Q: 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

(i) Given statement is 4 : 7 :: 12 : 21. This is true if $\frac{4}{7}=\frac{12}{21}$ or if $\frac{4}{7}=\frac{4}{7}$, which is true. ∴ The given statement is true. (ii) Given statement is 8 : 3 :: 24 : 6. This is true if $\frac{8}{3}=\frac{24}{6}$ or if $\frac {8}{3}$ = 4 , which is false. ∴ The given statement is not true. (iii) Given statement is 7 : 12 :: 12 : 7. This is true if $\frac{7}{12}=\frac{12}{7}$, which is false. ∴ The given statement is not true. (iv) Given statement is 21 : 6 :: 35 : 10. This is true if $\frac{21}{6}=\frac{35}{10}$ or if $\frac{7}{2}=\frac{7}{2}$, which is true. ∴ The given statement is true. (v) Given statement is 12 : 18 :: 28 : 12. This is true if $\frac{12}{18}=\frac{28}{12}$ or if $\frac{2}{3}=\frac{7}{3}$ or 2 = 7, which is false. ∴ The given statement is not true. (vi) Given statement is 24 : 8 :: 9 : 3. This is true if $\frac{24}{8}=\frac{9}{3}$ Or if 3 = 3, which is true. ∴ The given statement is true.

Q: Give 3 ratios that are proportional to 4 : 9. ________ : ________ ________ : ________ ________ : ________

The given ratio is 4 : 9. We have $\frac{4}{9}=\frac{4 \times 2}{9 \times 2}=\frac{4 \times 3}{9 \times 3}=\frac{4 \times 4}{9 \times 4}$ or $\frac{4}{9}=\frac{8}{18}=\frac{12}{27}=\frac{16}{36}$ ∴ 4 : 9 :: 8 : 18, 4 : 9 :: 12 : 27, and 4 : 9 :: 16 : 36 ∴ Ratios 8 : 18, 12 : 27 and 16 : 36 are proportional to the given ratio 4 : 9.

Question 1

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

Accepted Answer

(i) Given statement is 4 : 7 :: 12 : 21.
This is true if $\frac{4}{7}=\frac{12}{21}$
or if $\frac{4}{7}=\frac{4}{7}$, which is true.
∴ The given statement is true.
(ii) Given statement is 8 : 3 :: 24 : 6.
This is true if $\frac{8}{3}=\frac{24}{6}$
or if $\frac {8}{3}$ = 4 , which is false.
∴ The given statement is not true.
(iii) Given statement is 7 : 12 :: 12 : 7.
This is true if $\frac{7}{12}=\frac{12}{7}$, which is false.
∴ The given statement is not true.
(iv) Given statement is 21 : 6 :: 35 : 10.
This is true if $\frac{21}{6}=\frac{35}{10}$
or if $\frac{7}{2}=\frac{7}{2}$, which is true.
∴ The given statement is true.
(v) Given statement is 12 : 18 :: 28 : 12.
This is true if $\frac{12}{18}=\frac{28}{12}$
or if $\frac{2}{3}=\frac{7}{3}$
or 2 = 7, which is false.
∴ The given statement is not true.
(vi) Given statement is 24 : 8 :: 9 : 3.
This is true if $\frac{24}{8}=\frac{9}{3}$
Or if 3 = 3, which is true.
∴ The given statement is true.

Question 2

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

Accepted Answer

The given ratio is 4 : 9.
We have $\frac{4}{9}=\frac{4 \times 2}{9 \times 2}=\frac{4 \times 3}{9 \times 3}=\frac{4 \times 4}{9 \times 4}$
or $\frac{4}{9}=\frac{8}{18}=\frac{12}{27}=\frac{16}{36}$
∴ 4 : 9 :: 8 : 18, 4 : 9 :: 12 : 27, and 4 : 9 :: 16 : 36
∴ Ratios 8 : 18, 12 : 27 and 16 : 36 are proportional to the given ratio 4 : 9.

Question 3

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?

Accepted Answer

Ratio of blue and yellow paints = 3 : 5
Quantity of green paint = 40 mL
∴ Quantity of blue paint = $\frac{3}{3+5} \times 40$
= $\frac {3}{8}$ × 40
= 15 mL
∴ Quantity of yellow paint = $\frac{5}{3+5} \times 40$
= $\frac {5}{8}$ × 40
= 25 mL
Addition of yellow paint to the mixture = 20 mL
∴ New quantity of blue paint = 15 mL
∴ New quantity of yellow paint = 25 mL + 20 mL = 45 mL
∴ New ratio of blue and yellow paints = 15 : 45 = 1 : 3.

Question 4

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.

Accepted Answer

Using a scale, we measure the width and height of given rectangles.
Here, the ratio ‘Width : Height’ for given rectangles A, B, C, D, and E are respectively 1 : 3, 3 : 2, 9 : 4, 7 : 2 and 3 : 1.
These ratios are all distinct. The ratios of A and E are 1 : 3 and 3 : 1 respectively.
∴ For A and E, one side is 3 times the other side.
∴ Only rectangles A and E are similar.

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 notebook? 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?

Accepted Answer

For the given rectangle;
Width = 32 mm and height = 18 mm
∴ Ratio is 32 : 18.
We shall draw smaller and bigger rectangles and similar to the given rectangle by considering different ‘factors of change’.
Let the factor of change be $\frac {1}{2}$.
∴ New width = $\frac {1}{2}$ × 32 = 16 mm
and New height = $\frac {1}{2}$ × 18 = 9 mm
A new, similar rectangle is shown in the figure.
Let ‘factor of change’ be 2.
∴ New width = 2 × 32 = 64 mm and new height = 2 × 18 = 36 mm
A new, similar rectangle is shown in the figure. The rectangles drawn by other classmates are all different, but they are all similar to the given rectangle.

Question 6

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.

Accepted Answer

(a) We consider one set of patterns in the given wall.
Number of grey bricks in one set of pattern = 2 + 3 + 4 = 9
Number of coloured bricks in one set of pattern = 3 + 2 + 1 = 6
∴ Ratio of grey bricks to coloured bricks = 9 : 6
We have 9 : 6 = 3 : 2
∴ Ratio in the simplest form = 3 : 2
(b) We use one set of patterns on the given wall
Number of grey bricks in one set of pattern
= 3 + 2 + 2 + 2 + 2 + 2 + 3
= 16
Number of coloured bricks in one set of pattern = 1 + (1 + 1) + (1 + 1) + (1 + 1) + (1 + 1) + (1 + 1) + 1
= 1 + 2 + 2 + 2 + 2 + 2 + 1
= 12
∴ Ratio of grey bricks to coloured bricks = 16 : 12
We have 16 : 12 = 4 : 3
∴ Ratio in the simplest form = 4 : 3.
Trairasika – The Rule of Three
Example 8.
For the mid-day meal in a school with 120 students, the cook usually makes 15 kg of rice. On a rainy day, only 80 students came to school. How many kilograms of rice should the cook make so that the food is not wasted?
The ratio of the number of students to the amount of rice needs to be proportional. So, 120 : 15 :: 80 : ?
What is the factor of change in the first term? (Page 167)
Solution:
For 120 students, the rice required is 15 kg.
Let x kg of rice be required for 80 students.
∴ Ratios 120 : 15 and 80 : x are in proportion.
∴ 120 : 15 :: 80 : x
⇒ $\frac{120}{15}=\frac{80}{x}$
⇒ x = 10
∴ 10 kg of rice is required.
Also, factor of change in the first term = $\frac{80}{120}=\frac{2}{3}$
Alternative Method:
Factor of change in second term = $\frac {x}{15}$
Since the quality of food is the same, we have $\frac{2}{3}=\frac{x}{15}$
⇒ 3x = 30
⇒ x = 10
∴ 10 kg of rice is required.
Example 9.
(i) A car travels 90 km in 150 minutes. If it continues at the same speed, what distance will it cover in 4 hours?
If it continues at the same speed, the ratio of the time taken should be proportional to the ratio of the distance covered.
(ii) 150 : 90 :: 4 : x
Is this the right way to formulate the question?
(iii) How can you find the distance covered in 240 minutes? (Page 169)
Solution:
(i) We have, 4 hours = 4 × 60 = 240 minutes
In 150 minutes, the distance covered = 90 km
Let x km be covered in 4 hours, i.e., in 240 minutes.
∴ The ratios 150 : 90 and 240 : x are in proportion.
∴ 150 : 90 :: 240 : x
(ii) Since units must be the same in comparing ratios, the given proportion 150 : 90 :: 4 : ? is meaningless.
We have 4 hours = 240 minutes
∴ The proportion 150 : 90 :: 240 : ? is correct.
(iii) We have 150 : 90 :: 240 : x
⇒ $\frac{150}{90}=\frac{240}{x}$
⇒ $\frac{5}{3}=\frac{240}{x}$
⇒ 5x = 3 × 240
⇒ x = 144
∴ Distance covered in 4 hours = 144 km.
Example 10.
A small farmer in Himachal Pradesh sells each 200 g packet of tea for ₹ 200. A large estate in Meghalaya sells each 1 kg packet of tea for ₹ 800. Are the weight-to-price ratios in both places proportional? Which tea is more expensive? Why? (Page 169)
Solution:
We have 1 kg = 1000 g
In Himachal Pradesh, 200 g of tea costs ₹ 200.
∴ Ratio of weight to price in Himachal Prdesh = 200 : 200 = 1 : 1
In Meghalaya, 1000 g tea costs ₹ 800.
∴ Ratio of weight to price in Meghalaya = 1000 : 800 = 5 : 4
The ratios 1 : 1 and 5 : 4 are not proportional, because $\frac{1}{1} \neq \frac{5}{4}$
Price of 200 g tea in Himachal Pradesh = ₹ 200
Price of 1000 g tea in Himachal Pradesh = $\frac {200}{200}$ × 1000 = ₹ 1000
Since 1000 > 800, tea is more expensive in Himachal Pradesh.
Figure It Out (Pages 170-171)

Question 7

The mass of equal volumes of gold and water is in the ratio 37 : 2. If 1 litre of water is 1 kg in mass, what is the mass of liter of gold?

Accepted Answer

The ratio of masses of gold and water, when their volumes are the same, is 37 : 2.
Mass of 1 litre of water = 1 kg
Let the mass of 1 litre of gold = x kg
∴ With equal volumes, the ratio of masses of gold and water is x : 1.
These ratios are in proportion.
∴ 37 : 2 :: 1 : x
⇒ $\frac{37}{2}=\frac{x}{1}$
⇒ x = $\frac {37}{2}$
Thus, the mass of 1 litre of gold is $\frac {37}{2}$ kg.

Question 8

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).

Accepted Answer

We have 1 ton = 1,000 kg
∴ 10 tonnes = 10 × 1,000 = 10,000 kg
Also, 1 acre = 43,560 sq. ft.
∴ Ratio of cow manure to area of land in kg and sq. ft. = 10,000 : 43,560
Size of plot = 200 ft. by 500 ft.
∴ Area of plot = 200 × 500 = 1,00,000 sq. ft.
Let cow manure be x kg.
∴ Ratio of cow manure to area of plot = x : 1,00,000
These ratios are in proportion.
∴ 10,000 : 43,560 :: x : 1,00,000
⇒ $\frac{10,000}{43,560}=\frac{x}{1,00,000}$
⇒ 43,560x = 10,000 × 1,00,000 = 1,00,00,00,000
⇒ x = $\frac {1,00,00,00,000}{43,560}$
⇒ x = 22956.84
∴ Required cow manure = 22956.84 kg = 22.95684 tonnes.

Question 9

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?

Accepted Answer

Time taken by the tap for 500 mL of water = 15 seconds
∴ Ratio of volume to time = 500 : 15
We know 1 litre = 1,000 mL
10 litre = 10 × 1,000 = 10,000 mL
Let the time taken to fill a bucket of 10,000 mL be x seconds.
∴ Ratio of volume to time = 10,000 : x
These ratios are proportional.
∴ 500 : 15 :: 10,000 : x
⇒ $\frac{500}{15}=\frac{10,000}{x}$
⇒ 500x = 1,50,000
⇒ x = 300
∴ Time to fill bucket = 300 seconds
= $\frac {300}{60}$ minutes
= 5 minutes.

Question 10

One acre of land costs ₹ 15,00,000. What is the cost of 2,400 square feet of the same land?

Accepted Answer

We know that 1 acre = 43,560 square feet.
∴ Cost of 43,560 sq. ft. land = ₹ 15,00,000
∴ Ratio of area of land to cost = 43,560 : 15,00,000
Let the cost of 2,400 sq. ft. of land be ₹ x.
∴ Ratio of area of land to cost = 2,400 : x
These ratios are proportional.
∴ 43,560 : 15,00,000 :: 2,400 : x
⇒ $\frac{43,560}{15,00,000}=\frac{2,400}{x}$
⇒ 43,560x = 2,400 × 15,00,000
⇒ x = $\frac{2,400 \times 15,00,000}{43,560}$
⇒ x = 82,664.63
∴ Cost of land = ₹ 82,664.63.

Question 11

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?

Accepted Answer

Ratio of efficiency of a tractor to a pair of oxen = 4 : 1
Time taken by a pair of oxen to plough 1 acre of field = 6 hours
∴ Time taken by a tractor to plough 1 acre field = $\frac {6}{4}$ = 1.5 hours
∴ Time taken by a pair of oxen to plough 20 20-acre field = 20 × 6 = 120 hours
∴ Time taken by a tractor to plough a 20-acre field = 20 × 1.5 = 30 hours

Question 12

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?

Accepted Answer

Ratio of copper and nickel in ₹ 10 coin = 3 : 1
Mass of one ₹ 10 coin = 7.74 grams
∴ Mass of copper in one ₹ 10 coin = $\frac{3}{3+1}$ × 7.74
= $\frac {3}{4}$ × 7.74
= 5.805 grams
Mass of nickel in one ₹ 10 coin = $\frac{1}{3+1}$ × 7.74
= $\frac {1}{4}$ × 7.74
= 1.935 grams
Cost of 1 kg copper = ₹ 906
∴ Cost of 1000 grams of copper = ₹ 906
∴ Cost of 5.805 grams copper = $\frac {906}{1000}$ × 5.805 = ₹ 5.26
Cost of 1 kg nickel = ₹ 1341
∴ Cost of 1000 grams of nickel = ₹ 1341
∴ Cost of 1.935 grams nickel = $\frac {1341}{1000}$ × 1.935 = ₹ 2.59
∴ In one ₹ 10 coin, the cost of copper and the cost of nickel are respectively ₹ 5.26 and ₹ 2.59.

Proportional Reasoning 1