Addition and Subtraction of Fractions and Mixed Numbers for Primary 4

Addition and Subtraction of Fractions and Mixed Numbers

Performance Objectives

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

1. perform addition and subtraction of proper fractions;
2. perform addition and subtraction of improper fractions;
3. carry out addition and subtraction of mixed numbers;
4. carry out addition and subtraction of fractions in everyday life activities;
5. solve quantitative aptitude problems involving addition and subtraction fractions.

Understanding Fractions

A fraction represents a part of a whole. When a whole object or group is divided into equal parts, a fraction shows how many of those parts are being considered.

Every fraction has two main parts:

1. Numerator: This is the top number. It tells how many parts are being counted.
2. Denominator: This is the bottom number. It tells how many equal parts the whole has been divided into.

For example, in the fraction $\frac{3}{4}$, the numerator is 3 and the denominator is 4. This means 3 parts out of 4 equal parts.

Types of Fractions

Fractions can be grouped into different types based on their numerators and denominators. Identifying these types helps in performing operations correctly.

Proper Fractions

A proper fraction is a fraction where the numerator is smaller than the denominator.

Proper fractions represent a value less than one whole.

Examples of proper fractions include:

1. $\frac{1}{2}$
2. $\frac{3}{4}$
3. $\frac{5}{8}$
4. $\frac{2}{3}$

Improper Fractions

An improper fraction is a fraction where the numerator is equal to or greater than the denominator.

Improper fractions represent a value equal to or greater than one whole.

Examples of improper fractions include:

1. $\frac{5}{4}$
2. $\frac{7}{3}$
3. $\frac{8}{8}$
4. $\frac{10}{6}$

Mixed Numbers

A mixed number is a combination of a whole number and a proper fraction.

Mixed numbers represent a value greater than one whole.

Examples of mixed numbers include:

1. $1\frac{1}{2}$
2. $2\frac{3}{4}$
3. $3\frac{1}{5}$
4. $5\frac{2}{3}$

Converting Between Improper Fractions and Mixed Numbers

It is often necessary to convert between improper fractions and mixed numbers when performing addition and subtraction.

Converting Improper Fractions to Mixed Numbers

To convert an improper fraction to a mixed number, divide the numerator by the denominator.

The quotient becomes the whole number part, the remainder becomes the new numerator, and the denominator stays the same.

Example 1: Convert $\frac{7}{3}$ to a mixed number.

Divide 7 by 3:

$$
\begin{array}{r}
2 \\
3 \overline{) 7} \\
-6 \\
\hline
1
\end{array}
$$

The quotient is 2 (whole number), the remainder is 1 (new numerator), and the denominator is 3.

So, $\frac{7}{3} = 2\frac{1}{3}$.

Example 2: Convert $\frac{10}{4}$ to a mixed number.

Divide 10 by 4:

$$
\begin{array}{r}
2 \\
4 \overline{) 10} \\
-8 \\
\hline
2
\end{array}
$$

The quotient is 2, the remainder is 2, and the denominator is 4.

So, $\frac{10}{4} = 2\frac{2}{4}$. This can be simplified to $2\frac{1}{2}$.

Converting Mixed Numbers to Improper Fractions

To convert a mixed number to an improper fraction, multiply the whole number by the denominator, then add the numerator to the result.

The sum becomes the new numerator, and the denominator stays the same.

Example 1: Convert $1\frac{2}{3}$ to an improper fraction.

Multiply the whole number (1) by the denominator (3): $1 \times 3 = 3$.

Add the numerator (2) to the result: $3 + 2 = 5$.

The new numerator is 5, and the denominator is 3.

So, $1\frac{2}{3} = \frac{5}{3}$.

Example 2: Convert $3\frac{1}{4}$ to an improper fraction.

Multiply the whole number (3) by the denominator (4): $3 \times 4 = 12$.

Add the numerator (1) to the result: $12 + 1 = 13$.

The new numerator is 13, and the denominator is 4.

So, $3\frac{1}{4} = \frac{13}{4}$.

Adding and Subtracting Fractions

Adding and subtracting fractions requires careful steps, especially when the denominators are different.

Adding and Subtracting Fractions with the Same Denominator

When fractions have the same denominator, addition and subtraction are straightforward.

To add or subtract fractions with the same denominator:

1. Add or subtract the numerators.
2. Keep the denominator the same.
3. Simplify the answer if possible.

Example 1: Add $\frac{1}{5} + \frac{2}{5}$.

$$
\frac{1}{5} + \frac{2}{5} = \frac{1+2}{5} = \frac{3}{5}
$$

Example 2: Subtract $\frac{4}{7} – \frac{1}{7}$.

$$
\frac{4}{7} – \frac{1}{7} = \frac{4-1}{7} = \frac{3}{7}
$$

Adding and Subtracting Proper Fractions with Different Denominators

When fractions have different denominators, they must first be changed to equivalent fractions with a common denominator.

The common denominator is usually the Least Common Multiple (LCM) of the original denominators.

The procedure involves the following steps:

1. Find the LCM of the denominators. This will be the new common denominator.
2. Convert each fraction to an equivalent fraction with the new common denominator.
3. Add or subtract the new numerators.
4. Keep the common denominator.
5. Simplify the answer if possible.

Example 1: Add $\frac{1}{2} + \frac{1}{3}$.

1. Find the LCM of 2 and 3.

   Multiples of 2: 2, 4, 6, 8, …

   Multiples of 3: 3, 6, 9, 12, …

   The LCM of 2 and 3 is 6.

2. Convert to equivalent fractions with a denominator of 6.
   $$
   \frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}
   $$
   $$
   \frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}
   $$
3. Add the new numerators:
   $$
   \frac{3}{6} + \frac{2}{6} = \frac{3+2}{6} = \frac{5}{6}
   $$

So, $\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$.

Example 2: Subtract $\frac{3}{4} – \frac{1}{6}$.

1. Find the LCM of 4 and 6.

   Multiples of 4: 4, 8, 12, 16, …

   Multiples of 6: 6, 12, 18, …

   The LCM of 4 and 6 is 12.

2. Convert to equivalent fractions with a denominator of 12.
   $$
   \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}
   $$
   $$
   \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}
   $$
3. Subtract the new numerators:
   $$
   \frac{9}{12} – \frac{2}{12} = \frac{9-2}{12} = \frac{7}{12}
   $$

So, $\frac{3}{4} – \frac{1}{6} = \frac{7}{12}$.

Adding and Subtracting Improper Fractions

The process for adding and subtracting improper fractions is similar to proper fractions. If the denominators are different, find the LCM and convert to equivalent fractions.

After adding or subtracting, the result may be an improper fraction. It should be converted to a mixed number and simplified if possible.

Example 1: Add $\frac{5}{4} + \frac{7}{6}$.

1. Find the LCM of 4 and 6. The LCM is 12.
2. Convert to equivalent fractions:
   $$
   \frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{15}{12}
   $$
   $$
   \frac{7}{6} = \frac{7 \times 2}{6 \times 2} = \frac{14}{12}
   $$
3. Add the new numerators:
   $$
   \frac{15}{12} + \frac{14}{12} = \frac{15+14}{12} = \frac{29}{12}
   $$
4. Convert the improper fraction to a mixed number:

   Divide 29 by 12:

   $$
   \begin{array}{r}
   2 \\
   12 \overline{) 29} \\
   -24 \\
   \hline
   5
   \end{array}
   $$

   So, $\frac{29}{12} = 2\frac{5}{12}$.

Therefore, $\frac{5}{4} + \frac{7}{6} = 2\frac{5}{12}$.

Example 2: Subtract $\frac{9}{2} – \frac{7}{3}$.

1. Find the LCM of 2 and 3. The LCM is 6.
2. Convert to equivalent fractions:
   $$
   \frac{9}{2} = \frac{9 \times 3}{2 \times 3} = \frac{27}{6}
   $$
   $$
   \frac{7}{3} = \frac{7 \times 2}{3 \times 2} = \frac{14}{6}
   $$
3. Subtract the new numerators:
   $$
   \frac{27}{6} – \frac{14}{6} = \frac{27-14}{6} = \frac{13}{6}
   $$
4. Convert the improper fraction to a mixed number:

   Divide 13 by 6:

   $$
   \begin{array}{r}
   2 \\
   6 \overline{) 13} \\
   -12 \\
   \hline
   1
   \end{array}
   $$

   So, $\frac{13}{6} = 2\frac{1}{6}$.

Therefore, $\frac{9}{2} – \frac{7}{3} = 2\frac{1}{6}$.

Adding and Subtracting Mixed Numbers

There are two main methods for adding and subtracting mixed numbers.

Method 1: Convert to Improper Fractions

This method involves converting all mixed numbers to improper fractions first, then adding or subtracting them as improper fractions.

The procedure involves the following steps:

1. Convert all mixed numbers to improper fractions.
2. Find the LCM of the denominators and convert to equivalent fractions.
3. Add or subtract the numerators.
4. Convert the final improper fraction back to a mixed number and simplify if needed.

Example 1: Add $1\frac{1}{2} + 2\frac{1}{3}$.

1. Convert to improper fractions:
   $$
   1\frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{3}{2}
   $$
   $$
   2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{7}{3}
   $$
2. Find the LCM of 2 and 3. The LCM is 6.
3. Convert to equivalent fractions:
   $$
   \frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6}
   $$
   $$
   \frac{7}{3} = \frac{7 \times 2}{3 \times 2} = \frac{14}{6}
   $$
4. Add the numerators:
   $$
   \frac{9}{6} + \frac{14}{6} = \frac{9+14}{6} = \frac{23}{6}
   $$
5. Convert to a mixed number:

   Divide 23 by 6:

   $$
   \begin{array}{r}
   3 \\
   6 \overline{) 23} \\
   -18 \\
   \hline
   5
   \end{array}
   $$

   So, $\frac{23}{6} = 3\frac{5}{6}$.

Therefore, $1\frac{1}{2} + 2\frac{1}{3} = 3\frac{5}{6}$.

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

1. Convert to improper fractions:
   $$
   3\frac{1}{4} = \frac{(3 \times 4) + 1}{4} = \frac{13}{4}
   $$
   $$
   1\frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{3}{2}
   $$
2. Find the LCM of 4 and 2. The LCM is 4.
3. Convert to equivalent fractions:
   $$
   \frac{13}{4} \text{ (already has denominator 4)}
   $$
   $$
   \frac{3}{2} = \frac{3 \times 2}{2 \times 2} = \frac{6}{4}
   $$
4. Subtract the numerators:
   $$
   \frac{13}{4} – \frac{6}{4} = \frac{13-6}{4} = \frac{7}{4}
   $$
5. Convert to a mixed number:

   Divide 7 by 4:

   $$
   \begin{array}{r}
   1 \\
   4 \overline{) 7} \\
   -4 \\
   \hline
   3
   \end{array}
   $$

   So, $\frac{7}{4} = 1\frac{3}{4}$.

Therefore, $3\frac{1}{4} – 1\frac{1}{2} = 1\frac{3}{4}$.

Method 2: Add/Subtract Whole Numbers and Fractions Separately

This method involves adding or subtracting the whole number parts and the fractional parts separately.

The procedure for addition involves the following steps:

1. Add the whole numbers.
2. Add the proper fractions. If the fractional sum is an improper fraction, convert it to a mixed number.
3. Add the whole number from step 1 to the whole number from the converted fraction in step 2.
4. Simplify the final fraction if needed.

Example 1: Add $1\frac{1}{2} + 2\frac{1}{3}$.

1. Add the whole numbers: $1 + 2 = 3$.
2. Add the fractions: $\frac{1}{2} + \frac{1}{3}$.

   LCM of 2 and 3 is 6.

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

3. Combine the whole number and the fraction: $3 + \frac{5}{6} = 3\frac{5}{6}$.

Therefore, $1\frac{1}{2} + 2\frac{1}{3} = 3\frac{5}{6}$.

The procedure for subtraction involves the following steps:

1. Subtract the whole numbers.
2. Subtract the proper fractions. If the first fraction is smaller than the second, “borrow” 1 from the whole number part of the first mixed number. Convert the borrowed 1 into a fraction with the same denominator as the fractional part, then add it to the first fraction.
3. Simplify the final fraction if needed.

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

1. Subtract the whole numbers: $3 – 1 = 2$.
2. Subtract the fractions: $\frac{1}{4} – \frac{1}{2}$.

   LCM of 4 and 2 is 4.

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

   Since $\frac{1}{4}$ is smaller than $\frac{2}{4}$, borrowing is needed. Borrow 1 from the whole number 3 in $3\frac{1}{4}$.

   $3\frac{1}{4}$ becomes $2 + 1 + \frac{1}{4} = 2 + \frac{4}{4} + \frac{1}{4} = 2\frac{5}{4}$.

   Now subtract: $2\frac{5}{4} – 1\frac{2}{4}$.

   Subtract whole numbers: $2 – 1 = 1$.

   Subtract fractions: $\frac{5}{4} – \frac{2}{4} = \frac{3}{4}$.

3. Combine the whole number and the fraction: $1\frac{3}{4}$.

Therefore, $3\frac{1}{4} – 1\frac{1}{2} = 1\frac{3}{4}$.

Using Fraction Charts or Counting Blocks

Fraction charts and counting blocks can help visualise fraction operations, especially for proper fractions and simple mixed numbers.

For Addition

To add $\frac{1}{4} + \frac{2}{4}$ using a fraction chart:

1. Take a fraction strip or block representing $\frac{1}{4}$.
2. Take another fraction strip or block representing $\frac{2}{4}$.
3. Place them together. Observe that they combine to form $\frac{3}{4}$ of the whole.

To add $\frac{1}{2} + \frac{1}{4}$ using a fraction chart:

1. Take a fraction strip or block representing $\frac{1}{2}$.
2. Take another fraction strip or block representing $\frac{1}{4}$.
3. To combine them, find equivalent pieces. Replace $\frac{1}{2}$ with two $\frac{1}{4}$ pieces.
4. Now combine the two $\frac{1}{4}$ pieces with the original $\frac{1}{4}$ piece. This makes three $\frac{1}{4}$ pieces, which is $\frac{3}{4}$.

For Subtraction

To subtract $\frac{3}{4} – \frac{1}{4}$ using a fraction chart:

1. Take a fraction strip or block representing $\frac{3}{4}$.
2. Remove one $\frac{1}{4}$ piece from it.
3. Observe that two $\frac{1}{4}$ pieces remain, which is $\frac{2}{4}$ or $\frac{1}{2}$.

Fractions in Everyday Life Activities

Fractions are used in many daily activities, such as cooking, sharing, measuring, and telling time.

Example 1: Cooking

A recipe requires $1\frac{1}{2}$ cups of flour and $\frac{3}{4}$ cup of sugar. How much flour and sugar are needed in total?

To solve this, add the amounts:

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

1. Convert $1\frac{1}{2}$ to an improper fraction: $\frac{3}{2}$.
2. Find the LCM of 2 and 4. The LCM is 4.
3. Convert to equivalent fractions:
   $$
   \frac{3}{2} = \frac{3 \times 2}{2 \times 2} = \frac{6}{4}
   $$
   $$
   \frac{3}{4} \text{ (already has denominator 4)}
   $$
4. Add the fractions:
   $$
   \frac{6}{4} + \frac{3}{4} = \frac{9}{4}
   $$
5. Convert to a mixed number:

   Divide 9 by 4:

   $$
   \begin{array}{r}
   2 \\
   4 \overline{) 9} \\
   -8 \\
   \hline
   1
   \end{array}
   $$

   So, $\frac{9}{4} = 2\frac{1}{4}$.

Therefore, $2\frac{1}{4}$ cups of flour and sugar are needed in total.

Example 2: Sharing a Cake

A cake was cut into 8 equal slices. John ate $\frac{3}{8}$ of the cake, and Mary ate $\frac{2}{8}$ of the cake. What fraction of the cake is left?

1. First, find the total fraction of cake eaten:
   $$
   \frac{3}{8} + \frac{2}{8} = \frac{3+2}{8} = \frac{5}{8}
   $$

   So, $\frac{5}{8}$ of the cake was eaten.

2. The whole cake is represented by $\frac{8}{8}$ or 1. To find the fraction left, subtract the eaten portion from the whole:
   $$
   \frac{8}{8} – \frac{5}{8} = \frac{8-5}{8} = \frac{3}{8}
   $$

Therefore, $\frac{3}{8}$ of the cake is left.

Example 3: Distance

A cyclist rode $5\frac{1}{2}$ km on Monday and $3\frac{1}{4}$ km on Tuesday. How much further did the cyclist ride on Monday than on Tuesday?

To solve this, subtract the distance ridden on Tuesday from the distance ridden on Monday:

$$
5\frac{1}{2} – 3\frac{1}{4}
$$

1. Convert to improper fractions:
   $$
   5\frac{1}{2} = \frac{(5 \times 2) + 1}{2} = \frac{11}{2}
   $$
   $$
   3\frac{1}{4} = \frac{(3 \times 4) + 1}{4} = \frac{13}{4}
   $$
2. Find the LCM of 2 and 4. The LCM is 4.
3. Convert to equivalent fractions:
   $$
   \frac{11}{2} = \frac{11 \times 2}{2 \times 2} = \frac{22}{4}
   $$
   $$
   \frac{13}{4} \text{ (already has denominator 4)}
   $$
4. Subtract the fractions:
   $$
   \frac{22}{4} – \frac{13}{4} = \frac{22-13}{4} = \frac{9}{4}
   $$
5. Convert to a mixed number:

   Divide 9 by 4:

   $$
   \begin{array}{r}
   2 \\
   4 \overline{) 9} \\
   -8 \\
   \hline
   1
   \end{array}
   $$

   So, $\frac{9}{4} = 2\frac{1}{4}$.

Therefore, the cyclist rode $2\frac{1}{4}$ km further on Monday.

Quantitative Reasoning Problems Involving Fractions

Quantitative reasoning problems require applying fraction operations to solve problems, often presented in a way that tests understanding of relationships and logical thinking.

Example 1:

If a bucket is $\frac{1}{3}$ full of water, and then $\frac{1}{6}$ more water is added, what fraction of the bucket is now full?

This problem requires adding the two fractions:

$$
\frac{1}{3} + \frac{1}{6}
$$

1. Find the LCM of 3 and 6. The LCM is 6.
2. Convert to equivalent fractions:
   $$
   \frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}
   $$
   $$
   \frac{1}{6} \text{ (already has denominator 6)}
   $$
3. Add the fractions:
   $$
   \frac{2}{6} + \frac{1}{6} = \frac{3}{6}
   $$
4. Simplify the answer:
   $$
   \frac{3}{6} = \frac{1}{2}
   $$

Therefore, the bucket is now $\frac{1}{2}$ full.

Example 2:

A farmer planted maize on $\frac{3}{5}$ of his land. He later decided to plant yam on $\frac{1}{10}$ of the land that was originally for maize. What fraction of the land is now used for maize only?

This problem requires subtraction. The farmer used $\frac{3}{5}$ for maize, then took $\frac{1}{10}$ of that maize land for yam. So, subtract the yam portion from the original maize portion.

$$
\frac{3}{5} – \frac{1}{10}
$$

1. Find the LCM of 5 and 10. The LCM is 10.
2. Convert to equivalent fractions:
   $$
   \frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10}
   $$
   $$
   \frac{1}{10} \text{ (already has denominator 10)}
   $$
3. Subtract the fractions:
   $$
   \frac{6}{10} – \frac{1}{10} = \frac{5}{10}
   $$
4. Simplify the answer:
   $$
   \frac{5}{10} = \frac{1}{2}
   $$

Therefore, $\frac{1}{2}$ of the land is now used for maize only.