A water tank is completely filled. $\frac{1}{10}$ of the water is used for the garden and $\frac{1}{4}$ is used for bathing. We need to find the fraction of water used for garden and bathing, the fraction of the tank used to wash clothes (given $\frac{4}{13}$ of the remaining water is used for washing clothes), the remaining fraction of the tank, and the capacity of the tank when 500 litres are used for kitchen requirements and $\frac{1}{4}$ of the tank remains filled.

1. Problem Description

A water tank is completely filled. $\frac{1}{10}$ of the water is used for the garden and $\frac{1}{4}$ is used for bathing. We need to find the fraction of water used for garden and bathing, the fraction of the tank used to wash clothes (given $\frac{4}{13}$ of the remaining water is used for washing clothes), the remaining fraction of the tank, and the capacity of the tank when 500 litres are used for kitchen requirements and $\frac{1}{4}$ of the tank remains filled.

2. Solution Steps

(i) Fraction of water used for garden and bathing:

We need to add the fractions of water used for the garden and bathing.

$$\frac{1}{10} + \frac{1}{4}$$

To add these fractions, we need a common denominator. The least common multiple of 10 and 4 is

2

0. $$ \frac{1}{10} = \frac{1 \times 2}{10 \times 2} = \frac{2}{20} $$

$$\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}$$

$$\frac{2}{20} + \frac{5}{20} = \frac{2+5}{20} = \frac{7}{20}$$

(ii) Fraction of water used to wash clothes:

The fraction of water remaining after garden and bathing is:

$$1 - \frac{7}{20} = \frac{20}{20} - \frac{7}{20} = \frac{13}{20}$$

The fraction of the remaining water used for washing clothes is $\frac{4}{13}$. Therefore, the fraction of the whole tank used for washing clothes is:

$$\frac{4}{13} \times \frac{13}{20} = \frac{4 \times 13}{13 \times 20} = \frac{4}{20} = \frac{1}{5}$$

(iii) Fraction of tank remaining filled:

The fraction of the tank used for garden, bathing, and washing clothes is:

$$\frac{7}{20} + \frac{1}{5}$$

We need a common denominator, which is

2

0. $$ \frac{1}{5} = \frac{1 \times 4}{5 \times 4} = \frac{4}{20} $$

$$\frac{7}{20} + \frac{4}{20} = \frac{7+4}{20} = \frac{11}{20}$$

The fraction of the tank remaining is:

$$1 - \frac{11}{20} = \frac{20}{20} - \frac{11}{20} = \frac{9}{20}$$

(iv) Capacity of the tank:

After using 500 litres for kitchen requirements, $\frac{1}{4}$ of the tank remains. Before using the water for the kitchen, $\frac{9}{20}$ of the tank remained. This means that 500 litres represents the difference between $\frac{9}{20}$ and $\frac{1}{4}$.

$$\frac{9}{20} - \frac{1}{4}$$

We need a common denominator, which is

2

0. $$ \frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20} $$

$$\frac{9}{20} - \frac{5}{20} = \frac{4}{20} = \frac{1}{5}$$

Therefore, $\frac{1}{5}$ of the tank is equal to 500 litres. Let C be the capacity of the tank.

$$\frac{1}{5} C = 500$$

$$C = 500 \times 5 = 2500$$

The capacity of the tank is 2500 litres.

3. Final Answer

(i) The fraction of water used for garden and bathing is $\frac{7}{20}$.

(ii) The fraction of water used to wash clothes is $\frac{1}{5}$.

(iii) The fraction of the tank remaining filled is $\frac{9}{20}$.

(iv) The capacity of the tank is 2500 litres.