Multiplication and Division of Fractions and Word Problems for JSS 1

Multiplication and Division of Fractions

Performance Objectives

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

1. solve problems on multiplication of fractions;
2. solve problems on division of fractions;
3. solve word-problems involving multiplication and division of fractions.

Meaning of a Fraction

A fraction is a mathematical expression that represents a part of a whole or a ratio between two numbers.

Fractions consist of a numerator on top and a denominator at the bottom, separated by a horizontal division bar. In the fraction $\frac{a}{b}$, $a$ is the numerator (the number of parts being considered) and $b$ is the denominator (the total number of equal parts that make up the whole). Denominators cannot be zero.

Multiplication of Fractions

To multiply fractions, the numerators are multiplied together to form the new numerator, and the denominators are multiplied together to form the new denominator. The resulting fraction is then simplified to its lowest terms if necessary.

The general rule for multiplying fractions is expressed as:

$$ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} $$

Imagine a rectangular farm divided into equal sections to visualize this operation. If a farmer plants maize on $\frac{1}{2}$ of the farm, and then decides to apply fertilizer to $\frac{1}{3}$ of that maize section, the portion of the entire farm that receives fertilizer is $\frac{1}{3}$ of $\frac{1}{2}$, which is $\frac{1}{6}$.

Multiplying a Fraction by a Whole Number

When multiplying a fraction by a whole number, the whole number is written as a fraction with a denominator of $1$. The standard multiplication rule is then applied.

Worked Example 1: Multiply $\frac{3}{4}$ by $5$.

Solution:

First, write the whole number $5$ as a fraction:

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

Next, multiply the numerators and the denominators:

$$ \frac{3}{4} \times \frac{5}{1} = \frac{3 \times 5}{4 \times 1} $$
$$ = \frac{15}{4} $$

Convert the improper fraction to a mixed number:

$$ \frac{15}{4} = 3\frac{3}{4} $$

Worked Example 2: Find the product of $8$ and $\frac{2}{3}$.

Solution:

Write the whole number $8$ as a fraction and multiply:

$$ \frac{8}{1} \times \frac{2}{3} = \frac{8 \times 2}{1 \times 3} $$
$$ = \frac{16}{3} $$

Convert the improper fraction to a mixed number:

$$ \frac{16}{3} = 5\frac{1}{3} $$

Worked Example 3: Multiply $\frac{5}{12}$ by $6$.

Solution:

Write the whole number $6$ as a fraction and multiply:

$$ \frac{5}{12} \times \frac{6}{1} = \frac{5 \times 6}{12 \times 1} $$
$$ = \frac{30}{12} $$

Simplify the fraction by dividing both the numerator and denominator by their highest common factor, which is $6$:

$$ \frac{30 \div 6}{12 \div 6} = \frac{5}{2} $$

Convert the improper fraction to a mixed number:

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

Multiplying a Fraction by Another Fraction

When multiplying two proper or improper fractions, multiply the numerators together and the denominators together. Simplifying the fractions before multiplying can make the calculation easier.

Worked Example 1: Multiply $\frac{2}{3}$ by $\frac{4}{5}$.

Solution:

Multiply the numerators and denominators directly:

$$ \frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} $$
$$ = \frac{8}{15} $$

Since $8$ and $15$ share no common factors other than $1$, the fraction is in its simplest form.

Worked Example 2: Find the value of $\frac{3}{8} \times \frac{4}{9}$.

Solution:

Method 1: Multiply first, then simplify.

$$ \frac{3}{8} \times \frac{4}{9} = \frac{3 \times 4}{8 \times 9} $$
$$ = \frac{12}{72} $$

Divide the numerator and denominator by their highest common factor, which is $12$:

$$ \frac{12 \div 12}{72 \div 12} = \frac{1}{6} $$

Method 2: Simplify by cancelling common factors before multiplying.

Divide $3$ in the numerator and $9$ in the denominator by $3$:

$$ \frac{1}{8} \times \frac{4}{3} $$

Divide $4$ in the numerator and $8$ in the denominator by $4$:

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

Now multiply the remaining numbers:

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

Worked Example 3: Multiply $\frac{5}{7}$ by $\frac{14}{15}$.

Solution:

Simplify by cancelling common factors before multiplying. Divide $5$ and $15$ by $5$, and divide $14$ and $7$ by $7$:

$$ \frac{5}{7} \times \frac{14}{15} = \frac{1}{1} \times \frac{2}{3} $$
$$ = \frac{1 \times 2}{1 \times 3} $$
$$ = \frac{2}{3} $$

Multiplying Mixed Numbers

To multiply mixed numbers, they must first be converted into improper fractions. Once converted, the standard multiplication procedure is followed.

Worked Example 1: Multiply $1\frac{1}{2}$ by $2\frac{2}{3}$.

Solution:

First, convert the mixed numbers to improper fractions:

$$ 1\frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{3}{2} $$
$$ 2\frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{8}{3} $$

Next, multiply the improper fractions:

$$ \frac{3}{2} \times \frac{8}{3} $$

Simplify by cancelling the common factor of $3$ from the numerator and denominator, and dividing $8$ by $2$:

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

Worked Example 2: Find the product of $3\frac{3}{4}$ and $1\frac{1}{5}$.

Solution:

Convert the mixed numbers to improper fractions:

$$ 3\frac{3}{4} = \frac{(3 \times 4) + 3}{4} = \frac{15}{4} $$
$$ 1\frac{1}{5} = \frac{(1 \times 5) + 1}{5} = \frac{6}{5} $$

Multiply the improper fractions:

$$ \frac{15}{4} \times \frac{6}{5} $$

Simplify by cancelling common factors. Divide $15$ and $5$ by $5$:

$$ \frac{3}{4} \times \frac{6}{1} $$

Divide $6$ and $4$ by $2$:

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

Multiply the remaining numbers:

$$ \frac{3 \times 3}{2 \times 1} = \frac{9}{2} $$

Convert the improper fraction back to a mixed number:

$$ \frac{9}{2} = 4\frac{1}{2} $$

Division of Fractions

Division of fractions relies on the concept of a reciprocal.

A reciprocal is the multiplicative inverse of a number, obtained by swapping its numerator and denominator.

For any non-zero fraction $\frac{a}{b}$, its reciprocal is $\frac{b}{a}$. For any whole number $x$, its reciprocal is $\frac{1}{x}$ because the whole number can be written as $\frac{x}{1}$.

The rule for dividing fractions is to multiply the first fraction by the reciprocal of the second fraction. This is often described as the “Keep, Change, Flip” method:

1. Keep the first fraction exactly as it is.
2. Change the division sign ($\div$) to a multiplication sign ($\times$).
3. Flip the second fraction upside down to find its reciprocal.

The general rule for dividing fractions is expressed as:

$$ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c} $$

Imagine dividing $2$ whole loaves of bread into slices that are each $\frac{1}{3}$ of a loaf. To find how many slices can be made, divide $2$ by $\frac{1}{3}$. This is equivalent to multiplying $2$ by the reciprocal of $\frac{1}{3}$, which is $3$. The result is $6$ slices.

Dividing a Fraction by a Whole Number

To divide a fraction by a whole number, express the whole number as a fraction with a denominator of $1$, find its reciprocal, and multiply.

Worked Example 1: Divide $\frac{3}{4}$ by $2$.

Solution:

Write the whole number $2$ as a fraction:

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

Apply the division rule by multiplying by the reciprocal of $\frac{2}{1}$:

$$ \frac{3}{4} \div \frac{2}{1} = \frac{3}{4} \times \frac{1}{2} $$
$$ = \frac{3 \times 1}{4 \times 2} $$
$$ = \frac{3}{8} $$

Worked Example 2: Find the value of $\frac{5}{6} \div 10$.

Solution:

Write the divisor $10$ as $\frac{10}{1}$ and multiply by its reciprocal:

$$ \frac{5}{6} \div \frac{10}{1} = \frac{5}{6} \times \frac{1}{10} $$

Simplify by cancelling common factors. Divide $5$ and $10$ by $5$:

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

Dividing a Fraction by Another Fraction

To divide a fraction by another fraction, multiply the first fraction by the reciprocal of the second fraction.

Worked Example 1: Divide $\frac{2}{3}$ by $\frac{5}{6}$.

Solution:

Multiply $\frac{2}{3}$ by the reciprocal of $\frac{5}{6}$:

$$ \frac{2}{3} \div \frac{5}{6} = \frac{2}{3} \times \frac{6}{5} $$

Simplify by dividing $6$ in the numerator and $3$ in the denominator by $3$:

$$ \frac{2}{1} \times \frac{2}{5} = \frac{2 \times 2}{1 \times 5} $$
$$ = \frac{4}{5} $$

Worked Example 2: Find the value of $\frac{4}{7} \div \frac{8}{14}$.

Solution:

Multiply $\frac{4}{7}$ by the reciprocal of $\frac{8}{14}$:

$$ \frac{4}{7} \div \frac{8}{14} = \frac{4}{7} \times \frac{14}{8} $$

Simplify by cancelling common factors. Divide $4$ and $8$ by $4$:

$$ \frac{1}{7} \times \frac{14}{2} $$

Divide $14$ and $7$ by $7$:

$$ \frac{1}{1} \times \frac{2}{2} $$

Since $\frac{2}{2} = 1$:

$$ 1 \times 1 = 1 $$

Dividing Mixed Numbers

To divide mixed numbers, convert them to improper fractions first, then apply the division rule.

Worked Example 1: Divide $2\frac{1}{4}$ by $1\frac{1}{2}$.

Solution:

Convert the mixed numbers to improper fractions:

$$ 2\frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{9}{4} $$
$$ 1\frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{3}{2} $$

Set up the division problem:

$$ \frac{9}{4} \div \frac{3}{2} $$

Multiply by the reciprocal of the divisor:

$$ \frac{9}{4} \times \frac{2}{3} $$

Simplify by cancelling common factors. Divide $9$ and $3$ by $3$, and divide $2$ and $4$ by $2$:

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

Convert the improper fraction back to a mixed number:

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

Worked Example 2: Find the value of $4\frac{2}{5} \div 1\frac{1}{10}$.

Solution:

Convert the mixed numbers to improper fractions:

$$ 4\frac{2}{5} = \frac{(4 \times 5) + 2}{5} = \frac{22}{5} $$
$$ 1\frac{1}{10} = \frac{(1 \times 10) + 1}{10} = \frac{11}{10} $$

Set up the division problem:

$$ \frac{22}{5} \div \frac{11}{10} $$

Multiply by the reciprocal of the divisor:

$$ \frac{22}{5} \times \frac{10}{11} $$

Simplify by cancelling common factors. Divide $22$ and $11$ by $11$, and divide $10$ and $5$ by $5$:

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

Word Problems Involving Fractions

Real-world situations often require the multiplication or division of fractions. Identifying which operation to use depends on the language used in the problem.

Key indicators for operations include:

1. Multiplication: Used when finding a fraction “of” a quantity, scaling a value, or finding a total when a fractional part is repeated.
2. Division: Used when sharing a quantity into equal fractional parts, or finding how many fractional portions can fit into a given total.

Worked Example 1: A farmer has a piece of land measuring $12\frac{1}{2}$ hectares. If the farmer uses $\frac{3}{5}$ of the land to cultivate cassava, how many hectares are used for cassava?

Solution:

The problem asks for a fraction “of” a total quantity, which indicates multiplication.

$$ \text{Area for cassava} = \frac{3}{5} \text{ of } 12\frac{1}{2} \text{ hectares} $$
$$ = \frac{3}{5} \times 12\frac{1}{2} $$

Convert the mixed number to an improper fraction:

$$ 12\frac{1}{2} = \frac{25}{2} $$

Multiply the fractions:

$$ \frac{3}{5} \times \frac{25}{2} $$

Simplify by dividing $25$ and $5$ by $5$:

$$ \frac{3}{1} \times \frac{5}{2} = \frac{15}{2} $$

Convert the improper fraction to a mixed number:

$$ \frac{15}{2} = 7\frac{1}{2} \text{ hectares} $$

The farmer uses $7\frac{1}{2}$ hectares of land to cultivate cassava.

Worked Example 2: A student spends $\frac{2}{3}$ of an hour studying Mathematics every day. How many hours does the student spend studying Mathematics in $6$ days?

Solution:

The student repeats the same fractional activity over a number of days, which indicates multiplication.

$$ \text{Total study time} = \frac{2}{3} \times 6 $$

Write $6$ as a fraction and multiply:

$$ \frac{2}{3} \times \frac{6}{1} = \frac{12}{3} $$
$$ = 4 \text{ hours} $$

The student spends $4$ hours studying Mathematics in $6$ days.

Worked Example 3: A tailor has $10\frac{1}{2}$ metres of Ankara fabric. If each school uniform requires $1\frac{3}{4}$ metres of fabric, how many uniforms can the tailor make from the fabric?

Solution:

The problem asks how many smaller, equal portions of fabric can be cut from a larger total, which indicates division.

$$ \text{Number of uniforms} = 10\frac{1}{2} \div 1\frac{3}{4} $$

Convert both mixed numbers to improper fractions:

$$ 10\frac{1}{2} = \frac{21}{2} $$
$$ 1\frac{3}{4} = \frac{7}{4} $$

Set up the division and multiply by the reciprocal of the divisor:

$$ \frac{21}{2} \div \frac{7}{4} = \frac{21}{2} \times \frac{4}{7} $$

Simplify by cancelling common factors. Divide $21$ and $7$ by $7$, and divide $4$ and $2$ by $2$:

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

The tailor can make $6$ school uniforms from the fabric.

Worked Example 4: A container holds $5\frac{1}{4}$ litres of palm oil. If the oil is shared equally into small bottles that hold $\frac{3}{8}$ of a litre each, how many bottles can be filled completely?

Solution:

The total volume of oil is being shared into equal smaller volumes, which indicates division.

$$ \text{Number of bottles} = 5\frac{1}{4} \div \frac{3}{8} $$

Convert the mixed number to an improper fraction:

$$ 5\frac{1}{4} = \frac{21}{4} $$

Multiply by the reciprocal of the divisor:

$$ \frac{21}{4} \times \frac{8}{3} $$

Simplify by cancelling common factors. Divide $21$ and $3$ by $3$, and divide $8$ and $4$ by $4$:

$$ \frac{7}{1} \times \frac{2}{1} = \frac{14}{1} $$
$$ = 14 \text{ bottles} $$

The palm oil can fill $14$ small bottles completely.

Evaluation Exercises

Solve the following multiplication and division problems:

1. $\frac{2}{5} \times 15$
2. $\frac{4}{9} \times \frac{3}{8}$
3. $2\frac{1}{2} \times 1\frac{3}{5}$
4. $\frac{7}{10} \times \frac{5}{14}$
5. $3\frac{3}{4} \times 8$
6. $\frac{5}{8} \div 5$
7. $\frac{3}{5} \div \frac{9}{10}$
8. $4\frac{1}{2} \div 1\frac{1}{2}$
9. $\frac{8}{15} \div \frac{4}{5}$
10. $5\frac{1}{3} \div 2$

Solve the following word problems:

1. A bag of rice weighs $50$ kilograms. If a family consumes $\frac{2}{5}$ of the bag in one month, how many kilograms of rice did they consume?
2. A carpenter needs pieces of wood that are each $1\frac{1}{4}$ metres long. How many of these pieces can be cut from a single plank of wood that is $10$ metres long?
3. A school bus tank holds $60$ litres of fuel. If the bus uses $\frac{3}{8}$ of a tank of fuel for a field trip, how many litres of fuel were used?
4. A trader shares $7\frac{1}{2}$ kilograms of groundnuts equally into $5$ small bags. What is the mass of groundnuts in each bag?
5. A water tap fills a bucket at a rate of $2\frac{1}{3}$ litres per minute. How many litres of water will be in the bucket after $4\frac{1}{2}$ minutes?