How to add and subtract rational numbers?

When we add or subtract fractions or rational numbers, the key is to look at the bottom numbers or denominators. If they’re the same, we just add or subtract the top numbers or numerators. If they’re different, we first make them the same and then do addition or subtraction.

How to add and subtract rational numbers?

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. Examples of rational numbers include $2$, $-7$, $\frac{1}{2}$, and $-\frac{3}{4}$.

Addition of Rational Numbers

Adding rational numbers is a straightforward process if the denominators of the numbers are the same. If the denominators are different, we first need to find a common denominator.

Case 1: Same Denominator

If the denominators are the same, we simply add the numerators. For example, let’s add $\frac{1}{4}$ and $\frac{3}{4}$:

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

Let us look at one other example

Example: add $\frac{3}{5}$ and $\frac{2}{5}$:

1. Check the denominators. They are the same (5), so we can go ahead and add the numerators.
2. Add the numerators: $3 + 2 = 5$.
3. Write the result over the common denominator: $\frac{5}{5}$.
4. Simplify if possible. $\frac{5}{5}$ simplifies to $1$.

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

Case 2: Different Denominators

If the denominators are different, we first need to find a common denominator. The common denominator is usually the least common multiple (LCM) of the two denominators. For example, let’s add $\frac{1}{4}$ and $\frac{2}{3}$:

The LCM of $4$ and $3$ is $12$. So, we rewrite the fractions with the common denominator:

$$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$$
$$\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$$

Now, we can add the fractions:

$$\frac{3}{12} + \frac{8}{12} = \frac{3+8}{12} = \frac{11}{12}$$

Let us look at one other example

Example: Let’s add $\frac{3}{4}$ and $\frac{2}{3}$:

1. Check the denominators. They are different, so we need to find a common denominator. The least common multiple of 4 and 3 is 12.
2. Rewrite each fraction with the common denominator: $\frac{3}{4} = \frac{9}{12}$ and $\frac{2}{3} = \frac{8}{12}$.
3. Now that the denominators are the same, add the numerators: $9 + 8 = 17$.
4. Write the result over the common denominator: $\frac{17}{12}$.

So, $\frac{3}{4} + \frac{2}{3} = \frac{17}{12}$.

Subtraction of Rational Numbers

The subtraction of rational numbers is similar to addition.

Case 1: Same Denominator

If the denominators are the same, we simply subtract the numerators. For example, let’s subtract $\frac{3}{4}$ from $\frac{7}{4}$:

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

Let us look at one other example

Example: Let’s subtract $\frac{2}{7}$ from $\frac{5}{7}$:

1. Check the denominators. They are the same (7), so we can go ahead and subtract the numerators.
2. Subtract the numerators: $5 – 2 = 3$.
3. Write the result over the common denominator: $\frac{3}{7}$.

So, $\frac{5}{7} – \frac{2}{7} = \frac{3}{7}$.

Case 2: Different Denominators

If the denominators are different, we first need to find a common denominator. For example, let’s subtract $\frac{2}{3}$ from $\frac{1}{4}$:

The LCM of $4$ and $3$ is $12$. So, we rewrite the fractions with the common denominator:

$$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$$
$$\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$$

Now, we can subtract the fractions:

$$\frac{3}{12} – \frac{8}{12} = \frac{3-8}{12} = \frac{-5}{12}$$

The result is a negative rational number, $-\frac{5}{12}$.

Let us look at one other example

Example: Let’s subtract $\frac{1}{6}$ from $\frac{3}{4}$:

1. Check the denominators. They are different, so we need to find a common denominator. The least common multiple of 6 and 4 is 12.
2. Rewrite each fraction with the common denominator: $\frac{3}{4} = \frac{9}{12}$ and $\frac{1}{6} = \frac{2}{12}$.
3. Now that the denominators are the same, subtract the numerators: $9 – 2 = 7$.
4. Write the result over the common denominator: $\frac{7}{12}$.

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

Practice Problems

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

Answers to these questions

1. $\frac{3}{4}$
2. $\frac{1}{2}$
3. $\frac{13}{8}$ or $1\frac{5}{8}$
4. $\frac{3}{10}$
5. $\frac{5}{7}$
6. $-\frac{2}{9}$
7. $\frac{7}{12}$
8. $\frac{3}{5}$
9. $1$
10. $\frac{1}{3}$

Further Reading

For more practice and to deepen your understanding, you can read about more rational numbers and try out the exercises provided.

Remember, practice is key when it comes to mastering mathematical concepts. Happy learning!