RD Sharma solution class 7 chapter 2 Fractions Exercise 2.3

Exercise 2.3

Page-2.24

Question 1:

Find the reciprocal of each of the following fractions and classify them as proper, improper and whole numbers:
(i) $\frac{3}{7}$
(ii) $\frac{5}{8}$
(iii) $\frac{9}{7}$
(iv) $\frac{6}{5}$
(v) $\frac{12}{7}$
(vi) $\frac{1}{8}$

Answer 1:

Reciprocal of a non-zero fraction $\frac{\mathrm{a}}{\mathrm{b}} \mathrm{is} \frac{\mathrm{b}}{\mathrm{a}}$
(i)
$$\begin{gathered} \mathrm{Reciprocal} \mathrm{of} \frac{3}{7} \mathrm{is} \frac{7}{3} \\ \mathrm{It}\text{'}\mathrm{s} \mathrm{improper} \mathrm{fraction} \mathrm{because} \text{n}\mathrm{umerator} \mathrm{is} \mathrm{greater} \mathrm{than} \mathrm{denominator} \end{gathered}$$

(ii)
$$\begin{gathered} \mathrm{Reciprocal} \mathrm{of} \frac{5}{8} \mathrm{is} \frac{8}{5} \\ \mathrm{It}\text{'}\mathrm{s} \mathrm{improper} \mathrm{fraction} \mathrm{because} \mathrm{n}\mathrm{umerator} \mathrm{is} \mathrm{greater} \mathrm{than} \mathrm{denominator} \end{gathered}$$

(iii)
$$\begin{gathered} \mathrm{Reciprocal} \mathrm{of} \frac{9}{7} \mathrm{is} \frac{7}{9} \\ \mathrm{It}\text{'}\mathrm{s} \mathrm{proper} \mathrm{fraction} \mathrm{because} \text{n}\mathrm{umerator} \mathrm{is} \mathrm{less} \mathrm{than} \mathrm{denominator} \end{gathered}$$

(iv)
$$\begin{gathered} \mathrm{Reciprocal} \mathrm{of} \frac{6}{5} \mathrm{is} \frac{5}{6} \\ \mathrm{It}\text{'}\mathrm{s} \mathrm{proper} \mathrm{fraction} \mathrm{because} \text{n}\mathrm{umerator} \mathrm{is} \mathrm{less} \mathrm{than} \mathrm{denominator} \end{gathered}$$

(v)
$$\begin{gathered} \mathrm{Reciprocal} \mathrm{of} \frac{12}{7} \mathrm{is} \frac{7}{12} \\ \mathrm{It}\text{'}\mathrm{s} \mathrm{proper} \mathrm{fraction} \mathrm{because} \text{n}\mathrm{umerator} \mathrm{is} \mathrm{less} \mathrm{than} \mathrm{denominator} \end{gathered}$$

(vi)
$$\begin{gathered} \mathrm{Reciprocal} \mathrm{of} \frac{1}{8} \mathrm{is} \frac{8}{1}=8 \\ \mathrm{It} \mathrm{i}\mathrm{s} \mathrm{a} \text{w}\mathrm{hole} \mathrm{number} \end{gathered}$$

Page-2.25

Question 2:

Divide:
(i) $\frac{3}{8} \mathrm{by} \frac{5}{9}$
(ii) $3\frac{1}{4} \mathrm{by} \frac{2}{3}$
(iii) $\frac{7}{8} \mathrm{by} 4\frac{1}{2}$
(iv) $6\frac{1}{4} \mathrm{by} 2\frac{3}{5}$

Answer 2:

(i)
$$\begin{gathered} \frac{3}{8}÷\frac{5}{9}=\frac{3}{8}\times \frac{9}{5} \\ \Rightarrow \frac{3\times 9}{8\times 5}\iff \frac{27}{40} \end{gathered}$$

(ii)
$$\begin{gathered} 3\frac{1}{4}÷\frac{2}{3}=\frac{(3\times 4)+1}{4}\times \frac{3}{2} \\ \Rightarrow \frac{13}{4}\times \frac{3}{2}\iff \frac{39}{8}=4\frac{7}{8} \end{gathered}$$

(iii)
$$\begin{gathered} \frac{7}{8}÷4\frac{1}{2}=\frac{7}{8}÷\frac{(4\times 2)+1}{2} \\ \Rightarrow \frac{7}{\cancel{8^4}}\times \frac{\cancel{2}}{9} \\ \Rightarrow \frac{7}{4}\times \frac{1}{9}\iff \frac{7}{36} \end{gathered}$$
(iv)
$$\begin{gathered} 6\frac{1}{4}÷2\frac{3}{5}=\frac{(6\times 4)+1}{4}÷\frac{(2\times 5)+3}{5} \\ \Rightarrow \frac{25}{4}÷\frac{13}{5}=\frac{25}{4}\times \frac{5}{13} \\ \Rightarrow \frac{125}{52}=2\frac{21}{52} \end{gathered}$$

Question 3:

Divide:
(i) $\frac{3}{8} \mathrm{by} 4$
(ii) $\frac{9}{16} \mathrm{by} 6$
(iii) $9 \mathrm{by} \frac{3}{16}$
(iv) $10 \mathrm{by} \frac{100}{3}$

Answer 3:

(i)
$$\begin{gathered} \frac{3}{8}÷4=\frac{3}{8}\times \frac{1}{4} \\ =\frac{3}{32} \end{gathered}$$

(ii)
$$\begin{gathered} \frac{9}{16}÷6=\frac{9}{16}\times \frac{1}{6}=\frac{\cancel{9^3}}{16\times \cancel{6^2}} \\ =\frac{3}{32} \end{gathered}$$

(iii)
$$\begin{gathered} 9÷\frac{3}{16}=\frac{9}{1}\times \frac{16}{3}\iff \frac{\cancel{9^3}\times 16}{\cancel{3}} \\ =48 \end{gathered}$$

(iv)
$$\begin{gathered} 10÷\frac{100}{3}=\frac{10}{1}\times \frac{3}{100}\iff \frac{\cancel{10}\times 3}{\cancel{{100}^{10}}} \\ =\frac{3}{10} \end{gathered}$$

Question 4:

Simplify:
(i) $\frac{3}{10}÷\frac{10}{3}$
(ii) $4\frac{3}{5}÷\frac{4}{5}$
(iii) $5\frac{4}{7}÷1\frac{3}{10}$
(iv) $4÷2\frac{2}{5}$

Answer 4:

(i)
$$\begin{gathered} \frac{3}{10}÷\frac{10}{3}=\frac{3}{10}\times \frac{3}{10}\iff \frac{3\times 3}{10\times 10} \\ \Rightarrow \frac{9}{100} \end{gathered}$$
(ii)
$$\begin{gathered} 4\frac{3}{5}÷\frac{4}{5}=\frac{(4\times 5)+3}{5}÷\frac{4}{5} \\ \Rightarrow 4\frac{3}{5}÷\frac{4}{5}=\frac{23}{5}÷\frac{4}{5}\iff \frac{23}{5}\times \frac{5}{4} \\ \Rightarrow \frac{23}{4}=5\frac{3}{4} \end{gathered}$$

(iii)
$$\begin{gathered} 5\frac{4}{7}÷1\frac{3}{10}=\frac{(5\times 7)+4}{7}÷\frac{(1\times 10)+3}{10} \\ \Rightarrow 5\frac{4}{7}÷1\frac{3}{10}=\frac{39}{7}÷\frac{13}{10}\iff \frac{\cancel{{39}^3}}{7}\times \frac{10}{\cancel{13}} \\ \Rightarrow \frac{30}{7}=4\frac{2}{7} \end{gathered}$$

(iv)
$$\begin{gathered} 4÷2\frac{2}{5}=\frac{4}{1}÷\frac{(2\times 5)+2}{5}\iff \frac{\cancel{4}}{1}\times \frac{5}{\cancel{{12}^3}} \\ \Rightarrow \frac{5}{3}=1\frac{2}{3} \end{gathered}$$

Question 5:

A wire of length $12\frac{1}{2}\mathrm{m}$ is cut into 10 pieces of equal length. Find the length of each piece.

Answer 5:

$$\begin{gathered} 12\frac{1}{2}\mathrm{m}=\frac{(12\times 2)+1}{2}\mathrm{m} \\ =\frac{25}{2}\mathrm{m} \end{gathered}$$

Length of one piece = $\frac{\mathrm{Length} \mathrm{of} wire}{10}=\frac{25}{2}\times \frac{1}{10}=\frac{25}{20}$
Length of one piece = $\frac{\cancel{{25}^5}}{{\cancel{20}}^4}=\frac{5}{4}\mathrm{m}$

Question 6:

The length of a rectangular plot of area $65\frac{1}{3}{\mathrm{m}}^2 \mathrm{is} 12\frac{1}{4}\mathrm{m}$. What is the width of the plot?

Answer 6:

Area of rectangle = Length of rectangle $\times$ Width of rectangle
$$\begin{gathered} 65\frac{1}{3}=12\frac{1}{4}\times \mathrm{Width} \mathrm{of} \mathrm{the} \mathrm{rectangle} \\ \frac{(65\times 3)+1}{3}=\left[\frac{(12\times 4)+1}{4}\right]\times \mathrm{Width} \mathrm{of} \mathrm{the} \mathrm{rectangle} \\ \frac{196}{3}=\frac{49}{4}\times \mathrm{Width} \mathrm{of} \mathrm{the} \mathrm{rectangle} \\ \mathrm{Width} \mathrm{of} \mathrm{the} \mathrm{rectangle}=\frac{\cancel{{196}^4}}{3}\times \frac{4}{\cancel{49}}=\frac{16}{3}\mathrm{m} \end{gathered}$$

Question 7:

By what number should $6\frac{2}{9}$ be multiplied to get $4\frac{4}{9}$?

Answer 7:

Let the required number be x.

According to the question:

$$\begin{gathered} 6\frac{2}{9}\times x=4\frac{4}{9} \\ \frac{(6\times 9)+2}{9}\times x=\frac{(4\times 9)+4}{9} \\ \frac{56}{9}\times x=\frac{40}{9} \\ x=\frac{40}{\cancel{9}}\times \frac{\cancel{9}}{56} \\ x=\frac{40}{56}=\frac{5}{7 } \end{gathered}$$

Question 8:

The product of two numbers is $25\frac{5}{6}$. If one of the numbers is $6\frac{2}{3}$, find the other.

Answer 8:

Let the required number be x.

According to the question:

$$\begin{gathered} 6\frac{2}{3}\times x=25\frac{5}{6} \\ \frac{(6\times 3)+2}{3}\times x=\frac{(25\times 6)+5}{6} \\ \frac{20}{3}\times x=\frac{155}{6} \\ x=\frac{3}{20}\times \frac{155}{6}=\frac{\cancel{3}\times {\cancel{155}}^{31}}{\cancel{{20}^4}\times \cancel{6^2}}=\frac{31}{8} \\ =3\frac{7}{8} \end{gathered}$$

Question 9:

The cost of $6\frac{1}{4} \mathrm{kg}$ of apples is Rs 400. At what rate per kg are the apples being sold?

Answer 9:

$$\begin{gathered} 6\frac{1}{4}\mathrm{kg}=\frac{(6\times 4)+1}{4}\mathrm{kg} \\ \Rightarrow \frac{25}{4}\mathrm{kg} \end{gathered}$$
Cost of  $\frac{25}{4}\mathrm{kg}$ of apples = Rs. 400
Cost of 1 kg of apples = $400÷\frac{25}{4}=400\times \frac{25}{4} = \mathrm{Rs}. 64$

Question 10:

By selling oranges at the rate of Rs $5\frac{1}{4}$ per orange, a fruit-seller gets Rs 630. How many dozens of oranges does he sell?

Answer 10:

Cost of 1 orange = $\mathrm{Rs} 5\frac{1}{4}=\frac{(5\times 4)+1}{4} =\mathrm{Rs}\frac{21}{4}$
Number of oranges sold = $630÷\frac{21}{4}$
$\Rightarrow \cancel{{630}^{30}}\times \frac{4}{\cancel{21}}=120$
$\because$ 12 oranges = 1 dozen

$\therefore$ 120 oranges = $\frac{120}{12}=10 \mathrm{dozen}$

Question 11:

In mid-day meal scheme $\frac{3}{10}$ litre of milk is given to each student of a primary school. If 30 litres of milk is distributed every day in the school, how many students are there in the school?

Answer 11:

Number of students in the school = $\frac{\mathrm{Total} \mathrm{amount} \mathrm{of} \mathrm{milk} \mathrm{distributed} \mathrm{per} \mathrm{day}}{\mathrm{Amount} \mathrm{of} \mathrm{milk} \mathrm{given} \mathrm{to} \mathrm{one} \mathrm{student}}$
$$\begin{gathered} =30÷\frac{3}{10} \\ =\cancel{{30}^{10}}\times \frac{10}{\cancel{3}} \\ =100 \end{gathered}$$

Question 12:

In a charity show Rs 6496 were collected by selling some tickets. If the price of each ticket was Rs $50\frac{3}{4}$, how many tikets were sold?

Answer 12:

Number of tickets sold = $\frac{\mathrm{Total} \mathrm{amount} \mathrm{of} \mathrm{money} \mathrm{collected}}{\mathrm{Price} \mathrm{of} \mathrm{one} \mathrm{ticket}}$
Price of one ticket:

$$\begin{gathered} 50\frac{3}{4}\Rightarrow \frac{(50\times 4)+3}{4} \\ \Rightarrow \mathrm{Rs} \frac{203}{4} \end{gathered}$$

Number of tickets sold = $6496÷\frac{203}{4}$
$$\begin{gathered} \Rightarrow \cancel{{6496}^{32}}\times \frac{4}{\cancel{203}} \\ \Rightarrow 128 \end{gathered}$$