to a fraction with denominator 4: 7×22×2=144\frac{7 \times 2}{2 \times 2} = \frac{14}{4}.
  4. Subtract the fractions: 144−74=14−74=74\frac{14}{4} - \frac{7}{4} = \frac{14-7}{4} = \frac{7}{4}.
  5. Convert the improper fraction back to a mixed number: 74=134\frac{7}{4} = 1 \frac{3}{4}.

Explanation:

When dealing with mixed numbers, converting them to improper fractions first makes the subtraction process straightforward. After finding a common denominator and subtracting, the result is converted back to a mixed number for the final answer.