# RD Sharma Solutions Class 7 Operations On Rational Numbers Exercise 5.4

## RD Sharma Solutions Class 7 Chapter 5 Exercise 5.4

#### Exercise 5.4

Q1. Divide:

(i) 1 by $\frac{1}{2}$

$1\div\frac{1}{2}\\ =1\times2\\ =2$

(ii) 5 by $\frac{-5}{7}$

$5\div\frac{-5}{7}\\ =5\times\frac{-7}{5}\\ =-7$

(iii) $\frac{-3}{4}$ by $\frac{9}{-16}$

$\frac{-3}{4}\div\frac{9}{-16}\\ =\frac{-3}{4}\div\frac{-9}{16}\\ =\frac{-3}{4}\times\frac{-16}{9}\\ =\frac{-4}{-3}\\ =\frac{4}{3}\\$

(iv) $\frac{-7}{8}$ by $\frac{-21}{16}$

$\frac{-7}{8}\div\frac{-21}{16}\\ =\frac{-7}{8}\times\frac{-16}{21}\\ =\frac{2}{3}\\$

(v) $\frac{7}{-4}$ by $\frac{63}{64}$

$\frac{7}{-4}\div\frac{63}{64}\\ =\frac{7}{-4}\times\frac{64}{63}\\ =\frac{-16}{9}\\$

(vi) 0 by $\frac{-7}{5}$

$0\div\frac{-7}{5}\\ =0\times\frac{-5}{7}\\ =0$

(vii) $\frac{-3}{4}$ by -6

$\frac{-3}{4}\div -6\\ =\frac{-3}{4}\times\frac{-1}{6}\\ =\frac{1}{8}$

(viii) $\frac{2}{3}$ by $\frac{-7}{12}$

$\frac{2}{3}\div\frac{-7}{12}\\ =\frac{2}{3}\times\frac{-12}{7}\\ =\frac{-8}{7}$

Q2. Find the value and express as a rational number in standard form:

(i) $\frac{2}{5}\div\frac{26}{15}$

$\frac{2}{5}\div\frac{26}{15}\\ =\frac{2}{5}\times\frac{15}{26}\\ =\frac{3}{13}$

(ii) $\frac{10}{3}\div\frac{-35}{12}$

$\frac{10}{3}\div\frac{-35}{12}\\ \frac{10}{3}\times\frac{-12}{35}\\ =\frac{-40}{35}\\ =\frac{-8}{7}$

(iii) $-6\div\frac{-8}{17}$

$-6\div\frac{-8}{17}\\ =-6\times\frac{-17}{8}\\ =\frac{102}{8}\\ =\frac{51}{4}\\$

(iv) $\frac{40}{98}\div -20$

$\frac{40}{98}\div -20\\ =\frac{40}{98}\times\frac{-1}{20}\\ =\frac{-2}{98}\\ =\frac{-1}{49}\\$

Q3. The product of two rational numbers is 15. If one of the numbers is -10, find the other.

Let the number to be found be x

$x\times-10=15\\ x=\frac{15}{-10}\\ x=\frac{3}{-2}\\ x=\frac{-3}{2}\\$

Hence the number is $x=\frac{-3}{2}\\$

Q4. The product of two rational numbers is $\frac{-8}{9}$. If one of the numbers is $\frac{-4}{15}$, find the other.

Let the number to be found be x

$x\times\frac{-4}{15}=\frac{-8}{9}\\ x=\frac{-8}{9}\div\frac{-4}{15}\\ x=\frac{-8}{9}\times\frac{15}{-4}\\ x=\frac{-8\times15}{9\times-4}\\ x=\frac{-120}{-36}\\ x=\frac{120}{36}\\ x=\frac{10}{3}$

Hence the number is $x=\frac{10}{3}\\$

Q5. By what number should we multiply $\frac{-1}{6}$ so that the product may be $\frac{-23}{9}$?

Let the number to be found be x

$x\times\frac{-1}{6}=\frac{-23}{9}\\ -x=\frac{-23}{9}\times6\\ -x=\frac{-23\times6}{9}\\ -x=\frac{-138}{9}\\ x=\frac{138}{9}\\ x=\frac{46}{3}\\$

Hence the number is $x=\frac{46}{3}$

Q6. By what number should we multiply $\frac{-15}{28}$ so that the product may be $\frac{-5}{7}$?

Let the number to be found be x

$x\times\frac{-15}{28}=\frac{-5}{7}\\ x=\frac{-5}{7}\div\frac{-15}{28}\\ x=\frac{-5}{7}\times\frac{-28}{15}\\ x=\frac{-8}{9}\times\frac{15}{-4}\\ x=\frac{4}{3}\\$

Hence the number is $x=\frac{4}{3}$

Q7. By what number should we multiply $\frac{-8}{13}$ so that the product may be 24?

Let the number to be found be x

$x\times\frac{-8}{13}=24\\ x=24\div\frac{-8}{13}\\ x=24\times\frac{13}{-8}\\ x=-3\times13\\ x=-39$

Hence the number is $x=-39$

Q8. By what number should $\frac{-3}{4}$ be multiplied in order to produce $\frac{-2}{3}$?

Let the number to be found be x

$x\times\frac{-8}{13}=24\\ x=24\div\frac{-8}{13}\\ x=24\times\frac{13}{-8}\\ x=-3\times13\\ x=-39$

Hence the number is $x=-39$

Q9. Find (x + y)÷(x —y), if

(i) x= $\frac{2}{3}$ y= $\frac{3}{2}$

$\left(x+y \right )\div\left(x-y \right )\\ =\left(\frac{2}{3}+\frac{3}{2} \right )\div\left(\frac{2}{3}-\frac{3}{2} \right )\\ =\left(\frac{4+9}{6} \right )\div\left(\frac{4-9}{6} \right )\\ =\left(\frac{4+9}{6} \right )\times\left(\frac{6}{4-9} \right )\\ =\left(\frac{4+9}{4-9} \right )\\ =\left(\frac{13}{-5} \right )\\ =\left(\frac{-13}{5} \right )\\$

(ii) x= $\frac{2}{5}$ y= $\frac{1}{2}$

$\left(x+y \right )\div\left(x-y \right )\\ =\left(\frac{2}{5}+\frac{1}{2} \right )\div\left(\frac{2}{5}-\frac{1}{2} \right )\\ =\left(\frac{4+5}{16} \right )\div\left(\frac{4-5}{16} \right )\\ =\left(\frac{4+5}{16} \right )\times\left(\frac{16}{4-5} \right )\\ =\left(\frac{4+5}{4-5} \right )\\ =\left(\frac{9}{-1} \right )\\ =\left(\frac{-9}{1} \right )\\ =9$

(iii) x= $\frac{5}{4}$ y= $\frac{-1}{3}$

$\left(x+y \right )\div\left(x-y \right )\\ =\left(\frac{5}{4}+\frac{-1}{3} \right )\div\left(\frac{5}{4}-\frac{-1}{3} \right )\\ =\left(\frac{5\times3-1\times4}{12} \right )\div\left(\frac{5\times3+1\times4}{12} \right )\\ =\left(\frac{5\times3-1\times4}{12} \right )\times\left(\frac{12}{5\times3+1\times4} \right )\\ =\left(\frac{5\times3-1\times4}{5\times3+1\times4} \right )\\ =\left(\frac{11}{19} \right )\\$

Q10. The cost of $7\frac{2}{3}$ metres of rope is Rs. $12\frac{3}{4}$. Find its cost per metre.

$7\frac{2}{3}$ metres of rope cost= Rs. $12\frac{3}{4}$

=Rs.$\frac{51}{4}$

$7\frac{2}{3}$= $\frac{23}{3}$

Cost per metre= $\frac{51}{4}\div\frac{23}{3}\\ =\frac{51}{4}\times\frac{3}{23}\\ =\frac{153}{92}\\ =Rs.1\frac{61}{92}$

Q11. The cost of $2\frac{1}{3}$ metres of cloth is Rs.$75\frac{1}{4}$. Find the cost of cloth per metre.

$2\frac{1}{3}$ metres of rope cost= Rs. $75\frac{1}{4}$

=Rs.$\frac{301}{4}$

$2\frac{1}{3}$= $\frac{7}{3}$

Cost per metre= $\frac{301}{4}\div\frac{7}{3}\\ =\frac{301}{4}\times\frac{3}{7}\\ =\frac{43\times3}{4}\\ =\frac{129}{4}\\ =Rs.32\frac{1}{4}$

Q12. By what number should $\frac{-33}{16}$ be divided to get  $\frac{-11}{4}$?

$\frac{-33}{16}\div\;x=\frac{-11}{4}\\ x=\frac{-33}{16}\div\frac{-11}{4}\\ x=\frac{-33}{16}\times\frac{4}{-11}\\ x=\frac{3}{4}\\$

The number is $x=\frac{3}{4}\\$

Q13. Divide the sum of $\frac{-13}{5}$ and $\frac{12}{7}$ by the product of $\frac{-31}{7}$ and $\frac{-1}{2}$

$\left(\frac{-13}{5}+\frac{12}{7} \right )\div\left(\frac{-31}{7}\times\frac{-1}{2} \right )\\ =\left(\frac{-13\times7}{5\times7}+\frac{12\times5}{7\times5} \right )\div\left(\frac{-31}{7}\times\frac{-1}{2} \right )\\ =\left(\frac{-91}{35}+\frac{60}{35} \right )\div\left(\frac{31}{14}\right )\\ =\left(\frac{-91+60}{35} \right )\div\left(\frac{31}{14}\right )\\ =\left(\frac{-31}{35} \right )\div\left(\frac{31}{14}\right )\\ =\left(\frac{-31}{35} \right )\times\left(\frac{14}{31}\right )\\ =\frac{-14}{35}\\ =\frac{-2}{5}$589

Q14. Divide the sum of $\frac{65}{12}$ and $\frac{8}{3}$ by their difference.

$\left(\frac{65}{12}+\frac{8}{3} \right )\div\left(\frac{65}{12}-\frac{8}{3} \right ) \\ =\left(\frac{65}{12}+\frac{8\times4}{3\times4} \right )\div\left(\frac{65}{12}-\frac{8\times4}{3\times4} \right ) \\ =\left(\frac{65}{12}+\frac{32}{12} \right )\div\left(\frac{65}{12}-\frac{32}{12} \right ) \\ =\left(\frac{65+32}{12}\right )\div\left(\frac{65-32}{12} \right ) \\ =\left(\frac{65+32}{12}\right )\times\left(\frac{12}{65-32} \right ) \\ =\frac{65+32}{65-32}\\ =\frac{97}{33}\\$

Q15. If 24 trousers of equal size can be prepared in 54 metres of cloth, what length of cloth is required for each trouser?

Length of cloth required for each trouser=$\frac{Total\;length\;of\;cloth}{number\; of\; trousers}$

=$\frac{54}{24}$

=$\frac{9}{4}$metres

$\frac{9}{4}$mmetres of cloth is required to make each trouser