Question

1. Find :-



$\frac{1}{3} \: + \: \bigg( \frac{ - 4}{5} \bigg) \: + \: \bigg( \frac{- 3}{2} \bigg) \: + \: \frac{6}{7}\ \textless \ br /\ \textgreater$




2. Find :-




$\frac{5}{14} \: \times \: \frac{ - 2}{11} \: \times \: \frac{ - 7}{10} \: \times \: \frac{33}{16}$




3. Find :-



$\frac{2}{5} \: \times \: \frac{4}{7} \: - \: \frac{1}{3} \: + \: \frac{4}{7} \: \times \: \frac{8}{5}$




4. Find using distributivity :-



$\bigg( \frac{9}{16} \: \times \: \frac{4}{12} \bigg) \: + \: \bigg( \frac{9}{16} \: \times \: \frac{ - 3}{9} \bigg)\ \textless \ br /\ \textgreater$
​

Answer 1

★ How to do :-

Here, we are given with some fractions to add and subtract and also to multiply with each other. We are asked to find the answer for the same. In addition we are given with the fractions that are having a different denominators. So, first we should convert them into like fractions by taking the LCM of the denominators. By taking the LCM of all the denominators, we can make all the denominators as a same number and then directly add the numerators. In multiplication we can directly multiply the numerator with numerator and tye denominator with denominator. So, let's solve!!

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➤ Solution (1) :-

${\tt \leadsto \dfrac{1}{3} + \dfrac{(-4)}{5} + \dfrac{(-3)}{2} + \dfrac{6}{7}}$

LCM of 3, 5, 2 and 7 is 210.

${\tt \leadsto \dfrac{1 \times 70}{3 \times 70} + \dfrac{(-4) \times 42}{5 \times 42} + \dfrac{(-3) \times 105}{2 \times 105} + \dfrac{6 \times 30}{7 \times 30}}$

Multiply the numerators and denominators of all four fractions.

${\tt \leadsto \dfrac{70}{210} + \dfrac{(-168)}{210} + \dfrac{(-315)}{210} + \dfrac{180}{210}}$

Write all the numerators in one fraction with common denominator.

${\tt \leadsto \dfrac{70 + (-168) + (-315) + 180}{210}}$

Convert all the numbers so that all of them have only one sign.

${\tt \leadsto \dfrac{70 - 168 - 315 + 180}{210}}$

Simplify two numbers together each.

${\tt \leadsto \dfrac{(-98) - 135}{210}}$

Subtract both numbers to get the final answer.

${\tt \leadsto \pink{\underline{\boxed{\tt \dfrac{(-233)}{210}}}}}$

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➤ Solution (2) :-

${\tt \leadsto \dfrac{5}{14} \times \dfrac{(-2)}{11} \times \dfrac{(-7)}{10} \times \dfrac{33}{16}}$

${\tt \leadsto \dfrac{5}{\cancel{14}} \times \dfrac{\cancel{(-2)}}{11} \times \dfrac{(-7)}{10} \times \dfrac{33}{16}}$

Write the resulting fraction.

${\tt \leadsto \dfrac{5}{7} \times \dfrac{(-1)}{11} \times \dfrac{(-7)}{10} \times \dfrac{33}{16}}$

Multiply the first two fractions.

${\tt \leadsto \dfrac{(-5)}{\cancel{77}} \times \dfrac{\cancel{(-7)}}{10} \times \dfrac{33}{16}}$

Write the resulting answer.

${\tt \leadsto \dfrac{(-5)}{11} \times \dfrac{(-1)}{10} \times \dfrac{33}{16}}$

Cancel it further.

${\tt \leadsto \dfrac{\cancel{(-5)}}{11} \times \dfrac{(-1)}{\cancel{10}} \times \dfrac{33}{16}}$

Write the resulting fraction.

${\tt \leadsto \dfrac{(-1)}{11} \times \dfrac{(-1)}{2} \times \dfrac{33}{16}}$

Multiply them now.

${\tt \leadsto \dfrac{1}{22} \times \dfrac{33}{16}}$

Write them in lowest form by cancellation method.

${\tt \leadsto \dfrac{1}{\cancel{22}} \times \dfrac{\cancel{33}}{16}}$

Write the resulting fraction.

${\tt \leadsto \dfrac{1}{2} \times \dfrac{3}{16}}$

Multiply them to get the answer.

${\tt \leadsto \dfrac{1 \times 3}{2 \times 16} = \pink{\underline{\boxed{\tt \dfrac{3}{32}}}}}$

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➤ Solution (3) :-

${\tt \leadsto \dfrac{2}{5} \times \dfrac{4}{7} - \dfrac{1}{3} + \dfrac{4}{7} \times \dfrac{8}{5}}$

Multiply the fractions first.

${\tt \leadsto \dfrac{2 \times 4}{5 \times 7} - \dfrac{1}{3} + \dfrac{4 \times 8}{7 \times 5}}$

Multiply the numerators and denominators of all the fractions.

${\tt \leadsto \dfrac{8}{35} - \dfrac{1}{3} + \dfrac{32}{35}}$

LCM of 35 and 3 is 210.

${\tt \leadsto \dfrac{8 \times 6}{35 \times 6} - \dfrac{1 \times 70}{3 \times 70} + \dfrac{32 \times 6}{35 \times 6}}$

${\tt \leadsto \dfrac{48}{210} - \dfrac{70}{210} + \dfrac{192}{210}}$

Add the fractions first.

${\tt \leadsto \dfrac{48}{210} - \dfrac{70 + 192}{210}}$

Add the fractions in numerator.

${\tt \leadsto \dfrac{48}{210} - \dfrac{262}{210}}$

Subtract the fractions to get the answer.

${\tt \leadsto \dfrac{48 - 262}{210} - \dfrac{(-214)}{210}}$

Write the fraction in lowest form by cancellation method.

${\tt \leadsto \cancel \dfrac{(-214)}{210} = \pink{\underline{\boxed{\tt \dfrac{(-107)}{105}}}}}$

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➤ Solution (4) :-

${\tt \leadsto \bigg( \dfrac{9}{16} \times \dfrac{4}{12} \bigg) + \bigg( \dfrac{9}{16} \times \dfrac{(-3)}{9} \bigg)}$

We can see that we have two common fractions, so we can use distributive method.

${\tt \leadsto \dfrac{9}{16} \times \bigg( \dfrac{4}{12} + \dfrac{(-3)}{9} \bigg)}$

LCM of 12 and 9 is 108.

${\tt \leadsto \dfrac{9}{16} \times \bigg( \dfrac{4 \times 9}{12 \times 9} + \dfrac{(-3) \times 12}{9 \times 12} \bigg)}$

Multiply the numerators and denominators in bracket.

${\tt \leadsto \dfrac{9}{16} \times \bigg( \dfrac{36}{108} + \dfrac{(-36)}{108} \bigg)}$

Add the fractions and multiply.

${\tt \leadsto \dfrac{9}{16} \times \dfrac{0}{126}}$

Multiply them.

${\tt \leadsto 0}$

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Hence solved !!