Exercise 3.2

Chapter 3: Ratio and Proportion

Class 6 — Solutions Key

Exercise 3.2 — Practice Problems

1. Fill in the blanks for the given equivalent ratios.

(i) 3 : 5 = 9 : 15

Explanation: Since $3 \times 3 = 9$, we also multiply the second term by 3: $5 \times 3 = 15$.

(ii) 4 : 5 = 8 : 10

Explanation: Since $5 \times 2 = 10$, we also multiply the first term by 2: $4 \times 2 = 8$.

(iii) 6 : 12 = 1 : 2

Explanation: Since $1 \times 6 = 6$, we also multiply the second term by 6: $2 \times 6 = 12$.

2. Complete the table.

(i) Feet to Inch Conversion (Ratio of 1 : 12):

Feet	1	2	3	6
Inch	12	24	36	72

Calculations:
• For 3 feet: $3 \times 12 = \mathbf{36}$ inches.
• For 72 inches: $72 \div 12 = \mathbf{6}$ feet.

(ii) Days to Weeks Conversion (Ratio of 7 : 1):

Days	28	21	14	63
Weeks	4	3	2	9

Calculations:
• For 2 weeks: $2 \times 7 = \mathbf{14}$ days.
• For 63 days: $63 \div 7 = \mathbf{9}$ weeks.

3. Say True or False.

(i) 5 : 7 is equivalent to 21 : 15

Answer: False

Reasoning: Simplifying $21 : 15$ by dividing both terms by 3 gives $7 : 5$. The ratio $5 : 7$ is not equal to $7 : 5$.

(ii) If 40 is divided in the ratio 3 : 2, then the larger part is 24.

Answer: True

Reasoning: Total parts = $3 + 2 = 5$. One part value = $40 \div 5 = 8$. The larger part is $3 \times 8 = 24$.

4. Give two equivalent ratios for each of the following.

Note: Equivalent ratios are found by multiplying or dividing both terms by the same non-zero number.

(i) 3 : 2

• Multiply by 2: $(3\times2) : (2\times2) = \mathbf{6 : 4}$
• Multiply by 3: $(3\times3) : (2\times3) = \mathbf{9 : 6}$
• Answer: 6 : 4 and 9 : 6

(ii) 1 : 6

• Multiply by 2: $(1\times2) : (6\times2) = \mathbf{2 : 12}$
• Multiply by 3: $(1\times3) : (6\times3) = \mathbf{3 : 18}$
• Answer: 2 : 12 and 3 : 18

(iii) 5 : 4

• Multiply by 2: $(5\times2) : (4\times2) = \mathbf{10 : 8}$
• Multiply by 3: $(5\times3) : (4\times3) = \mathbf{15 : 12}$
• Answer: 10 : 8 and 15 : 12

5. Which of the two ratios is larger?

(i) 4 : 5 or 8 : 15

• Convert to fractions: $\frac{4}{5}$ and $\frac{8}{15}$
• Make denominators equal (LCM of 5 and 15 is 15): $\frac{4\times3}{5\times3} = \frac{12}{15}$
• Compare: $\frac{12}{15} > \frac{8}{15}$ ⇒ 4 : 5 is larger

(ii) 3 : 4 or 7 : 8

• Convert to fractions: $\frac{3}{4}$ and $\frac{7}{8}$
• Make denominators equal (LCM is 8): $\frac{3\times2}{4\times2} = \frac{6}{8}$
• Compare: $\frac{7}{8} > \frac{6}{8}$ ⇒ 7 : 8 is larger

(iii) 1 : 2 or 2 : 1

• Convert to fractions: $\frac{1}{2}$ ($0.5$) and $\frac{2}{1}$ ($2.0$)
• Compare: $\frac{2}{1} > \frac{1}{2}$ ⇒ 2 : 1 is larger

6. Divide the numbers given below in the required ratio.

(i) 20 in the ratio 3 : 2

• Total parts = $3 + 2 = 5$
• Value of 1 part = $20 \div 5 = 4$
• First part = $3 \times 4 = 12$; Second part = $2 \times 4 = 8$
• Answer: 12 and 8

(ii) 27 in the ratio 4 : 5

• Total parts = $4 + 5 = 9$
• Value of 1 part = $27 \div 9 = 3$
• First part = $4 \times 3 = 12$; Second part = $5 \times 3 = 15$
• Answer: 12 and 15

(iii) 40 in the ratio 6 : 14

• Total parts = $6 + 14 = 20$
• Value of 1 part = $40 \div 20 = 2$
• First part = $6 \times 2 = 12$; Second part = $14 \times 2 = 28$
• Answer: 12 and 28

7. In a family, the amount spent in a month for buying Provisions and Vegetables are in the ratio 3 : 2. If the allotted amount is ₹4000, then what will be the amount spent for (i) Provisions and (ii) Vegetables?

• Ratio = $3 : 2$ (Provisions : Vegetables)
• Total parts = $3 + 2 = 5$ parts
• Total amount = ₹4000
• Value of 1 part = $4000 \div 5 = ₹800$
• (i) Amount spent for Provisions: $3 \times 800 = \mathbf{₹2400}$
• (ii) Amount spent for Vegetables: $2 \times 800 = \mathbf{₹1600}$

8. A line segment 63 cm long is to be divided into two parts in the ratio 3 : 4. Find the length of each part.

• Total length = $63$ cm
• Ratio = $3 : 4$
• Total parts = $3 + 4 = 7$ parts
• Length of 1 part = $63 \div 7 = 9$ cm
• First part: $3 \times 9 = \mathbf{27\text{ cm}}$
• Second part: $4 \times 9 = \mathbf{36\text{ cm}}$