Proportional Reasoning - 1 | FIO

Question 16

Last year, we hired 3 buses for the school trip. We had a total of 162 students and teachers who went on that trip and all the buses were full. This year we have 204 students. How many buses will we need? Will all the buses be full?

Solution

Understand the Question

The capacity of a single bus can be found from last year's trip, where 162 people filled 3 buses completely.
Dividing the total number of students this year (204) by the capacity per bus will give the number of full buses and any remaining students.
Any remaining students will require an additional bus, and if there is a remainder, not all buses will be full.

Step 1 · Find the Capacity of One Bus

Given that 162 people filled 3 buses completely: Diagram 1

\begin{aligned} C &= \dfrac{\text{Total people last year}}{\text{Number of buses last year}} \\[0.6em] &= \dfrac{162}{3} \\[0.6em] &= 54 \text{ people per bus} \end{aligned}

Step 2 · Calculate the Number of Buses Needed

For 204 students with a bus capacity of 54:

\begin{aligned} N &= \dfrac{\text{Total students this year}}{\text{Capacity per bus}} \\[0.6em] &= \dfrac{204}{54} \end{aligned}

Dividing 204 by 54 gives: $204 \div 54 = 3 \text{ with a remainder of } 42$

Students in 3 full buses $= 3 \times 54 = 162$
Remaining students $= 204 - 162 = 42$

Since 42 students require an additional bus: $\text{Total buses needed} = 3 + 1 = 4$

Step 3 · Check if All Buses Will Be Full

From the division:

The first 3 buses will be full with 54 students each.
The 4th bus will carry only the remaining 42 students.

Since $42 < 54$ , the 4th bus will not be full.

Answer

(i) We will need $4$ buses.

(ii) No, not all the buses will be full.

Common Mistakes

Rounding Down the Remainder: Dividing $204 \div 54 \approx 3.78$ and rounding down to $3$ buses, leaving $42$ students without transportation. Any non-zero remainder requires an additional bus.
Assuming All Buses Are Full: Forgetting to check whether $204$ is an exact multiple of $54$ before deciding if all buses will be completely occupied.

More questions in FIO

Circle the following statements of proportion that are true.

(i) $4 : 7 :: 12 : 21$

(ii) $8 : 3 :: 24 : 6$

(iii) $7 : 12 :: 12 : 7$

(iv) $21 : 6 :: 35 : 10$

(v) $12 : 18 :: 28 : 12$

(vi) $24 : 8 :: 9 : 3$

Give 3 ratios that are proportional to $4 : 9$ .

Fill in the missing numbers for these ratios that are proportional to $18 : 24$ .

Look at the following rectangles. Which rectangles are similar to each other? You can verify this by measuring the width and height using a scale and comparing their ratios.

Look at the following rectangle. Can you draw a smaller rectangle and a bigger rectangle with the same width to height ratio in your notebooks? Compare your rectangles with your classmates' drawings.

Are all of them the same? If they are different from yours, can you think why? Are they wrong?

The following figure shows a small portion of a long brick wall with patterns made using coloured bricks. Each wall continues this pattern throughout the wall. What is the ratio of grey bricks to coloured bricks? Try to give the ratios in their simplest form.

Let us draw some human figures. Measure your friend's body—the lengths of their head, torso, arms, and legs. Write the ratios as mentioned below—

Now, draw a figure with head, torso, arms, and legs with equivalent ratios as above.

The Earth travels approximately 940 million kilometres around the Sun in a year. How many kilometres will it travel in a week?

A mason is building a house in the shape shown in the diagram. He needs to construct both the outer walls and the inner wall that separates two rooms. To build a wall of $10\text{ ft}$ , he requires approximately $1450$ bricks. How many bricks would he need to build the house? Assume all walls are of the same height and thickness.

Q10

Divide ₹4,500 into two parts in the ratio $2 : 3$ .

Q11

In a science lab, acid and water are mixed in the ratio of $1 : 5$ to make a solution. In a bottle that has $240\text{ mL}$ of the solution, how much acid and water does the solution contain?

Q12

Blue and yellow paints are mixed in the ratio of $3 : 5$ to produce green paint. To produce $40\text{ mL}$ of green paint, how much of these two colours are needed? To make the paint a lighter shade of green, I added $20\text{ mL}$ of yellow to the mixture. What is the new ratio of blue and yellow in the paint?

Q13

To make soft idlis, you need to mix rice and urad dal in the ratio of $2 : 1$ . If you need 6 cups of this mixture to make idlis tomorrow morning, how many cups of rice and urad dal will you need?

Q14

I have one bucket of orange paint that I made by mixing red and yellow paints in the ratio of $3 : 5$ . I added another bucket of yellow paint to this mixture. What is the ratio of red paint to yellow paint in the new mixture?

Q15

Anagh mixes $600\text{ mL}$ of orange juice with $900\text{ mL}$ of apple juice to make a fruit drink. Write the ratio of orange juice to apple juice in its simplest form.

Q16

Q17

The area of Delhi is 1,484 sq. km and the area of Mumbai is 550 sq. km. The population of Delhi is approximately 30 million and that of Mumbai is 20 million people. Which city is more crowded? Why do you say so?

Q18

A crane of height $155 \text{ cm}$ has its neck and the rest of its body in the ratio $4 : 6$ . For your height, if your neck and the rest of the body also had this ratio, how tall would your neck be?

Q19

Let us try an ancient problem from Lilavati. At that time weights were measured in a unit named palas and niskas was a unit of money. “If $2\dfrac{1}{2}$ palas of saffron costs $\dfrac{3}{7}$ niskas, O expert businessman! tell me quickly what quantity of saffron can be bought for 9 niskas?”

Q20

Harmain is a 1-year-old girl. Her elder brother is 5 years old. What will be Harmain’s age when the ratio of her age to her brother’s age is $1 : 2$ ?

Q21

The mass of equal volumes of gold and water are in the ratio $37 : 2$ . If 1 litre of water is $1\text{ kg}$ in mass, what is the mass of 1 litre of gold?

Q22

It is good farming practice to apply 10 tonnes of cow manure for 1 acre of land. A farmer is planning to grow tomatoes in a plot of size 200 ft by 500 ft. How much manure should he buy? (Please refer to the section on Unit Conversions earlier in this chapter).

Q23

A tap takes 15 seconds to fill a mug of water. The volume of the mug is 500 mL. How much time does the same tap take to fill a bucket of water if the bucket has a 10-litre capacity?

Q24

One acre of land costs ₹15,000,000. What is the cost of 2,400 square feet of the same land?

Q25

A tractor can plough the same area of a field 4 times faster than a pair of oxen. A farmer wants to plough his 20-acre field. A pair of oxen takes 6 hours to plough an acre of land. How much time would it take if the farmer used a pair of oxen to plough the field? How much time would it take him if he decides to use a tractor instead?

Q26

The ₹10 coin is an alloy of copper and nickel called ‘cupro-nickel’. Copper and nickel are mixed in a $3 : 1$ ratio to get this alloy. The mass of the coin is $7.74\text{ grams}$ . If the cost of copper is ₹906 per kg and the cost of nickel is ₹1,341 per kg, what is the cost of these metals in a ₹10 coin?