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# Subtraction in Rational Numbers

John found the difference of two rational numbers$\frac{5}{7}$ and $\frac{3}{8}$ in this way:

$\frac{5}{7}$ - $\frac{3}{8}$

= $\frac{40-21}{56}$

= $\frac{19}{56}$

David knew that for two integers a and b she could write a – b = a +(-b)

He tried this for rational numbers also and found,

John found the difference of two rational numbers $\frac57$ and $\frac38$ in this way:

$\frac57$ - $\frac38$

= $\frac40-2156$

= $\frac1956$

David knew that for two integers a and b she could write a – b = a +(-b)

He tried this for rational numbers also and found,

$\frac57$ - $\frac38$ = $\frac57$ + $\frac-38$ = $\frac1956$

Both obtained the same difference.

Try to find $\frac78$ - $\frac59$, $\frac311$ - $\frac87$ in both ways. Did you get the same answer?

So, we say while subtracting two rational numbers, we add the additive inverse of the rational number that is being subtracted, to the other rational number.

Thus, 1$\frac23$ - 2$\frac45$ = $\frac53$ - $\frac145$ = $\frac53$ + additive inverse of $\frac145$ = $\frac53$ + $\frac(-14)5$

= $\frac-1715$ =-1$\frac215$

What will be $\frac27$ - ($\frac-56$)?

$\frac27$ - ($\frac-56$) = $\frac27$ + additive inverse of ($\frac-56$) = $\frac27$ + $\frac56$ = $\frac4742$ = 1$\frac542$

## Examples on Subtraction of Rational Numbers

Example: find

1. $\frac34$ -$\frac14$
2. $\frac918$ - $\frac936$
3. $\frac724$ - $\frac1736$
4. $\frac563$ - ($\frac-621$)
5. $\frac-613$ - ($\frac-715$)
6. $\frac-38$ - $\frac711$
7. -2$\frac19$ - 6

Solution:

1. $\frac34$ - $\frac14$

As the denominator of both these rational numbers are same.

Hence, $\frac3-14$

= $\frac24$

2. $\frac918$ - $\frac936$

Since the denominator of both these rational numbers are different. Hence, we will find the LCD of both the denominators.

Therefore, $\frac9 x 218 x 2$ - $\frac936$

= $\frac18 - 936$

= $\frac936$

3. $\frac724$ - $\frac1736$

Since the denominator of both these rational numbers are different. Hence, we will find the LCD of both the denominators.

Therefore, $\frac7 x 3- 17 x 272$

= $\frac21-3472$

= $\frac-1372$

4. $\frac563$- ($\frac-621$)

= $\frac5 + 6 x 363$

Since the denominator of both these rational numbers are different. Hence, we will find the LCD of both the denominators.

= $\frac5+1863$

=$\frac2363$

5. $\frac-613$ - ($\frac-715$)

$\frac-613$ + $\frac715$

Since the denominator of both these rational numbers are different. Hence, we will find the LCD of both the denominators.

= $\frac-6 x 15 + 7 x 13195$

= $\frac-90 + 91195$

= $\frac1195$

6. $\frac-38$ - $\frac711$

Since the denominator of both these rational numbers are different. Hence, we will find the LCD of both the denominators.

= $\frac-3 x 11 – 7 x 888$

= $\frac-33-5688$

= $\frac-8988$

7. -2$\frac19$ - 6

We can write it as

$\frac-199$ - $\frac61$

Since the denominator of both these rational numbers are different. Hence, we will find the LCD of both the denominators.

=$\frac-19 -6 x 99$

= $\frac-19-54$

= $\frac-739$