Proportional Reasoning - 2 | FIO

Question 20

A small pump can fill a tank in 3 hours, while a large pump can fill the same tank in 2 hours. If both pumps are used together, how long will the tank take to fill?

Solution

We can solve this problem by figuring out how much of the tank each pump fills in one hour.

Step 1 — Find individual pump rates

Let us imagine the tank has a total capacity of 1 unit (like 1 whole tank). The small pump fills the tank in 3 hours. So, in one hour, the small pump fills a fraction of the tank.

$\text{Small pump rate} = \frac{1 \text{ tank}}{3 \text{ hours}}$

$= \frac{1}{3} \text{ tank per hour}$

The large pump fills the tank in 2 hours. So, in one hour, the large pump fills a fraction of the tank.

$\text{Large pump rate} = \frac{1 \text{ tank}}{2 \text{ hours}}$

$\boxed{\frac{1}{2} \text{ tank per hour}}$

Diagram 1

Step 2 — Find combined pump rate

When both pumps work together, their filling rates add up. We will add the amount each pump fills in one hour.

$\text{Combined rate} = \text{Small pump rate} + \text{Large pump rate}$

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

To add these fractions, we find a common denominator, which is 6.

$= \frac{1 \times 2}{3 \times 2} + \frac{1 \times 3}{2 \times 3}$

$= \frac{2}{6} + \frac{3}{6}$

$\boxed{\frac{5}{6} \text{ tank per hour}}$

Step 3 — Calculate total time

The combined rate tells us that both pumps together fill $\frac{5}{6}$ of the tank in one hour. We want to find out how long it takes to fill the entire 1 unit tank. We can use the formula: Time = Total Work / Rate.

$\text{Time taken} = \frac{\text{Total tank capacity}}{\text{Combined rate}}$

$= \frac{1 \text{ tank}}{\frac{5}{6} \text{ tank per hour}}$

To divide by a fraction, we multiply by its reciprocal.

$= 1 \times \frac{6}{5} \text{ hours}$

$= \frac{6}{5} \text{ hours}$

We can convert this fraction to a decimal or to hours and minutes.

$= 1.2 \text{ hours}$

To convert the decimal part to minutes, we multiply by 60.

$0.2 \text{ hours} = 0.2 \times 60 \text{ minutes}$

$= 12 \text{ minutes}$

So, the total time is 1 hour and 12 minutes.

$\boxed{1.2 \text{ hours or 1 hour 12 minutes}}$

Answer

The tank will take 1.2 hours (or 1 hour 12 minutes) to fill if both pumps are used together.

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Q10

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Q11

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(iii) The distance a car can travel and the amount of petrol in the tank.

(iv) The speed of a cyclist and the time taken to cover a fixed route.

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(vi) The number of pages in a book and the time required to read it at a fixed reading speed.

Q12

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Q13

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Q14

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Q15

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(i) What is the most common mode of transport?

(ii) What fraction of children travel by car?

(iii) If 18 children travel by car, how many children took part in the survey? How many children use taxis to travel to school?

(iv) By which two modes of transport are equal numbers of children travelling?

Q16

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Q17

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Q18

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Q19

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Q20

A small pump can fill a tank in 3 hours, while a large pump can fill the same tank in 2 hours. If both pumps are used together, how long will the tank take to fill?

Q21

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Q22

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