Proportional Reasoning - 2 | FIO

Question 13

A tank on a building has enough water to supply 20 families living there for 6 days. If 10 more families move in there, how long will the water last? What assumptions do you need to make to work out this problem?

Solution

Understand the Question

The number of families and the number of days the water lasts are in inverse proportion (more families $\implies$ water lasts for fewer days).
The total water capacity is constant, meaning the product of the number of families and the number of days remains constant:
$\text{Number of families} \times \text{Number of days} = \text{Constant (family-days)}$

Step 1 · Calculate Total Water Capacity

Given:

Initial number of families $= 20$
Number of days $= 6$

\begin{aligned} \text{Total water units} &= \text{Number of families} \times \text{Number of days} \\[0.6em] &= 20 \times 6 \\[0.6em] &= 120 \text{ family-days} \end{aligned}

Step 2 · Calculate Days for the New Number of Families

When $10$ more families move in:

\begin{aligned} \text{New number of families} &= 20 + 10 \\[0.6em] &= 30 \end{aligned}

Let $x$ be the number of days the water will last. Since total water capacity is constant:

\begin{aligned} \text{New number of families} \times x &= \text{Total water units} \\[0.6em] 30 \times x &= 120 \\[0.6em] x &= \dfrac{120}{30} \\[0.6em] &= 4 \end{aligned}

Step 3 · State the Assumptions

The assumptions needed to solve this problem are:

Each family uses the same amount of water per day.
The daily water consumption rate per family remains constant.
No extra water is added to the tank and there is no leakage.

Answer

The water will last for $4$ days.

Assumptions needed:

All families consume an equal amount of water per day.
Daily consumption per family remains constant.
No water is added or lost due to leakage.

Common Mistakes

Direct Proportion Error: Using direct variation ( $\frac{20}{6} = \frac{30}{x} \implies x = 9\text{ days}$ ) instead of inverse variation. More families consume the water faster, so the number of days must decrease.
Incorrect Total Families: Calculating with $10$ families instead of adding them to the existing $20$ families to get a total of $30$ families.

More questions in FIO

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Q10

Fill in the empty cells if $x$ and $y$ are in inverse proportion.

Q11

Which of the following pairs of quantities are in inverse proportion?

(i) The number of taps filling a water tank and the time taken to fill it.

(ii) The number of painters hired and the days needed to paint a wall of fixed size.

(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.

(v) The length of cloth bought and the price paid at a fixed rate per metre.

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

Q12

If 24 pencils cost ₹120, how much will 20 such pencils cost?

Q13

Q14

Fill in the average number of hours each living being sleeps in a day by looking at the charts. Select the appropriate hours from this list: 15, 2.5, 20, 8, 3.5, 13, 10.5, 18.

Q15

The pie chart on the right shows the result of a survey carried out to find the modes of transport used by children to go to school. Study the pie chart and answer the following questions.

(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

Three workers can paint a fence in 4 days. If one more worker joins the team, how many days will it take them to finish the work? What are the assumptions you need to make?

Q17

It takes 6 hours to fill 2 tanks of the same size with a pump. How long will it take to fill 5 such tanks with the same pump?

Q18

A given set of chairs are arranged in 25 rows, with 12 chairs in each row. If the chairs are rearranged with 20 chairs in each row, how many rows does this new arrangement have?

Q19

A school has 8 periods a day, each of 45 minutes duration. How long is each period, if the school has 9 periods a day, assuming that the number of school hours per day stays the same?

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

A factory requires 42 machines to produce a given number of toys in 63 days. How many machines are required to produce the same number of toys in 54 days?

Q22

A car takes 2 hours to reach a destination, travelling at a speed of $60 \text{ km/h}$ . How long will the car take if it travels at a speed of $80 \text{ km/h}$ ?