# Subtracting rational numbers

In this chapter we will learn to subtract two of more rational numbers with examples.

To understand the chapter you should have basic understanding of the concept of rational number. Click the red link to learn in detail.

## How to subtract rational numbers ?

In this chapter will discuss three different cases;

(a) Subtraction of rational number with same denominator

(b) Subtraction of decimal number with different denominator

(c) Decimal method of subtraction

We will discuss each of the cases in detail.

### Subtraction of rational number with same denominator

As the denominator is same, we simply subtract the numerator and come up with final solution.

Let a/b and c/b are the two rational numbers.

The subtraction is given as;

\mathtt{\Longrightarrow \ \frac{a}{b} \ -\ \frac{c}{b} \ }\\\ \\ \mathtt{\Longrightarrow \frac{a-c}{b}}

The method is simple & straight forward and requires basic subtraction skills.

Let us see some solved examples;

Example 01
Subtract \mathtt{\frac{10}{3} \ -\ \frac{7}{3}}

Solution
Note that both the rational numbers have same denominator. So we will simply subtract the numerator and retain the denominator.

\mathtt{\Longrightarrow \ \frac{10}{3} \ -\ \frac{7}{3} \ }\\\ \\ \mathtt{\Longrightarrow \frac{10-7}{3} \ }\\\ \\ \mathtt{\Longrightarrow \frac{3}{3}}

Hence, 3/3 is the solution.

We can further reduce the fraction by dividing numerator and denominator by 3.

\mathtt{\Longrightarrow \frac{3\div 3}{3\div 3}}\\\ \\ \mathtt{\Longrightarrow \ 1}

Hence, 1 is the solution.

Example 02
Subtract \mathtt{\frac{17}{15} \ -\ \frac{4}{15} \ }

Solution
Since the denominator is same, we will simply subtract the numerator.

\mathtt{\Longrightarrow \ \frac{17}{15} \ -\ \frac{4}{15} \ }\\\ \\ \mathtt{\Longrightarrow \frac{17-4}{15} \ }\\\ \\ \mathtt{\Longrightarrow \frac{14}{15}}

Hence, 14/15 is the solution of given subtraction.

### Subtraction of rational number with different denominator

In this case we will try to make all the denominators same by mathematical computation and then do the subtraction.

To do the subtraction, follow the below steps;

(a) Find LCM of denominators

(b) Multiply each rational number to make denominator equal to LCM

(c) Now simply subtract the numerator since we have rational number with same denominator.

I hope you understood the process, let us see some examples for further clarity.

Example 01
Subtract \mathtt{\ \frac{11}{4} \ -\ \frac{5}{6} \ }

Solution
Note that the given rational numbers have different denominators.

To do the subtraction, follow the below steps;

(a) Find LCM of denominator

LCM ( 4, 6 ) = 12

(b) Multiply each rational number to make denominator 12.

(i) Rational number 11/ 4

Multiply numerator and denominator by 3.

\mathtt{\Longrightarrow \frac{11}{4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{11\times 3}{4\times 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{33}{12}}

(ii) Rational number 5 / 6

Multiply numerator and denominator by 2.

\mathtt{\Longrightarrow \frac{5}{6}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5\times 2}{6\times 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10}{12}}

Now we have got rational number with same denominator. We can do the calculation by simply subtracting the numerator.

(c) Subtract the numerator.

\mathtt{\Longrightarrow \ \frac{33}{12} \ -\ \frac{10}{12} \ }\\\ \\ \mathtt{\Longrightarrow \frac{33-10}{12} \ }\\\ \\ \mathtt{\Longrightarrow \frac{23}{12}}

Hence, 23/12 is the final solution.

Example 02
Subtract \mathtt{\frac{10}{7} \ -\ \frac{18}{5} \ }

Solution
To subtract the given rational numbers, follow the below steps.

(a) Find LCM of denominator

LCM (7, 5) = 35

(b) Multiply the rational numbers to make denominator 35.

(i) Fraction 10 / 7

Multiply numerators & denominators by 5.

\mathtt{\Longrightarrow \frac{10}{7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10\times 5}{7\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{50}{35}}

(ii) Rational number 18/5

Multiply numerator and denominator by 7

\mathtt{\Longrightarrow \frac{18}{5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{18\times 7}{5\times 7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{126}{35}}

Hence, we got the rational number with same denominator.

(c) Now subtract the numerator.

\mathtt{\Longrightarrow \ \frac{50}{35} \ -\ \frac{126}{35} \ }\\\ \\ \mathtt{\Longrightarrow \frac{50-126}{35} \ }\\\ \\ \mathtt{\Longrightarrow \frac{-76}{35}}

Hence, -76/35 is the solution.

### Subtracting rational number using decimal method

In this method you find the decimal value of given rational number and then do the direct subtraction.

To use this method effectively, you should have good math division skills.

Given below are some examples for better clarity of concept.

Example 01
Subtract \mathtt{\frac{19}{3} \ -\ \ \frac{16}{5}}

Solution
Find the decimal value of given rational number.

19/3 ⟹ 6.33

16/5 ⟹ 3.2

Now do the subtraction.

\mathtt{\Longrightarrow \ \frac{19}{3} \ -\ \ \frac{16}{5}}\\\ \\ \mathtt{\Longrightarrow \ 6.33\ -3.2}\\\ \\ \mathtt{\Longrightarrow \ 3.13}

Hence, 3.13 is the solution.

Example 02
Subtract \mathtt{\frac{21}{10} \ -\ \ \frac{15}{4}}

Solution
Find the decimal value of each fraction.

21/10 ⟹ 2.1

15/4 ⟹ 3.75

Now do the subtraction.

\mathtt{\Longrightarrow \ \frac{21}{10} \ -\ \ \frac{15}{4}}\\\ \\ \mathtt{\Longrightarrow \ 2.1\ -3.75}\\\ \\ \mathtt{\Longrightarrow \ -1.65}

Hence, -1.65 is the solution