Proportional Reasoning - 1 | FIO

Question 5

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?

Solution

We need to draw rectangles that have the same shape but different sizes.

Step 1 — Find the ratio of the given rectangle

First, we measure the width and height of the given rectangle. Let's say the width of the given rectangle is $W_1$ . Let's say the height of the given rectangle is $H_1$ . The width to height ratio is $W_1 : H_1$ . For example, if we measure the given rectangle and find its width is $4 \text{ cm}$ and its height is $3 \text{ cm}$ . The ratio of width to height is:

$W_1 : H_1 = 4 \text{ cm} : 3 \text{ cm}$

$= 4 : 3$

$\boxed{\text{Ratio} = 4:3}$

Diagram 1

Step 2 — Draw smaller and bigger rectangles

To draw a smaller or bigger rectangle with the same ratio, we need to multiply both the width and height by the same number. This number is called the scale factor.

For a smaller rectangle, we use a scale factor less than 1. Let's use a scale factor of $0.5$ . New width $W_2$ :

$W_2 = W_1 \times 0.5$

$= 4 \text{ cm} \times 0.5$

$= 2 \text{ cm}$

New height $H_2$ :

$H_2 = H_1 \times 0.5$

$= 3 \text{ cm} \times 0.5$

$= 1.5 \text{ cm}$

So, a smaller rectangle could be $2 \text{ cm}$ wide and $1.5 \text{ cm}$ high.

For a bigger rectangle, we use a scale factor greater than 1. Let's use a scale factor of $2$ . New width $W_3$ :

$W_3 = W_1 \times 2$

$= 4 \text{ cm} \times 2$

$= 8 \text{ cm}$

New height $H_3$ :

$H_3 = H_1 \times 2$

$= 3 \text{ cm} \times 2$

$= 6 \text{ cm}$

So, a bigger rectangle could be $8 \text{ cm}$ wide and $6 \text{ cm}$ high.

Step 3 — Compare rectangles

When comparing our rectangles with classmates' drawings, they might look different.

This is because:

Different sizes: Some might choose a larger scale factor, making their "bigger" rectangle much larger than yours. Others might choose a smaller scale factor for their "smaller" rectangle.
Different orientations: Some might draw their rectangles vertically (height > width), while others draw them horizontally (width > height). As long as the ratio of the longer side to the shorter side (or vice versa) is the same, the shape is similar.
Different initial measurements: If classmates measured the original rectangle slightly differently, their base ratio might be slightly different.

All these rectangles are correct as long as they maintain the same width to height ratio as the original rectangle. For example, if the original ratio is $4:3$ , then rectangles with dimensions $8:6$ , $12:9$ , or $2:1.5$ are all correct. They are all similar shapes.

A rectangle would be wrong only if its width to height ratio is different from the original rectangle's ratio. For instance, a rectangle with dimensions $5:3$ would be a different shape from a $4:3$ rectangle.

Answer

(i) Yes, we can draw a smaller rectangle and a bigger rectangle with the same width to height ratio. (ii) No, not all rectangles drawn by classmates will look exactly the same as yours. (iii) They may look different because they can be of different sizes (scaled differently) or drawn in different orientations (horizontal or vertical). They are not wrong as long as they maintain the correct width to height ratio.

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-feet, 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 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 mL of green paint, how much of these two colours are needed? To make the paint a lighter shade of green, I added 20 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 mL of orange juice with 900 mL of apple juice to make a fruit drink. Write the ratio of orange juice to apple juice in its simplest form.

Q16

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?

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 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 $2\frac{1}{2}$ palas of saffron costs $\frac{3}{7}$ 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 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 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?