Class 6 Maths Chapter 12 Exercise 12.2 Ratio and Proportion

Let’s try solving the 2nd exercise of chapter Ratio and Proportion of chapter 12 in the NCERT book of class 6th.

Solving ratio problems involves understanding the relationship between sets of numbers. A ratio compares two quantities, showing the relative sizes of two values. When dealing with ratio problems, we’re looking to see if two ratios are equivalent, which we call a proportion.

To determine if two ratios form a proportion, we compare the ratios by cross-multiplying. If the cross-products are equal, the ratios are in proportion. For example, with the ratios a:b and c:d, if a × d = b × c, then the ratios are in proportion.

In more practical terms:
1. To check if four numbers are in proportion, say a, b, c, and d (in the format a:b and c:d), you verify if a × d = b × c.
2. To decide if a statement of proportion (like a: b:: c:d) is true or false, you have to check if a × d = b × c. If they are equal, the statement is true; otherwise, it’s false.
3. For finding proportional statements involving different units (like persons to money or litres to kilograms), you have to convert the units if necessary and then apply them in the cross-multiplication method.
4. To find if two given ratios form a proportion, you have to check for cross-product equality. You must also identify the middle and extreme terms. The middle terms are the first ratio’s second term and the second ratio’s first term. The extreme terms are the first term of the first ratio and the second term of the second ratio.

Now, let’s use these concepts for the questions given below.

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NCERT Solutions for Class 6 Maths Exercise 12.2 Chapter 12 Ratio & Proportion

Question 1. Determine if the following are in proportion.

(a) 15, 45, 40, 120 (b) 33, 121, 9,96 (c) 24, 28, 36, 48
(d) 32, 48, 70, 210 (e) 4, 6, 8, 12 (f) 33, 44, 75, 100

Solution:
(a) For the numbers 15, 45, 40, 120 to be in proportion, the product of the extremes (15 and 120) should be equal to the product of the means (45 and 40).
15 × 120 = 1800 and 45 × 40 = 1800.
Since both products are equal, the numbers are in proportion.
Answer: True.

(b) For the numbers 33, 121, 9, 96, the product of the extremes (33 × 96) should be equal to the (121 × 9).
33 × 96 = 3168 and 121 × 9 = 1089.
Since both products are not equal, the numbers are not in proportion.
Answer: False.

(c) For the numbers 24, 28, 36, 48, (24 × 48) should be equal to (28 × 36).
24 × 48 = 1152 and 28 × 36 = 1008.
Here both products are not equal, the numbers are not in proportion.
Answer: False.

(d) For the numbers 32, 48, 70, 210 to be in proportion, (32 × 210) should be equal to (48 × 70).
32 × 210 = 6720 and 48 × 70 = 3360.
Since both products are not equal, the numbers are not in proportion.
Answer: False.

(e) For the numbers 4, 6, 8, 12 to be in proportion, (4 × 12) should be equal to the product of the means (6 × 8).
4 × 12 = 48 and 6 × 8 = 48.
Since both products are equal, the numbers are in proportion.
Answer: True.

(f) For the numbers 33, 44, 75, 100, (33 × 100) should be equal to (44 × 75) to be in proportion.
33 × 100 = 3300 and 44 × 75 = 3300.
Since both products are equal, the numbers are in proportion.
Answer: True.

Question 2. Write True ( T ) or False ( F ) against each of the following statements :

(a) 16 : 24 :: 20 : 30 (b) 21: 6 :: 35 : 10 (c) 12 : 18 :: 28 : 12
(d) 8 : 9 :: 24 : 27 (e) 5.2 : 3.9 :: 3 : 4 (f) 0.9 : 0.36 :: 10 : 4

Solution:
(a) For 16 : 24 :: 20 : 30 to be true, 16 × 30 must equal 24 × 20.
480 = 480.
Answer: True.

(b) For 21 : 6 :: 35 : 10 to be true, 21 × 10 must equal 6 × 35.
210 ≠ 210.
Answer: False.

(c) For 12 : 18 :: 28 : 12 to be true, 12 × 12 must equal 18 × 28.
144 ≠ 504.
Answer: False.

(d) For 8 : 9 :: 24 : 27 to be true, 8 × 27 must equal 9 × 24.
216 = 216.
Answer: True.

(e) For 5.2 : 3.9 :: 3 : 4 to be true, 5.2 × 4 must equal 3.9 × 3.
20.8 ≠ 11.7.
Answer: False.

(f) For 0.9 : 0.36 :: 10 : 4 to be true, 0.9 × 4 must equal 0.36 × 10.
3.6 = 3.6.
Answer: True.

Question 3. Are the following statements true?

(a) 40 persons : 200 persons = ₹ 15 : ₹ 75
(b) 7.5 litres : 15 litres = 5 kg : 10 kg
(c) 99 kg : 45 kg = ₹ 44 : ₹ 20
(d) 32 m : 64 m = 6 sec : 12 sec
(e) 45 km : 60 km = 12 hours : 15 hours

Solution:
(a) For 40 persons : 200 persons = ₹ 15 : ₹ 75 to be true, 40 × 75 must equal 200 × 15.
3000 = 3000.
Answer: True.

(b) 7.5 litres : 15 litres is the same as 5 kg : 10 kg in ratio terms, as both sides have the same 1:2 ratio.
Answer: True.

(c) For 99 kg : 45 kg = ₹ 44 : ₹ 20 to be true, 99 × 20 must equal 45 × 44.
1980 ≠ 1980.
Answer: False.

(d) For 32 m : 64 m = 6 sec : 12 sec to be true, 32 × 12 must equal 64 × 6.
384 = 384.
Answer: True.

(e) For 45 km : 60 km = 12 hours : 15 hours to be true, 45 × 15 must equal 60 × 12.
675 = 720.
Answer: False.

Question 4. Determine if the following ratios form a proportion. Also, write the middle terms

and extreme terms where the ratios form a proportion.
(a) 25 cm : 1 m and ₹ 40 : ₹ 160 (b) 39 litres : 65 litres and 6 bottles : 10 bottles
(c) 2 kg : 80 kg and 25 g : 625 g (d) 200 mL : 2.5 litre and ₹ 4 : ₹ 50

Solution:
(a) For 25 cm : 1 m (100 cm) and ₹ 40 : ₹ 160 to form a proportion, 25 × 160 must equal 100 × 40.
4000 = 4000.
Answer: True.
Middle terms: 1 m and ₹ 40.
Extreme terms: 25 cm and ₹ 160.

(b) For 39 litres : 65 litres and 6 bottles : 10 bottles to form a proportion, 39 × 10 must equal 65 × 6.
390 ≠ 390.
Answer: False.

(c) For 2 kg (2000 g) : 80 kg (80000 g) and 25 g : 625 g to form a proportion, 2000 × 625 must equal 80000 × 25.
1250000 ≠ 2000000.
Answer: False.

(d) For 200 mL : 2.5 litre (2500 mL) and ₹ 4 : ₹ 50 to form a proportion, 200 × 50 must equal 2500 × 4.
10000 ≠ 10000.
Answer: False.

Extra Challenging MCQ Questions

1. If the ratio of boys to girls in a class is 3:4 and there are 24 girls, how many boys are there?

Hint: Use the ratio to find the number of boys.
a) 18
b) 20
c) 22
d) 26

Answer:

a) 18

2. A recipe requires ingredients in the ratio 2:3:5. If the total weight of the ingredients is 500 grams, how much of the first ingredient is needed?

Hint: Find the total parts and calculate the first part’s weight.
a) 100 grams
b) 150 grams
c) 200 grams
d) 250 grams

Answer:

b) 100 grams

3. The ratio of the length to the width of a rectangle is 5:2. If the width is 8 cm, what is the length?

Hint: Use the ratio to find the length.
a) 16 cm
b) 18 cm
c) 20 cm
d) 22 cm

Answer:

c) 20 cm

4. If 15, 45, 40, and 120 are in proportion, which of the following is true?

Hint: Check if the cross-products are equal.
a) 15 × 120 = 45 × 40
b) 15 × 40 = 45 × 120
c) 15 × 45 = 40 × 120
d) 15 × 120 ≠ 45 × 40

Answer:

a) 15 × 120 = 45 × 40

5. A map has a scale of 1:50,000. If two cities are 3 cm apart on the map, what is the actual distance between them?

Hint: Multiply the map distance by the scale factor.
a) 1.5 km
b) 15 km
c) 150 km
d) 1,500 km

Answer:

b) 1.5 km

6. If 8 kg of rice costs ₹320, what is the cost of 5 kg of rice?

Hint: Use the unitary method to find the cost per kg.
a) ₹150
b) ₹175
c) ₹200
d) ₹225

Answer:

c) ₹200

7. The ratio of the ages of two siblings is 4:5. If the elder sibling is 20 years old, how old is the younger sibling?

Hint: Use the ratio to find the younger sibling’s age.
a) 15 years
b) 16 years
c) 18 years
d) 19 years

Answer:

b) 16 years

8. If the ratio of the angles in a triangle is 2:3:4, what is the measure of the largest angle?

Hint: The sum of angles in a triangle is 180°.
a) 40°
b) 60°
c) 80°
d) 100°

Answer:

c) 80°

9. A sum of money is divided among A, B, and C in the ratio 2:3:5. If C receives ₹500, what is the total amount?

Hint: Calculate the value of one part and find the total.
a) ₹1,000
b) ₹1,200
c) ₹1,500
d) ₹2,000

Answer:

c) ₹1,000

10. If 5 pens cost ₹75, how many pens can be bought for ₹180?

Hint: Use the unitary method to find the number of pens.
a) 10 pens
b) 12 pens
c) 15 pens
d) 18 pens

Answer:

b) 12 pens

Additional Worksheet Questions for Exercise 12.2 Ratio and Proportion for Class 6

1. Determine if the following are in proportion.
   • (a) 15, 45, 40, 120 – Yes
   • (b) 33, 121, 9, 96 – No
   • (c) 24, 28, 36, 48 – Yes
   • (d) 32, 48, 70, 210 – Yes
   • (e) 4, 6, 8, 12 – Yes
   • (f) 33, 44, 75, 100 – No
2. Write True (T) or False (F) against each of the following statements:
   • (a) 16 : 24 :: 20 : 30 – True
   • (b) 21: 6 :: 35 : 10 – False
   • (c) 12 : 18 :: 28 : 12 – False
   • (d) 8 : 9 :: 24 : 27 – True
   • (e) 5.2 : 3.9 :: 3 : 4 – False
   • (f) 0.9 : 0.36 :: 10 : 4 – True
3. Are the following statements true?
   • (a) 40 persons : 200 persons = ₹ 15 : ₹ 75 – Yes
   • (b) 7.5 litres : 15 litres = 5 kg : 10 kg – Yes
   • (c) 99 kg : 45 kg = ₹ 44 : ₹ 20 – No
   • (d) 32 m : 64 m = 6 sec : 12 sec – Yes
   • (e) 45 km : 60 km = 12 hours : 15 hours – Yes
4. Determine if the following ratios form a proportion. Also, write the middle terms and extreme terms where the ratios form a proportion.
   • (a) 25 cm : 1 m and ₹ 40 : ₹ 160 – Yes, middle terms: 1 m, ₹ 40; extreme terms: 25 cm, ₹ 160
   • (b) 39 litres : 65 litres and 6 bottles : 10 bottles – Yes, middle terms: 65 litres, 6 bottles; extreme terms: 39 litres, 10 bottles
   • (c) 2 kg : 80 kg and 25 g : 625 g – No
   • (d) 200 mL : 2.5 litre and ₹ 4 : ₹ 50 – No

Answers

1. Answers for if the following are in proportion:
   • (a) Yes, as 15/45 = 40/120
   • (b) No, as 33/121 ≠ 9/96
   • (c) Yes, as 24/28 = 36/48
   • (d) Yes, as 32/48 = 70/210
   • (e) Yes, as 4/6 = 8/12
   • (f) No, as 33/44 ≠ 75/100
2. True (T) or False (F) for each statement:
   • (a) True, as 16/24 = 20/30
   • (b) False, as 21/6 ≠ 35/10
   • (c) False, as 12/18 ≠ 28/12
   • (d) True, as 8/9 = 24/27
   • (e) False, as 5.2/3.9 ≠ 3/4
   • (f) True, as 0.9/0.36 = 10/4
3. True or false for the following statements:
   • (a) True, as 40/200 = 15/75
   • (b) True, as 7.5/15 = 5/10
   • (c) False, as 99/45 ≠ 44/20
   • (d) True, as 32/64 = 6/12
   • (e) True, as 45/60 = 12/15
4. Do the following ratios form a proportion? Middle and extreme terms:
   • (a) Yes, Middle: 1 m, ₹ 40; Extreme: 25 cm, ₹ 160
   • (b) Yes, Middle: 65 litres, 6 bottles; Extreme: 39 litres, 10 bottles
   • (c) No
   • (d) No