Fractions — Case Study Based Questions

A man spends $\dfrac{2}{5}$ of his salary on house-rent, $\dfrac{3}{10}$ of his salary on food and $\dfrac{1}{8}$ of his salary on conveyance. He saves the remaining ₹ 10,500.

1. The total salary of the man is :
   (a) ₹ 52,500
   (b) ₹ 55,000
   (c) ₹ 57,500
   (d) ₹ 60,000

2. The amount spent on house-rent is :
   (a) ₹ 30,000
   (b) ₹ 26,000
   (c) ₹ 24,000
   (d) ₹ 28,000

3. The amount spent on food is :
   (a) ₹ 18,000
   (b) ₹ 20,000
   (c) ₹ 22,500
   (d) ₹ 24,000

4. The amount spent on conveyance is :
   (a) ₹ 12,000
   (b) ₹ 10,500
   (c) ₹ 7,500
   (d) ₹ 15,000

Answer

1. Given,

Part spent on house-rent = $\dfrac{2}{5}$

Part spent on food = $\dfrac{3}{10}$

Part spent on conveyance = $\dfrac{1}{8}$

Savings = ₹ 10,500.

$\Rightarrow \text{Total fraction spent } = \dfrac{2}{5} + \dfrac{3}{10} + \dfrac{1}{8}\\[1em] \Rightarrow \dfrac{2 \times 8}{5 \times 8} + \dfrac{3 \times 4}{10 \times 4} + \dfrac{1 \times 5}{8 \times 5}\\[1em] \Rightarrow \dfrac{16}{40} + \dfrac{12}{40} + \dfrac{5}{40}\\[1em] \Rightarrow \dfrac{16 + 12 + 5}{40}\\[1em] \Rightarrow \dfrac{33}{40}\\[1em]$

Savings fraction = $1 - \dfrac{33}{40} = \dfrac{40 - 33}{40} = \dfrac{7}{40}$.

Savings = Savings fraction × Total salary

$\Rightarrow 10500 = \dfrac{7}{40} \times \text{Total salary} \\[1em] \Rightarrow \text{Total salary} = \dfrac{10500 \times 40}{7} \\[1em] \Rightarrow \text{Total salary} = 1500 \times 40 = ₹60,000.$

Hence, option (d) is the correct option.

2. Amount spent on house rent = $\dfrac{2}{5}$ of the salary

$= \dfrac{2}{5} \times 60,000 \\[1em] = \dfrac{2 \times 60,000}{5}\\[1em] = \dfrac{1,20,000}{5}\\[1em] = ₹24,000.$

Hence, option (c) is the correct option.

3. Amount spent on food = $\dfrac{3}{10}$ of the salary

$= \dfrac{3}{10} \times 60,000 \\[1em] = \dfrac{3 \times 60,000}{10}\\[1em] = \dfrac{1,80,000}{10}\\[1em] = ₹18,000$

Hence, option (a) is the correct option.

4. Amount spent on conveyance = $\dfrac{1}{8}$ of the salary

$= \dfrac{1}{8} \times 60,000 \\[1em] = \dfrac{1 \times 60,000}{8}\\[1em] = \dfrac{60,000}{8}\\[1em] = ₹7,500.$

Hence, option (c) is the correct option.

A drum of kerosene is $\dfrac{3}{4}$ full. When 30 litres of kerosene is drawn from it, it remains $\dfrac{7}{12}$ full.

1. The capacity of the drum is :
   (a) 120 l
   (b) 150 l
   (c) 180 l
   (d) 168 l

2. The quantity of kerosene in the drum is :
   (a) 135 l
   (b) 115 l
   (c) 125 l
   (d) 108 l

3. If 15 l of kerosene is added to the drum, what fraction of the capacity would be filled?

(a) $\dfrac{1}{2}$

(b) $\dfrac{2}{3}$

(c) $\dfrac{5}{6}$

(d) $\dfrac{5}{12}$

4. If 35 l of kerosene is drawn from the drum, what fraction of the capacity would be filled?

(a) $\dfrac{5}{9}$

(b) $\dfrac{7}{9}$

(c) $\dfrac{11}{18}$

(d) $\dfrac{13}{18}$

Answer

1. Given,

Initial capacity of drum = $\dfrac{3}{4}$ full

After removing 30 litres: Capacity of drum = $\dfrac{7}{12}$ full

Let the total capacity of drum be x litres.

$\Rightarrow \dfrac{3}{4}x - 30 = \dfrac{7}{12}x\\[1em] \Rightarrow \dfrac{3}{4}x - \dfrac{7}{12}x = 30\\[1em] \Rightarrow \dfrac{3 \times 3}{4 \times 3}x - \dfrac{7}{12}x = 30\\[1em] \Rightarrow \dfrac{9}{12}x - \dfrac{7}{12}x = 30\\[1em] \Rightarrow \dfrac{9 - 7}{12}x = 30\\[1em] \Rightarrow \dfrac{2}{12}x = 30\\[1em] \Rightarrow \dfrac{1}{6}x = 30\\[1em] \Rightarrow x = 30 \times 6\\[1em] \Rightarrow x = 180 \text{ l}.$

Hence, option (c) is the correct option.

2. Initial quantity of kerosene = $\dfrac{3}{4}$ of the capacity of drum

$\Rightarrow \dfrac{3}{4} \times 180\\[1em] \Rightarrow \dfrac{3 \times 180}{4}\\[1em] \Rightarrow \dfrac{540}{4}\\[1em] \Rightarrow 135$

Hence, option (a) is the correct option.

3. If 15 l of kerosene is added to the drum.

So, final quantity of kerosene = 135 + 15 = 150

Fraction of drum filled = $\dfrac{150}{180} = \dfrac{5}{6}$

Hence, option (c) is the correct option.

4. If 35 l of kerosene is drawn from the drum.

So, final quantity of kerosene = 135 - 35 = 100

Fraction of drum filled = $\dfrac{100}{180} = \dfrac{5}{9}$.

Hence, option (a) is the correct option.