Exercise 1.4
Question 1
Find the value of the following:
(i) $-\frac{3}{7} \div 4$
(ii) $4 \frac{5}{8} \div\left(\frac{-4}{9}\right)$
(iii) $\frac{-8}{9} \div \frac{-3}{5}$
Sol :
(i) $-\frac{3}{7} \div 4$
=$-\frac{3}{7} \div \frac{4}{1}$
=$\frac{-3}{7} \times \frac{1}{4}$
=$\frac{-3 \times 1}{7 \times 4}$
=$\frac{-3}{28}$
(ii) $4 \frac{5}{8} \div\left(\frac{-4}{9}\right)$
=$\frac{37}{8} \div\left(\frac{-4}{9}\right)$
=$\frac{37}{8} \times\left(\frac{9}{-4}\right)$
=$\frac{37 \times 9}{8 \times(-4)}$
=$-\frac{333}{32}$
=$-10 \frac{13}{32}$
(iii) $\frac{-8}{9} \div \frac{-3}{5}$
=$\frac{-8}{9} \times \frac{5}{-3}$
=$\frac{-8 \times 5}{9 x-3}$
=$\frac{-40}{-27}$
=$\frac{40}{27}=1 \frac{13}{27}$
Question 2
State whether the following statements are true or false:(i) $\frac{-9}{13} \div \frac{2}{7}$ is a rational number(ii) $\frac{4}{13} \div \frac{11}{12}=\frac{11}{12} \div \frac{4}{13}$(iii) $\frac{-3}{4} \div \left(\frac{5}{9}\div \frac{-4}{11}\right)=\left(\frac{-3}{4}\div \frac{5}{9}\right) \div \frac{-4}{11}$(iv) $\frac{13}{14} \div \frac{-5}{7} \neq \frac{-5}{7} \div \frac{13}{14}$(v) $\left(-7 \div \frac{4}{5}\right) \div \frac{-9}{10} \neq -7 \div \left(\frac{4}{5} \div \frac{-9}{10}\right)$(vi) $\frac{-7}{24} \div \frac{6}{11}$ is not a rational number.
Sol :(i) true
(ii) false
(iii) false
(iv) true
(v) true
(vi)false
Question 3
The product of two rational numbers is $\frac{-11}{12}$. If one of them is $2\frac{4}{9}$ , find the other.Sol :Let unknown number be x
⇒$x \times 2 \frac{4}{9}=\frac{-11}{12}$
⇒$x \times \frac{22}{9}=\frac{-11}{12}$
$x=\frac{-11}{12}$ x $\frac{22}{9}$
⇒$\frac{-11}{12} \times \frac{9}{22}$
$x=\frac{-3}{8}$
Other number $=\frac{-3}{8}$
Question 4
By what rational number should $\frac{-7}{12}$ be multiplied to get the product as $\frac{5}{14}$ ?Sol :Let unknown number be x
$x \times\left(\frac{-7}{12}\right)=\frac{5}{14}$
$x=\frac{5}{14} \div\left(\frac{-7}{12}\right)$
$=\frac{5}{14} \times \frac{12}{-7}$
$=\frac{5 \times 12}{14 \times(-7)}$
$=-\frac{60}{98}$
$=\frac{-30}{49}$
Question 5
By what rational number should -3 is divided to get $\frac{-9}{13}$Sol :
Let unknown number be x
$\frac{-3}{x}=\frac{-9}{13}$
$x=-\frac{3}{1} \div\left(\frac{-9}{13}\right)$
$=\frac{-3}{1} \times \frac{13}{-9}$
$x=\frac{13}{3}=4 \frac{1}{3}$
Question 6
Divide the sum of $\frac{-13}{8}$ and $\frac{5}{12}$ by their difference.Sol :
Sum of number = $\frac{-13}{8}+\frac{5}{12}$
$=\frac{(-13 \times 3)+(5 \times 2)}{24} \quad$ LCM of $812=24$
⇒$\frac{-39+10}{24}$
⇒$\frac{-29}{24}$
sum of numbers = $\frac{-29}{24}$
difference of numbers= $\frac{(-13 \times 3)-(5 \times 2)}{24}$
$\frac{-39-10}{24}$
Difference of numbers $=\frac{-49}{24}$
Required product = sum of numbers ÷ difference of numbers
$\begin{aligned} &=\frac{-29}{24} \div\left(-\frac{49}{24}\right) \\=& \frac{-29}{24} \times \frac{24}{-49} \\ &=\frac{29}{49} \end{aligned}$
Question 7
Divide the sum of $\frac{8}{3}$ and $\frac{4}{7}$ by the product of $\frac{-3}{7}$ and $\frac{14}{9}$
Sol :
$\begin{aligned} \text { Sum of two numbers } &=\frac{8}{3}+\frac{4}{7} \\ &=\frac{(8 \times 7)+(3 \times 4)}{21} \text { LCM Of } 3,7=21 \\ &=\frac{56+12}{21} \\ \text { Sum of two numbers } &=\frac{68}{21} \end{aligned}$
$\begin{aligned} \text { Product of given numbers } &=\frac{-3}{7} \times \frac{14}{9} \\ &=\frac{-2}{3} \end{aligned}$
$\begin{aligned} \text { Required product } &=\frac{\text { Sum of } \frac{8}{3} \text { and } \frac{4}{7}}{\text { product of }\frac{3}{1} \text { and } \frac{14}{9}} \\ &=\frac{68}{21} \div\left(\frac{-2}{3}\right) \\ &=\frac{68}{21} \times \frac{3}{-2} \\ &=\frac{-34}{7} \end{aligned}$
Question 8
If $p=\frac{-3}{2},q=\frac{4}{5}$ and $r=\frac{-7}{12}$, then verify that (p ÷ q) ÷ r ≠ p ÷ (q ÷ r).Sol :
Given $\quad P=\frac{-3}{2}, \quad q=\frac{4}{5}, \quad \ r=\frac{-7}{12}$
$(p \div q) \div r=p \div(q \div \ r)$
$\begin{aligned} \text { L.H.S } &=(p \div q) \div r \\ &=\left(\frac{-3}{2} \div \frac{4}{5}\right) \div\left(\frac{-7}{12}\right) \\ &=\left(\frac{-3}{2} \times \frac{5}{4}\right) \div\left(\frac{-7}{12}\right) \\ &=\left(\frac{-15}{8}\right) \div\left(-\frac{7}{12}\right) \\ &=\frac{-15}{8} \times \frac{12}{-7} \\ &=\frac{-15}{2} \times \frac{3}{-7} \end{aligned}$
$\begin{aligned} \text { L.H.S } &=+\frac{45}{14} \\ \text { R.H.S } &=p \div(q \div \ r) \\ &=\frac{-3}{2} \div\left(\frac{4}{5} \div\left(\frac{-7}{12}\right)\right) \\ &=\frac{-3}{2} \div\left(\frac{4}{5} \times \frac{12}{-7}\right) \\ &=\frac{-3}{2} \div\left(-\frac{48}{35}\right) \\ &=\frac{-3}{2} \times-\frac{35}{48} \\ R \cdot H S &=\frac{35}{32} \\ L \cdot H \cdot S \neq R \cdot H \cdot S \\(p \div q) \div r \neq P \div(q \div \tau) \end{aligned}$
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