Subtracting Different types of Rational Numbers Subtracting Different types of Rational Numbers (Definition, Types and Examples) - BYJUS

# Subtracting Different types of Rational Numbers

Rational numbers are a set of numbers that include natural numbers, whole numbers, integers, and certain fractions that follow a specific rule. We can perform basic operations on rational numbers just like we have learned with natural numbers and whole numbers. Learn some properties and math models that will help you subtract rational numbers easily....Read MoreRead Less

## Rational number

In mathematics, a rational number is a number that can be written as  $$\frac{a}{b}$$, where a and b are integers and b ≠ 0. A rational number can be represented as a fraction with non-zero denominators.

$$\frac{2}{5}$$ ,  $$\frac{1}{3}$$ ,  $$\frac{2}{3}$$  are a few examples of rational numbers. These numbers can also be expressed in decimal form.

## Properties of the rational numbers

Since a rational number is a subset of real numbers, it will obey all of the properties of the real number system. The following are some of the most important properties of rational numbers:

• If we multiply, add, or subtract any two rational numbers, the result is always a rational number.
• If we divide or multiply the numerator and denominator with the same factor, the rational number remains the same.

## How to subtract rational numbers?

There are two methods to subtract rational numbers, which are using the number line and using absolute values.

## Subtraction of rational numbers using the number line

Distance between the two rational numbers after plotting the rational numbers on a number line gives the result of subtraction.

Firstly we have to plot the values on the number line and then we have to count on the difference between higher and lower rational numbers. Then that will give you the result of subtraction.

The absolute value of the difference between any two numbers on a number line is the distance between them.

$$\left|a~-~b \right|$$ = $$\left|b~-~a \right|$$

Let’s understand this by using an example.

For example,  $$\frac{1}{4}~-~\left(~-~\frac{3}{4}\right)$$ = $$\frac{1}{4}$$ + $$\frac{3}{4}$$

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

= $$\frac{4}{4}$$ =  1

Solved Example:

• Find the distance between  $$\frac{-1}{2}$$  and -2 on a number line.

Solution: Find the absolute value of the difference of the numbers to determine the distance.

$$\left|-~2~-~(\frac{-1}{2}) \right|~=~\left|-~2~+~\frac{1}{2} \right|$$          ( Adding the opposite of $$\frac{-1}{2}$$ )

=  $$\left|-~1~\frac{1}{2} \right|$$                ( Adding – 2 + $$\frac{1}{2}$$ )

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

So, the distance between  $$\frac {-1}{2}$$  and -2 is  $$1\frac {1}{2}$$.

## Subtraction of rational numbers using the absolute values

Absolute value is the distance between 0 to the number on the number line. In other words, it is a number’s magnitude or size that is calculated using a number line. The absolute value (or modulus)  $$\left|a\right|$$  of a real number “a” is its non-negative value, regardless of its sign.

We have to add the opposite of a rational number to subtract a rational number.

To subtract rational numbers with different signs, subtract the lesser absolute value from the greater absolute value.

Now, let’s find the subtraction between two rational numbers  $$\frac{1}{3}$$  and  $$-\frac{2}{3}$$  using absolute values. Let’s estimate 0.4 – ( – 0.6 ) = 1

By adding the opposite, rewrite the difference as a sum.

$$\frac{1}{3}$$ – ($$~-\frac{2}{3}$$) = $$\frac{1}{3}$$ + ( $$\frac{2}{3}$$ )

Because the signs are the same, add  $$\left|\frac{1}{3}\right|$$  and  $$\left|~-\frac{2}{3}\right|$$.

$$\left|\frac{1}{3}\right|~+~\left|~-\frac{2}{3}\right|~=~\frac{1}{3}~+~\frac{2}{3}$$         (Finding the absolute values)

=  $$\frac{1}{3}~+~\frac{2}{3}$$     ( Writing)

= $$\frac{3}{3}$$              ( Adding the fractions and simplifying )

= 1.

Because  $$-\frac{2}{3}$$  has a negative sign, use a negative sign in the difference.

So, $$\frac{1}{3}~-~(-\frac{2}{3})$$  =  1.

Check: Estimated value  = solution (   1  =   1 )

Solved Examples:

• Find – 3 $$\frac{3}{4} ~-~ \frac{1}{4}$$  = ?

Solution: Let’s estimate – 3 – 1 = – 4

By adding the opposite, rewrite the difference as a sum.

– 3 $$\frac{3}{4} ~-~ \frac{1}{4}$$ = – 3 $$\frac{3}{4} ~+~ ( -\frac{1}{4} )$$

Because the signs are the same, add $$\left|-3~\frac{3}{4}\right|$$ and $$\left|-~\frac{1}{4}\right|$$.

$$\left|-3~\frac{3}{4}\right|$$ + $$\left|-~\frac{1}{4}\right|$$ = $$~3~\frac{3}{4}$$ + $$\frac{1}{4}$$                    (Finding the absolute values)

= 3 + $$\frac{3}{4}$$ + $$\frac{1}{4}$$            ( Writing  3 $$\frac{3}{4}$$ as 3 + $$\frac{3}{4}$$  )

= 3 + $$\frac{4}{4}$$                   ( Adding the fractions and simplifying )

= 3 + 1 = 4.

Because – 3 $$\frac{3}{4}$$  and – $$\frac{1}{4}$$  both are negative, use a negative sign in the difference.

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

Check: Estimated value = solution ( – 4 = – 4 )

## Subtracting the rational numbers in decimal form

Follow the steps given below to subtract the rational numbers in decimal form.

Step 1: We need to add the opposite of a number to be subtracted which means we have to rewrite the difference as the sum.

Step 2: Find the absolute values of both the rational numbers in the form of decimals before subtracting.

Step 3: After subtracting the higher decimal number with the lower decimal number we have to decide the positive sign or negative sign to put with the result number.

For example, To find the subtraction between two rational numbers 2.3 and 5.4 in decimal form.

By adding the opposite, rewrite the difference as a sum.

2.3 – 5.4 = 2.3 + (-5.4)

Because the signs are different and [late]\left|-5.4 \right|[/latex] > $$\left|2.3 \right|$$, subtract $$\left|2.3 \right|$$ from $$\left|-5.4 \right|$$.

$$\left|-5.4 \right|$$ – $$\left|2.3 \right|$$ = 5.4 – 2.3    (Finding the absolute values)

= 3.1             (Subtracted)

Because  $$\left|-5.4 \right|$$ > $$\left|2.3 \right|$$, use the sign of  -5.4.

So, 2.3 – 5.4 = -3.1.

Solved examples:

1)  Find  2.1 – 3.9 = ?

Solution: By adding the opposite, rewrite the difference as a sum.

2.1 – 3.9 = 2.1+ ( -3.9 )

Because the signs are different and $$\left|3.9 \right|$$ > $$\left|2.1 \right|$$, subtract $$\left|2.1 \right|$$ from $$\left|-3.9 \right|$$.

$$\left|-3.19\right|$$ – $$\left|2.1 \right|$$ = 3.9 – 2.1      (Finding the absolute values)

= 1.8                         (Subtracted)

Because $$\left|-3.9 \right|$$  >  $$\left|2.1 \right|$$, use the sign of -3.9.

So, 2.3 – 3.9= -1.8.

2) Find 0.05 – 0.45 = ?

Solution: By adding the opposite, rewrite the difference as a sum.

0.05 – 0.45 = 0.05 + (-0.45)

Because the signs are different and  $$\left|-0.45 \right|$$  >  $$\left|-0.05 \right|$$, subtract $$\left|-0.05 \right|$$  from $$\left|-0.45 \right|$$.

$$\left|-0.45 \right|$$ – $$\left|-0.05 \right|$$ = 0.45 – 0.05              (Finding the absolute values)

= 0.4                                (Subtracted)

Because  $$\left|-0.45 \right|$$ > $$\left|-0.05 \right|$$, use the sign of -0.45.

So, 0.05 – 0.45 = -0.4.

## Subtracting the rational numbers using properties of addition

In the fields of mathematics and statistics, addition is a very important and common operation. The addition operation is denoted by the plus ( + ) sign. Addends are the numbers that need to be added together. The sum is the value that results from this summation step. You can add and sum any digit with any number of units.

We use the addition property while subtracting rational numbers to rewrite the given numbers as a sum of terms to subtract easily.

For example, To evaluate $$-2~\frac{3}{8}$$ $$-7~\frac{1}{2}$$ – ( $$-7~\frac{7}{8}$$ )

To group mixed numbers that include fractions with the same denominator, use addition properties.

$$-2~\frac{3}{8}$$ – $$7~\frac{7}{8}$$ – (- $$-7~\frac{7}{8}$$ ) = $$-2~\frac{3}{8}$$ + ($$-7~\frac{1}{2}$$ ) + $$7~\frac{7}{8}$$              (Rewriting as a sum of terms)

= $$-2~\frac{3}{8}$$ + $$-7~\frac{7}{8}$$  + ( $$-7~\frac{1}{2}$$)             (Common properties of addition)

=  $$\frac{54}{8}$$ + ($$-7 \frac{1}{2}$$ )                          ( Add  $$-2\frac{3}{8}$$  and $$7\frac{7}{8}$$)

=  $$\frac{54}{8}$$ – $$-7\frac{1}{2}$$

= + 3.25

So, $$-2\frac{3}{8}$$  $$-7\frac{1}{2}$$ – ($$-7\frac{7}{8}$$) = + 3.25 .

Solved example:

• Evaluate $$-3\frac{3}{8}$$  $$-6\frac{1}{2}$$  –  ($$-7\frac{7}{8}$$)

Solution: To group mixed numbers that include fractions with the same denominator, use addition properties.

-$$3\frac{3}{8}$$ – $$6\frac{1}{2}$$ – ($$-7\frac{7}{8}$$) = $$-3\frac{3}{8}$$  + ($$-6\frac{1}{2}$$) + $$7\frac{7}{8}$$           (Rewriting as a sum of terms)

= $$-3\frac{3}{8}$$ + $$7\frac{7}{8}$$ + ($$-6\frac{3}{2}$$)         (Common properties of addition)

=  $$\frac{42}{8}$$ + ($$-6\frac{3}{2}$$)                     ( Add $$-2\frac{3}{8}$$  and  $$7\frac{7}{8}$$)

=  $$\frac{42}{8}$$ – $$6\frac{1}{2}$$

= + 2.25

So, $$-3\frac{3}{8}$$ –  $$6\frac{1}{2}$$ – ( $$-7\frac{7}{8}$$ ) = + 2.25.