GSEB Class 6 Maths Solutions Chapter 12 Ratio and Proportion Exercise 12.2

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
Answer:
(a) For 15, 45, 40, 120:
Ratio of 15 and 45 is \( 15:45 \).
\( \frac{15}{45} = \frac{15 \div 15}{45 \div 15} = \frac{1}{3} = 1:3 \)
(The HCF of 15 and 45 is 15).
Ratio of 40 and 120 is \( 40:120 \).
\( \frac{40}{120} = \frac{40 \div 40}{120 \div 40} = \frac{1}{3} = 1:3 \)
(The HCF of 40 and 120 is 40).
Since both ratios are equal, 15, 45, 40, and 120 are in proportion.
In simple words: We check if the ratio of the first two numbers is the same as the ratio of the last two numbers. If they are equal, the numbers are in proportion.

Exam Tip: To find if numbers are in proportion, always simplify both ratios to their simplest form and then compare them directly.

Answer:
(b) For 33, 121, 9, 96:
Ratio of 33 and 121 is \( 33:121 \).
\( \frac{33}{121} = \frac{33 \div 11}{121 \div 11} = \frac{3}{11} = 3:11 \)
(The HCF of 33 and 121 is 11).
Ratio of 9 and 96 is \( 9:96 \).
\( \frac{9}{96} = \frac{9 \div 3}{96 \div 3} = \frac{3}{32} = 3:32 \)
(The HCF of 9 and 96 is 3).
Since \( 3:11 \neq 3:32 \), the numbers 33, 121, 9, and 96 are not in proportion.
In simple words: We compare the first ratio (33 to 121) and the second ratio (9 to 96). Since these two ratios are not the same, the numbers are not proportional.

Exam Tip: Be careful with simplification; ensure you divide by the correct HCF (Highest Common Factor) for each pair to get the simplest ratio.

Answer:
(c) For 24, 28, 36, 48:
Ratio of 24 and 28 is \( 24:28 \).
\( \frac{24}{28} = \frac{24 \div 4}{28 \div 4} = \frac{6}{7} = 6:7 \)
(The HCF of 24 and 28 is 4).
Ratio of 36 and 48 is \( 36:48 \).
\( \frac{36}{48} = \frac{36 \div 12}{48 \div 12} = \frac{3}{4} = 3:4 \)
(The HCF of 36 and 48 is 12).
Since \( 6:7 \neq 3:4 \), the numbers 24, 28, 36, and 48 are not in proportion.
In simple words: We check if 24 compared to 28 is the same as 36 compared to 48. Because these comparisons give different results, the numbers are not in proportion.

Exam Tip: Even if one ratio seems simpler, always reduce both ratios to their lowest terms for accurate comparison.

Answer:
(d) For 32, 48, 70, 210:
Ratio of 32 and 48 is \( 32:48 \).
\( \frac{32}{48} = \frac{32 \div 16}{48 \div 16} = \frac{2}{3} = 2:3 \)
(The HCF of 32 and 48 is 16).
Ratio of 70 and 210 is \( 70:210 \).
\( \frac{70}{210} = \frac{70 \div 70}{210 \div 70} = \frac{1}{3} = 1:3 \)
(The HCF of 70 and 210 is 70).
Since \( 2:3 \neq 1:3 \), the numbers 32, 48, 70, and 210 are not in proportion.
In simple words: The first pair (32 and 48) has a different relationship than the second pair (70 and 210), so they are not in proportion.

Exam Tip: Double-check your HCF calculation to ensure the ratios are simplified correctly before making the final comparison.

Answer:
(e) For 4, 6, 8, 12:
Ratio of 4 and 6 is \( 4:6 \).
\( \frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3} = 2:3 \)
(The HCF of 4 and 6 is 2).
Ratio of 8 and 12 is \( 8:12 \).
\( \frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3} = 2:3 \)
(The HCF of 8 and 12 is 4).
Since both ratios are equal, 4, 6, 8, and 12 are in proportion.
In simple words: Both pairs of numbers, 4 and 6, and 8 and 12, show the same relationship when simplified, which means they are in proportion.

Exam Tip: Remember that two ratios form a proportion if, after simplification, their values are identical.

Answer:
(f) For 33, 44, 75, 100:
Ratio of 33 and 44 is \( 33:44 \).
\( \frac{33}{44} = \frac{33 \div 11}{44 \div 11} = \frac{3}{4} = 3:4 \)
(The HCF of 33 and 44 is 11).
Ratio of 75 and 100 is \( 75:100 \).
\( \frac{75}{100} = \frac{75 \div 25}{100 \div 25} = \frac{3}{4} = 3:4 \)
(The HCF of 75 and 100 is 25).
Since both ratios are equal, 33, 44, 75, and 100 are in proportion.
In simple words: When we simplify the ratio of 33 to 44, and the ratio of 75 to 100, both give us the same result. This indicates that these numbers are proportional.

Exam Tip: Proportionality requires both pairs of numbers to have the exact same ratio after full reduction.

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) \( 0.9:0.36 :: 10:4 \)
Answer:
(a) For \( 16:24 :: 20:30 \):
First ratio: \( 16:24 = \frac{16}{24} = \frac{16 \div 8}{24 \div 8} = \frac{2}{3} = 2:3 \)
(The HCF of 16 and 24 is 8).
Second ratio: \( 20:30 = \frac{20}{30} = \frac{20 \div 10}{30 \div 10} = \frac{2}{3} = 2:3 \)
(The HCF of 20 and 30 is 10).
Since \( 16:24 = 20:30 \), the statement \( 16:24 :: 20:30 \) is True.
In simple words: The simplified ratio of 16 to 24 is the same as the simplified ratio of 20 to 30, so the statement is correct.

Exam Tip: Remember that '::' means 'is in proportion to', so you are verifying if the two given ratios are equal.

Answer:
(b) For \( 21:6 :: 35:10 \):
First ratio: \( 21:6 = \frac{21}{6} = \frac{21 \div 3}{6 \div 3} = \frac{7}{2} = 7:2 \)
(The HCF of 21 and 6 is 3).
Second ratio: \( 35:10 = \frac{35}{10} = \frac{35 \div 5}{10 \div 5} = \frac{7}{2} = 7:2 \)
(The HCF of 35 and 10 is 5).
Since \( 21:6 = 35:10 \), the statement \( 21:6 :: 35:10 \) is True.
In simple words: After simplifying, both the ratio of 21 to 6 and the ratio of 35 to 10 are identical. Therefore, the statement claiming they are proportional is accurate.

Exam Tip: Always reduce fractions to their simplest form to avoid errors in comparison, especially with numbers that share common factors.

Answer:
(c) For \( 12:18 :: 28:12 \):
First ratio: \( 12:18 = \frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3} = 2:3 \)
(The HCF of 12 and 18 is 6).
Second ratio: \( 28:12 = \frac{28}{12} = \frac{28 \div 4}{12 \div 4} = \frac{7}{3} = 7:3 \)
(The HCF of 28 and 12 is 4).
Since \( 2:3 \neq 7:3 \), the statement \( 12:18 :: 28:12 \) is not True (False).
In simple words: The ratio of 12 to 18 does not match the ratio of 28 to 12. This means the statement that they are proportional is incorrect.

Exam Tip: Clearly show the simplification steps for both ratios to demonstrate your working and avoid losing marks.

Answer:
(d) For \( 8:9 :: 24:27 \):
First ratio: \( 8:9 = \frac{8}{9} \)
(8 and 9 have an HCF of 1, so this ratio is already in its simplest form).
Second ratio: \( 24:27 = \frac{24}{27} = \frac{24 \div 3}{27 \div 3} = \frac{8}{9} = 8:9 \)
(The HCF of 24 and 27 is 3).
Since \( 8:9 = 24:27 \), the statement \( 8:9 :: 24:27 \) is True.
In simple words: Both the ratio of 8 to 9 and the simplified ratio of 24 to 27 are the same. Thus, the given statement is true.

Exam Tip: If one ratio is already in its simplest form, you only need to simplify the second ratio to compare them.

Answer:
(e) For \( 5.2:3.9 :: 3:4 \):
First ratio: \( 5.2:3.9 = \frac{5.2}{3.9} = \frac{52}{39} = \frac{52 \div 13}{39 \div 13} = \frac{4}{3} = 4:3 \)
(The HCF of 52 and 39 is 13).
Second ratio: \( 3:4 = \frac{3}{4} \)
Since \( 4:3 \neq 3:4 \), the statement \( 5.2:3.9 :: 3:4 \) is not True (False).
In simple words: The first ratio, when simplified, is 4 to 3. The second ratio is 3 to 4. Since these are different, the statement is incorrect.

Exam Tip: When dealing with decimals in ratios, multiply both numbers by a power of 10 to make them whole numbers before simplifying.

Answer:
(f) For \( 0.9:0.36 :: 10:4 \):
First ratio: \( 0.9:0.36 \)
To remove decimals, multiply both by 100:
\( 0.9 \times 100 : 0.36 \times 100 = 90:36 \)
\( \frac{90}{36} = \frac{90 \div 18}{36 \div 18} = \frac{5}{2} = 5:2 \)
(The HCF of 90 and 36 is 18).
Second ratio: \( 10:4 = \frac{10}{4} = \frac{10 \div 2}{4 \div 2} = \frac{5}{2} = 5:2 \)
(The HCF of 10 and 4 is 2).
Since \( 0.9:0.36 = 10:4 \), the statement \( 0.9:0.36 :: 10:4 \) is True.
In simple words: We changed the decimal numbers into whole numbers and then simplified both ratios. Since both simplified ratios are the same, the statement is correct.

Exam Tip: Always convert decimal ratios into whole number ratios by multiplying by the appropriate power of 10 before simplifying, to make calculations easier.

Question 3. Are the following statements true?
(a) 40 persons : 200 persons = Rs. 15 : Rs. 75
Answer:
(a) For 40 persons : 200 persons = Rs. 15 : Rs. 75:
Ratio of persons: \( \frac{40 \text{ persons}}{200 \text{ persons}} = \frac{40}{200} = \frac{40 \div 40}{200 \div 40} = \frac{1}{5} = 1:5 \)
(The HCF of 40 and 200 is 40).
Ratio of money: \( \frac{\text{Rs. } 15}{\text{Rs. } 75} = \frac{15}{75} = \frac{15 \div 15}{75 \div 15} = \frac{1}{5} = 1:5 \)
(The HCF of 15 and 75 is 15).
Since both ratios are equal, the given statement 40 persons : 200 persons = Rs. 15 : Rs. 75 is true.
In simple words: When we compare the number of people and the amount of money, both comparisons simplify to the exact same ratio. This means the statement is true.

Exam Tip: Ensure that units are consistent or cancel out correctly when calculating ratios, and simplify each ratio completely before comparing.

Answer:
(b) For 7.5 litres : 15 litres = 5 kg : 10 kg:
Ratio of litres: \( \frac{7.5 \text{ litres}}{15 \text{ litres}} = \frac{7.5}{15} = \frac{75}{150} = \frac{75 \div 75}{150 \div 75} = \frac{1}{2} = 1:2 \)
(The HCF of 75 and 150 is 75).
Ratio of kg: \( \frac{5 \text{ kg}}{10 \text{ kg}} = \frac{5}{10} = \frac{5 \div 5}{10 \div 5} = \frac{1}{2} = 1:2 \)
(The HCF of 5 and 10 is 5).
Since the two ratios are equal, the statement 7.5 litres : 15 litres = 5 kg : 10 kg is true.
In simple words: We compare the ratio of volumes in litres and the ratio of masses in kilograms. Both comparisons give the same simplified ratio, so the statement is correct.

Exam Tip: When dealing with decimals, convert them to whole numbers for easier simplification of the ratio.

Answer:
(c) For 99 kg : 45 kg = Rs. 44 : Rs. 20:
Ratio of kg: \( \frac{99 \text{ kg}}{45 \text{ kg}} = \frac{99}{45} = \frac{99 \div 9}{45 \div 9} = \frac{11}{5} = 11:5 \)
(The HCF of 99 and 45 is 9).
Ratio of money: \( \frac{\text{Rs. } 44}{\text{Rs. } 20} = \frac{44}{20} = \frac{44 \div 4}{20 \div 4} = \frac{11}{5} = 11:5 \)
(The HCF of 44 and 20 is 4).
Since the two ratios are equal, the statement 99 kg : 45 kg = Rs. 44 : Rs. 20 is true.
In simple words: The ratio of the masses (kilograms) and the ratio of the money amounts both simplify to the identical result. This verifies that the statement is true.

Exam Tip: Be mindful of the units in each part of the proportion; while they don't have to be the same between the first and second ratio, they must be consistent within each ratio pair (e.g., kg to kg, Rs. to Rs.).

Answer:
(d) For 32 m : 64 m = 6 sec : 12 sec:
Ratio of metres: \( \frac{32 \text{ m}}{64 \text{ m}} = \frac{32}{64} = \frac{32 \div 32}{64 \div 32} = \frac{1}{2} = 1:2 \)
(The HCF of 32 and 64 is 32).
Ratio of seconds: \( \frac{6 \text{ sec}}{12 \text{ sec}} = \frac{6}{12} = \frac{6 \div 6}{12 \div 6} = \frac{1}{2} = 1:2 \)
(The HCF of 6 and 12 is 6).
Since the two ratios are equal, the statement 32 m : 64 m = 6 sec : 12 sec is true.
In simple words: Both the comparison of distances and the comparison of times simplify to the same numerical relationship. This confirms that the statement is correct.

Exam Tip: Recognize that while the units (meters and seconds) are different, the *ratios* themselves are being compared numerically.

Answer:
(e) For 45 km : 60 km = 12 hours : 15 hours:
Ratio of km: \( \frac{45 \text{ km}}{60 \text{ km}} = \frac{45}{60} = \frac{45 \div 15}{60 \div 15} = \frac{3}{4} = 3:4 \)
(The HCF of 45 and 60 is 15).
Ratio of hours: \( \frac{12 \text{ hours}}{15 \text{ hours}} = \frac{12}{15} = \frac{12 \div 3}{15 \div 3} = \frac{4}{5} = 4:5 \)
(The HCF of 12 and 15 is 3).
Since \( 3:4 \neq 4:5 \), the statement 45 km : 60 km = 12 hours : 15 hours is false.
In simple words: The ratio of distances is 3 to 4, but the ratio of times is 4 to 5. Since these are not the same, the statement is incorrect.

Exam Tip: Always make sure both ratios are simplified to their absolute lowest terms before concluding whether they are equal or not.

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 Rs. 40 : Rs. 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 litres and Rs. 4 : Rs. 50
Answer:
(a) For 25 cm : 1 m and Rs. 40 : Rs. 160:
First, convert units so they are the same: 1 m = 100 cm.
Ratio of lengths: \( \frac{25 \text{ cm}}{100 \text{ cm}} = \frac{25}{100} = \frac{25 \div 25}{100 \div 25} = \frac{1}{4} = 1:4 \)
(The HCF of 25 and 100 is 25).
Ratio of money: \( \frac{\text{Rs. } 40}{\text{Rs. } 160} = \frac{40}{160} = \frac{40 \div 40}{160 \div 40} = \frac{1}{4} = 1:4 \)
(The HCF of 40 and 160 is 40).
Since both ratios are equal, they form a proportion.
The middle terms are 1 m and Rs. 40.
The extreme terms are 25 cm and Rs. 160.
In simple words: We made sure the units were consistent and then simplified both ratios. Since both ratios were the same, they form a proportion. The numbers in the middle are the middle terms, and the numbers at the ends are the extreme terms.

Exam Tip: Always convert all quantities to the same unit within each ratio before simplifying (e.g., cm to cm, ml to ml) to ensure accurate comparison.

Answer:
(b) For 39 litres : 65 litres and 6 bottles : 10 bottles:
Ratio of litres: \( \frac{39 \text{ litres}}{65 \text{ litres}} = \frac{39}{65} = \frac{39 \div 13}{65 \div 13} = \frac{3}{5} = 3:5 \)
(The HCF of 39 and 65 is 13).
Ratio of bottles: \( \frac{6 \text{ bottles}}{10 \text{ bottles}} = \frac{6}{10} = \frac{6 \div 2}{10 \div 2} = \frac{3}{5} = 3:5 \)
(The HCF of 6 and 10 is 2).
Since both ratios are equal, they form a proportion.
The middle terms are 65 litres and 6 bottles.
The extreme terms are 39 litres and 10 bottles.
In simple words: We simplified both the ratio of litres and the ratio of bottles. Since both results were the same, these quantities are in proportion. We then identified which numbers were in the middle and which were at the ends.

Exam Tip: Remember that in a proportion \( a:b :: c:d \), 'b' and 'c' are the middle terms, and 'a' and 'd' are the extreme terms.

Answer:
(c) For 2 kg : 80 kg and 25 g : 625 g:
Ratio of kg: \( \frac{2 \text{ kg}}{80 \text{ kg}} = \frac{2}{80} = \frac{2 \div 2}{80 \div 2} = \frac{1}{40} = 1:40 \)
(The HCF of 2 and 80 is 2).
Ratio of g: \( \frac{25 \text{ g}}{625 \text{ g}} = \frac{25}{625} = \frac{25 \div 25}{625 \div 25} = \frac{1}{25} = 1:25 \)
(The HCF of 25 and 625 is 25).
Since \( 1:40 \neq 1:25 \), the given ratios do not form a proportion.
In simple words: We simplified the ratio of kilograms and the ratio of grams separately. Because the final simplified ratios were different, these quantities are not in proportion.

Exam Tip: Always make sure to simplify each ratio fully, even if the numbers are large, to guarantee an accurate comparison.

Answer:
(d) For 200 ml : 2.5 litres and Rs. 4 : Rs. 50:
First, convert units: 2.5 litres = \( 2.5 \times 1000 \) ml = 2500 ml.
Ratio of ml: \( \frac{200 \text{ ml}}{2500 \text{ ml}} = \frac{200}{2500} = \frac{200 \div 100}{2500 \div 100} = \frac{2}{25} = 2:25 \)
(The HCF of 200 and 2500 is 100).
Ratio of money: \( \frac{\text{Rs. } 4}{\text{Rs. } 50} = \frac{4}{50} = \frac{4 \div 2}{50 \div 2} = \frac{2}{25} = 2:25 \)
(The HCF of 4 and 50 is 2).
Since the two ratios are equal, they form a proportion.
The middle terms are 2.5 litres and Rs. 4.
The extreme terms are 200 ml and Rs. 50.
In simple words: After converting units so they matched, we simplified both pairs of numbers into ratios. Since both ratios came out to be the same, they form a proportion. We then identified the middle and end terms.

Exam Tip: Pay close attention to unit conversions (e.g., litres to ml) as a common source of error. Always convert to the smaller unit to avoid decimals in the ratio calculation.