Addition and Subtraction of Fractions and Word Problems for JSS 1

Addition and Subtraction of Fractions

Performance Objectives

By the end of the lesson, students should be able to:

1. solve given problems on addition and subtraction of fractions;
2. solve word-problems involving addition and subtraction of fractions.

Concept of Fractions

A fraction is a part of a whole number or a quantity.

Fractions are written in the form $\frac{a}{b}$, where $a$ is the numerator (the top number representing the parts being considered) and $b$ is the denominator (the bottom number representing the total number of equal parts the whole is divided into). The denominator must not be zero.

The three main types of fractions are:

1. Proper fractions: Fractions where the numerator is smaller than the denominator, such as $\frac{2}{3}$, $\frac{5}{8}$ and $\frac{11}{12}$.
2. Improper fractions: Fractions where the numerator is equal to or larger than the denominator, such as $\frac{7}{4}$, $\frac{15}{8}$ and $\frac{9}{2}$.
3. Mixed numbers: Numbers that consist of a whole number and a proper fraction, such as $2\frac{1}{2}$, $5\frac{3}{4}$ and $1\frac{2}{3}$.

Addition and Subtraction of Fractions with Like Denominators

Fractions with like denominators have the same bottom number. To add or subtract these fractions, the numerators are added or subtracted while the denominator remains unchanged.

The general rules are:

$$ \frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} $$

$$ \frac{a}{c} – \frac{b}{c} = \frac{a – b}{c} $$

Worked Examples

Example 1: Simplify $\frac{3}{8} + \frac{2}{8}$

Solution:

$$ \frac{3}{8} + \frac{2}{8} = \frac{3 + 2}{8} = \frac{5}{8} $$

Example 2: Simplify $\frac{7}{10} – \frac{3}{10}$

Solution:

$$ \frac{7}{10} – \frac{3}{10} = \frac{7 – 3}{10} = \frac{4}{10} $$

The fraction $\frac{4}{10}$ can be simplified to its lowest terms by dividing both the numerator and the denominator by their highest common factor, which is 2:

$$ \frac{4 \div 2}{10 \div 2} = \frac{2}{5} $$

Addition and Subtraction of Fractions with Unlike Denominators

Fractions with unlike denominators have different bottom numbers. To add or subtract these fractions, they must first be converted into equivalent fractions with a common denominator. This common denominator is the Lowest Common Multiple (LCM) of the original denominators.

The procedure involves the following steps:

1. Find the LCM of the denominators.
2. Convert each fraction into an equivalent fraction with the LCM as the new denominator.
3. Add or subtract the numerators.
4. Simplify the resulting fraction to its lowest terms if necessary.

Worked Examples

Example 1: Find the sum of $\frac{1}{4}$ and $\frac{2}{5}$

Solution:

The denominators are 4 and 5. The LCM of 4 and 5 is 20.

Convert each fraction to have a denominator of 20:

$$ \frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20} $$

$$ \frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20} $$

Now, add the numerators:

$$ \frac{5}{20} + \frac{8}{20} = \frac{5 + 8}{20} = \frac{13}{20} $$

Example 2: Simplify $\frac{5}{6} – \frac{1}{3}$

Solution:

The denominators are 6 and 3. The LCM of 6 and 3 is 6.

Convert $\frac{1}{3}$ to an equivalent fraction with a denominator of 6:

$$ \frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6} $$

Now, subtract the numerators:

$$ \frac{5}{6} – \frac{2}{6} = \frac{5 – 2}{6} = \frac{3}{6} $$

Simplify the result to its lowest terms:

$$ \frac{3 \div 3}{6 \div 3} = \frac{1}{2} $$

Example 3: Simplify $\frac{3}{4} + \frac{1}{6} – \frac{1}{3}$

Solution:

The denominators are 4, 6 and 3. The LCM of 4, 6 and 3 is 12.

Convert each fraction to have a denominator of 12:

$$ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} $$

$$ \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12} $$

$$ \frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12} $$

Perform the operations from left to right:

$$ \frac{9}{12} + \frac{2}{12} – \frac{4}{12} = \frac{9 + 2 – 4}{12} = \frac{7}{12} $$

Observe the diagram below, which illustrates the addition of fractions with unlike denominators by converting them to a common denominator.

Addition and Subtraction of Mixed Numbers

Mixed numbers can be added or subtracted using two main methods.

The two methods are:

1. Method 1: Convert the mixed numbers to improper fractions, find the common denominator, perform the operation, and convert the final improper fraction back to a mixed number.
2. Method 2: Group the whole numbers and the fractions separately, perform the operations on each group, and combine the results.

Worked Examples

Example 1: Simplify $2\frac{1}{3} + 1\frac{2}{5}$

Solution using Method 1 (Improper Fractions):

Convert the mixed numbers to improper fractions:

$$ 2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{7}{3} $$

$$ 1\frac{2}{5} = \frac{(1 \times 5) + 2}{5} = \frac{7}{5} $$

The LCM of 3 and 5 is 15. Convert and add:

$$ \frac{7}{3} + \frac{7}{5} = \frac{7 \times 5}{15} + \frac{7 \times 3}{15} = \frac{35}{15} + \frac{21}{15} = \frac{56}{15} $$

Convert the improper fraction back to a mixed number:

$$ \frac{56}{15} = 3\frac{11}{15} $$

Solution using Method 2 (Separate Whole Numbers and Fractions):

Group the whole numbers and the fractions:

$$ 2\frac{1}{3} + 1\frac{2}{5} = (2 + 1) + \left(\frac{1}{3} + \frac{2}{5}\right) $$

Add the whole numbers:

$$ 2 + 1 = 3 $$

Add the fractions (LCM of 3 and 5 is 15):

$$ \frac{1}{3} + \frac{2}{5} = \frac{5}{15} + \frac{6}{15} = \frac{11}{15} $$

Combine the whole number and the fraction:

$$ 3 + \frac{11}{15} = 3\frac{11}{15} $$

Example 2: Simplify $4\frac{3}{4} – 2\frac{1}{2}$

Solution using Method 2:

Group the whole numbers and the fractions:

$$ 4\frac{3}{4} – 2\frac{1}{2} = (4 – 2) + \left(\frac{3}{4} – \frac{1}{2}\right) $$

Subtract the whole numbers:

$$ 4 – 2 = 2 $$

Subtract the fractions (LCM of 4 and 2 is 4):

$$ \frac{3}{4} – \frac{1}{2} = \frac{3}{4} – \frac{2}{4} = \frac{1}{4} $$

Combine the results:

$$ 2 + \frac{1}{4} = 2\frac{1}{4} $$

Example 3: Simplify $3\frac{1}{4} – 1\frac{5}{6}$

Solution using Method 1 (Improper Fractions):

Convert to improper fractions:

$$ 3\frac{1}{4} = \frac{13}{4} $$

$$ 1\frac{5}{6} = \frac{11}{6} $$

The LCM of 4 and 6 is 12. Convert and subtract:

$$ \frac{13}{4} – \frac{11}{6} = \frac{13 \times 3}{12} – \frac{11 \times 2}{12} = \frac{39}{12} – \frac{22}{12} = \frac{17}{12} $$

Convert back to a mixed number:

$$ \frac{17}{12} = 1\frac{5}{12} $$

Word Problems on Addition and Subtraction of Fractions

Word problems require translating real-world situations into mathematical expressions. Key terms help identify which operation to perform.

Common key terms include:

1. Addition terms: sum, total, altogether, combined, increase, in all.
2. Subtraction terms: difference, left, remaining, decrease, how much more, spent, lost.

Worked Examples

Example 1: A farmer plants cassava on $\frac{1}{3}$ of his farmland and maize on $\frac{2}{5}$ of the land. What fraction of the land is used for both crops?

Solution:

The term “both crops” indicates addition.

$$ \text{Fraction of land used} = \frac{1}{3} + \frac{2}{5} $$

The LCM of 3 and 5 is 15.

$$ \frac{1}{3} = \frac{5}{15} $$

$$ \frac{2}{5} = \frac{6}{15} $$

$$ \frac{5}{15} + \frac{6}{15} = \frac{11}{15} $$

The fraction of the land used for both crops is $\frac{11}{15}$.

Example 2: A container holds $5\frac{1}{2}$ litres of water. If $2\frac{3}{4}$ litres of water are poured out, how many litres of water remain in the container?

Solution:

The term “remain” indicates subtraction.

$$ \text{Water remaining} = 5\frac{1}{2} – 2\frac{3}{4} $$

Convert the mixed numbers to improper fractions:

$$ 5\frac{1}{2} = \frac{11}{2} $$

$$ 2\frac{3}{4} = \frac{11}{4} $$

The LCM of 2 and 4 is 4. Convert and subtract:

$$ \frac{11}{2} – \frac{11}{4} = \frac{22}{4} – \frac{11}{4} = \frac{11}{4} $$

Convert the improper fraction back to a mixed number:

$$ \frac{11}{4} = 2\frac{3}{4} \text{ litres} $$

The volume of water remaining in the container is $2\frac{3}{4}$ litres.

Example 3: A student spends $\frac{1}{4}$ of her pocket money on books and $\frac{1}{2}$ on food. What fraction of her pocket money is left?

Solution:

First, find the total fraction spent by adding the two parts:

$$ \text{Total spent} = \frac{1}{4} + \frac{1}{2} $$

The LCM of 4 and 2 is 4.

$$ \frac{1}{4} + \frac{2}{4} = \frac{3}{4} $$

The entire pocket money is represented as 1 whole. Subtract the total spent from 1 to find the fraction left:

$$ \text{Fraction left} = 1 – \frac{3}{4} = \frac{4}{4} – \frac{3}{4} = \frac{1}{4} $$

The fraction of her pocket money left is $\frac{1}{4}$.

Look at the diagram below, which represents the water tank word problem visually.

Practice Exercises

Solve the following problems on addition and subtraction of fractions:

1. $\frac{5}{12} + \frac{3}{12}$
2. $\frac{9}{14} – \frac{2}{14}$
3. $\frac{2}{3} + \frac{1}{6}$
4. $\frac{7}{8} – \frac{3}{4}$
5. $3\frac{2}{5} + 1\frac{1}{10}$
6. $5\frac{1}{3} – 2\frac{2}{3}$
7. $\frac{1}{2} + \frac{1}{3} – \frac{1}{4}$
8. $4\frac{3}{5} – 1\frac{1}{2}$
9. $2\frac{3}{4} + 3\frac{5}{6}$
10. $6 – 2\frac{3}{7}$

Solve the following word problems:

1. A tailor uses $2\frac{1}{4}$ metres of lace for a blouse and $3\frac{1}{2}$ metres of the same lace for a skirt. What is the total length of lace used?
2. A school bag contains books weighing $3\frac{1}{5}$ kilograms and a lunchbox weighing $\frac{4}{5}$ kilograms. What is the total weight of the bag’s contents?
3. A piece of wood is $8\frac{2}{3}$ metres long. If a piece measuring $3\frac{1}{4}$ metres is cut off, what is the length of the remaining wood?
4. A boy walked $\frac{3}{10}$ of a journey and ran $\frac{2}{5}$ of it. What fraction of the journey has he covered altogether?
5. A tin of paint was $\frac{5}{6}$ full. After painting a room, it was $\frac{1}{4}$ full. What fraction of the paint was used?