GSEB Class 6 Maths Solutions Chapter 12 ગુણોત્તર અને પ્રમાણ Exercise 12.2

Question 1. Determine whether the given numbers are in proportion or not:
(a) 15, 45, 40, 120
Answer:
Ratio of 15 to 45 \( = 15 : 45 \)
\( = \frac{15}{45} = \frac{15 \div 15}{45 \div 15} \) (HCF of 15 and 45 is 15)
\( = \frac{1}{3} = 1:3 \)
Ratio of 40 to 120 \( = 40 : 120 \)
\( = \frac{40}{120} = \frac{40 \div 40}{120 \div 40} \) (HCF of 40 and 120 is 40)
\( = \frac{1}{3} = 1:3 \)
So, \( 15 : 45 :: 40 : 120 \)
Therefore, 15, 45, 40, and 120 are in proportion.
In simple words: First, find the simplest form of the ratio for the first two numbers, then do the same for the last two. If these two simplified ratios match, then the four numbers are in proportion.

Exam Tip: Remember to find the Highest Common Factor (HCF) for each pair of numbers to simplify the ratios correctly. Show your working for HCF division.

Question 1. Determine whether the given numbers are in proportion or not:
(b) 33, 121, 9, 96
Answer:
Ratio of 33 to 121 \( = 33 : 121 \)
\( = \frac{33}{121} = \frac{33 \div 11}{121 \div 11} \) (HCF of 33 and 121 is 11)
\( = \frac{3}{11} = 3:11 \)
Ratio of 9 to 96 \( = 9 : 96 \)
\( = \frac{9}{96} = \frac{9 \div 3}{96 \div 3} \) (HCF of 9 and 96 is 3)
\( = \frac{3}{32} = 3:32 \)
Here, \( 3 : 11 \neq 3 : 32 \)
So, \( 33 : 121 \neq 9 : 96 \)
Therefore, 33, 121, 9, and 96 are not in proportion.
In simple words: When you simplify the first ratio and the second ratio, if they are not the same, it means the numbers are not in proportion.

Exam Tip: Pay close attention to unit conversions if given, though not relevant in this specific numerical question. Always verify if the simplified ratios are truly identical.

Question 1. Determine whether the given numbers are in proportion or not:
(c) 24, 28, 36, 48
Answer:
Ratio of 24 to 28 \( = 24 : 28 \)
\( = \frac{24}{28} = \frac{24 \div 4}{28 \div 4} \) (HCF of 24 and 28 is 4)
\( = \frac{6}{7} = 6:7 \)
Ratio of 36 to 48 \( = 36 : 48 \)
\( = \frac{36}{48} = \frac{36 \div 12}{48 \div 12} \) (HCF of 36 and 48 is 12)
\( = \frac{3}{4} = 3:4 \)
Here, \( 6 : 7 \neq 3 : 4 \)
So, \( 24 : 28 \neq 36 : 48 \)
Therefore, 24, 28, 30 (OCR error, should be 36) and 48 are not in proportion.
In simple words: If the reduced form of the first pair's ratio is different from the reduced form of the second pair's ratio, then the numbers do not form a proportion.

Exam Tip: Be careful with transcription errors or OCR issues in the question. Always check the original numbers carefully before solving. The number provided in OCR `30` in "24, 28, 30 and 48" should be `36` from the original question statement.

Question 1. Determine whether the given numbers are in proportion or not:
(d) 32, 48, 70, 210
Answer:
Ratio of 32 to 48 \( = 32 : 48 \)
\( = \frac{32}{48} = \frac{32 \div 16}{48 \div 16} \) (HCF of 32 and 48 is 16)
\( = \frac{2}{3} = 2:3 \)
Ratio of 70 to 210 \( = 70 : 210 \)
\( = \frac{70}{210} = \frac{70 \div 70}{210 \div 70} \) (HCF of 70 and 210 is 70)
\( = \frac{1}{3} = 1:3 \)
Here, \( 2 : 3 \neq 1 : 3 \)
So, \( 32 : 48 \neq 70 : 210 \)
Therefore, 32, 48, 70, and 210 are not in proportion.
In simple words: To check if four numbers are in proportion, you just need to compare the simplified ratios of the first two and the last two numbers. If they are unequal, the numbers are not proportional.

Exam Tip: Practice finding the HCF quickly for different pairs of numbers. This skill helps you simplify ratios faster and more accurately.

Question 1. Determine whether the given numbers are in proportion or not:
(e) 4, 6, 8, 12
Answer:
Ratio of 4 to 6 \( = 4 : 6 \)
\( = \frac{4}{6} = \frac{4 \div 2}{6 \div 2} \) (HCF of 4 and 6 is 2)
\( = \frac{2}{3} = 2:3 \)
Ratio of 8 to 12 \( = 8 : 12 \)
\( = \frac{8}{12} = \frac{8 \div 4}{12 \div 4} \) (HCF of 8 and 12 is 4)
\( = \frac{2}{3} = 2:3 \)
So, \( 4 : 6 :: 8 : 12 \)
Therefore, 4, 6, 8, and 12 are in proportion.
In simple words: When the simplest ratios of the first two numbers and the last two numbers are exactly the same, it means all four numbers maintain a proportional relationship.

Exam Tip: Remember that two ratios form a proportion if their cross-products are equal after simplification. In this case \( 4 \times 12 = 48 \) and \( 6 \times 8 = 48 \).

Question 1. Determine whether the given numbers are in proportion or not:
(f) 33, 44, 75, 100
Answer:
Ratio of 33 to 44 \( = 33 : 44 \)
\( = \frac{33}{44} = \frac{33 \div 11}{44 \div 11} \) (HCF of 33 and 44 is 11)
\( = \frac{3}{4} = 3:4 \)
Ratio of 75 to 100 \( = 75 : 100 \)
\( = \frac{75}{100} = \frac{75 \div 25}{100 \div 25} \) (HCF of 75 and 100 is 25)
\( = \frac{3}{4} = 3:4 \)
So, \( 33 : 44 :: 75 : 100 \)
Therefore, 33, 44, 75, and 100 are in proportion.
In simple words: If both pairs of numbers, when simplified, have the same ratio, they are considered to be in proportion, showing a consistent relationship.

Exam Tip: Always show the division by the HCF in your working to demonstrate how you arrived at the simplest form of each ratio. This helps earn full marks.

Question 2. State whether each of the following statements is true or false:
(a) \( 16:24::20:30 \)
Answer:
\( 16:24 = \frac{16}{24} = \frac{16 \div 8}{24 \div 8} \) (HCF of 16 and 24 is 8)
\( = \frac{2}{3} = 2:3 \)
\( 20:30 = \frac{20}{30} = \frac{20 \div 10}{30 \div 10} \) (HCF of 20 and 30 is 10)
\( = \frac{2}{3} = 2:3 \)
Since, \( 16:24 = 20:30 \)
Therefore, \( 16:24::20:30 \) is true.
In simple words: The first ratio and the second ratio, when simplified, are equal, which means the statement that they are in proportion is correct.

Exam Tip: For true/false questions involving proportions, always simplify both ratios to their lowest terms and then compare them. Do not assume equality without calculation.

Question 2. State whether each of the following statements is true or false:
(b) \( 21:6::35:10 \)
Answer:
\( 21:6 = \frac{21}{6} = \frac{21 \div 3}{6 \div 3} \) (HCF of 21 and 6 is 3)
\( = \frac{7}{2} = 7:2 \)
\( 35:10 = \frac{35}{10} = \frac{35 \div 5}{10 \div 5} \) (HCF of 35 and 10 is 5)
\( = \frac{7}{2} = 7:2 \)
Since, \( 21:6 = 35:10 \)
Therefore, \( 21:6::35:10 \) is true.
In simple words: The simplified forms of both ratios are the same, which confirms that the statement about them being in proportion is correct.

Exam Tip: Clearly show the HCF used for simplification. This step is important for demonstrating your understanding and getting partial marks even if there's a minor calculation error.

Question 2. State whether each of the following statements is true or false:
(c) \( 12:18:28:12 \)
Answer:
\( 12:18 = \frac{12}{18} = \frac{12 \div 6}{18 \div 6} \) (HCF of 12 and 18 is 6)
\( = \frac{2}{3} = 2:3 \)
\( 28:12 = \frac{28}{12} = \frac{28 \div 4}{12 \div 4} \) (HCF of 28 and 12 is 4)
\( = \frac{7}{3} = 7:3 \)
Here, \( 2 : 3 \neq 7 : 3 \)
Therefore, \( 12:18::28:12 \) is false.
In simple words: Because the two ratios are not equal after being reduced, the given statement that they are in proportion is incorrect.

Exam Tip: Ensure that you simplify each ratio correctly. A mistake in finding the HCF or performing division will lead to an incorrect comparison and answer.

Question 2. State whether each of the following statements is true or false:
(d) \( 8:9::24:27 \)
Answer:
\( 8:9 = \frac{8}{9} \) (This is already in its simplest form.)
\( 24:27 = \frac{24}{27} = \frac{24 \div 3}{27 \div 3} \) (HCF of 24 and 27 is 3)
\( = \frac{8}{9} = 8:9 \)
Since, \( 8:9 = 24:27 \)
Therefore, \( 8:9::24:27 \) is true.
In simple words: The first ratio is already as simple as it can be, and the second ratio, when simplified, matches it, so the statement is correct.

Exam Tip: Always check if a given ratio is already in its simplest form before attempting to simplify it further. This saves time and avoids errors.

Question 2. State whether each of the following statements is true or false:
(e) \( 5.2: 3.9::3:4 \)
Answer:
\( 5.2:3.9 = \frac{5.2}{3.9} = \frac{52}{39} = \frac{52 \div 13}{39 \div 13} \) (HCF of 52 and 39 is 13)
\( = \frac{4}{3} = 4:3 \)
The second ratio is \( 3:4 \).
Here, \( 4 : 3 \neq 3 : 4 \)
Therefore, \( 5.2:3.9::3:4 \) is false.
In simple words: The first ratio simplifies to 4:3, but the second ratio is 3:4, and since these are not the same, the statement is incorrect.

Exam Tip: When dealing with decimals in ratios, convert them to whole numbers first by multiplying by an appropriate power of 10 before simplifying. This makes HCF calculation easier.

Question 2. State whether each of the following statements is true or false:
(f) \( 0.9: 0.36::10:4 \)
Answer:
\( 0.9:0.36 = \frac{0.9}{0.36} = \frac{9}{10} \times \frac{100}{36} = \frac{90}{36} = \frac{5}{2} = 5:2 \)
\( 10:4 = \frac{10}{4} = \frac{10 \div 2}{4 \div 2} \) (HCF of 10 and 4 is 2)
\( = \frac{5}{2} = 5:2 \)
Since, \( 0.9:0.36 = 10:4 \)
Therefore, \( 0.9:0.36::10:4 \) is true.
In simple words: Both ratios, after converting decimals to whole numbers and simplifying, turn out to be equal, confirming that the statement is correct.

Exam Tip: To simplify ratios with decimals, multiply both sides by a power of 10 (e.g., 10, 100, 1000) until both numbers become integers, then simplify as usual.

Question 3. Are the following statements true?
(a) 40 persons: 200 persons = Rs.15 : Rs. 75
Answer:
Ratio of 40 persons to 200 persons \( = \frac{40 \text{ persons}}{200 \text{ persons}} = \frac{40 \div 40}{200 \div 40} \) (HCF of 40 and 200 is 40)
\( = \frac{1}{5} = 1:5 \)
Ratio of Rs.15 to Rs.75 \( = \frac{15}{75} = \frac{15 \div 15}{75 \div 15} \) (HCF of 15 and 75 is 15)
\( = \frac{1}{5} = 1:5 \)
Both ratios are equal.
Thus, 40 persons: 200 persons = Rs.15 : Rs.75 is true.
In simple words: When we simplify the ratio of people and the ratio of money, both give 1:5, which means the statement is correct.

Exam Tip: Remember that ratios must be between quantities of the same type. However, for a proportion, you are comparing two *different* ratios (e.g., persons to persons and money to money). The units cancel out within each ratio.

Question 3. Are the following statements true?
(b) 7.5 litres: 15 litres = 5 kg: 10 kg
Answer:
Ratio of 7.5 litres to 15 litres \( = \frac{7.5 \text{ litres}}{15 \text{ litres}} = \frac{75}{150} = \frac{75 \div 75}{150 \div 75} \) (HCF of 75 and 150 is 75)
\( = \frac{1}{2} = 1:2 \)
Ratio of 5 kg to 10 kg \( = \frac{5 \text{ kg}}{10 \text{ kg}} = \frac{5 \div 5}{10 \div 5} \) (HCF of 5 and 10 is 5)
\( = \frac{1}{2} = 1:2 \)
Both ratios are equal.
Thus, 7.5 litres: 15 litres = 5 kg: 10 kg is true.
In simple words: After simplifying the ratio for the liquid volume and the ratio for the weight, we get the same result for both, making the statement correct.

Exam Tip: When dealing with decimal numbers in ratios, it is often helpful to convert them into whole numbers by multiplying both parts of the ratio by 10, 100, etc., before simplifying.

Question 3. Are the following statements true?
(c) 99 kg : 45 kg = Rs. 44 : Rs. 20
Answer:
Ratio of 99 kg to 45 kg \( = \frac{99 \text{ kg}}{45 \text{ kg}} = \frac{99 \div 9}{45 \div 9} \) (HCF of 99 and 45 is 9)
\( = \frac{11}{5} = 11:5 \)
Ratio of Rs. 44 to Rs. 20 \( = \frac{44}{20} = \frac{44 \div 4}{20 \div 4} \) (HCF of 44 and 20 is 4)
\( = \frac{11}{5} = 11:5 \)
Both ratios are equal.
Thus, 99 kg : 45 kg = Rs. 44 : Rs. 20 is true.
In simple words: The ratio of the masses, when simplified, is equal to the ratio of the money values when simplified, so the statement is correct.

Exam Tip: Ensure that you are consistent with units when setting up ratios. Although the units cancel out within a ratio, it's crucial to acknowledge the original units if they were mixed.

Question 3. Are the following statements true?
(d) 32 m : 64 m = 6 seconds : 12 seconds
Answer:
Ratio of 32 m to 64 m \( = \frac{32 \text{ m}}{64 \text{ m}} = \frac{32 \div 32}{64 \div 32} \) (HCF of 32 and 64 is 32)
\( = \frac{1}{2} = 1:2 \)
Ratio of 6 seconds to 12 seconds \( = \frac{6 \text{ seconds}}{12 \text{ seconds}} = \frac{6 \div 6}{12 \div 6} \) (HCF of 6 and 12 is 6)
\( = \frac{1}{2} = 1:2 \)
Both ratios are equal.
Thus, 32 m : 64 m = 6 seconds : 12 seconds is true.
In simple words: The simplified ratio of lengths is 1:2, and the simplified ratio of times is also 1:2, so the statement is true.

Exam Tip: Ratios are always expressed in their simplest form. Make sure you divide by the largest common factor to avoid extra steps.

Question 3. Are the following statements true?
(e) 45 km : 60 km = 12 hours : 15 hours
Answer:
Ratio of 45 km to 60 km \( = \frac{45 \text{ km}}{60 \text{ km}} = \frac{45 \div 15}{60 \div 15} \) (HCF of 45 and 60 is 15)
\( = \frac{3}{4} = 3:4 \)
Ratio of 12 hours to 15 hours \( = \frac{12 \text{ hours}}{15 \text{ hours}} = \frac{12 \div 3}{15 \div 3} \) (HCF of 12 and 15 is 3)
\( = \frac{4}{5} = 4:5 \)
Both ratios are not equal.
Thus, 45 km : 60 km \( \neq \) 12 hours : 15 hours is false.
In simple words: The ratio of distances simplifies to 3:4, but the ratio of times simplifies to 4:5; since these are different, the statement is incorrect.

Exam Tip: Even if the numbers look similar, always calculate and simplify the ratios to confirm whether a statement about proportion is true or false. Visual inspection can be misleading.

Question 4. Determine whether the given ratios are in proportion or not. If the ratios are in proportion, write their middle terms and extreme terms.
(a) 25 cm : 1 m and Rs. 40 : Rs. 160
Answer:
Ratio of 25 cm to 1 m:
First, convert units: 1 m = 100 cm.
So, 25 cm : 100 cm \( = \frac{25 \text{ cm}}{100 \text{ cm}} = \frac{25 \div 25}{100 \div 25} \) (HCF of 25 and 100 is 25)
\( = \frac{1}{4} = 1:4 \)
Ratio of Rs. 40 to Rs. 160:
\( = \frac{40}{160} = \frac{40 \div 40}{160 \div 40} \) (HCF of 40 and 160 is 40)
\( = \frac{1}{4} = 1:4 \)
Here, both ratios are equal.
Thus, 25 cm : 1 m and Rs. 40 : Rs. 160 are in proportion.
The middle terms are 1 m and Rs. 40.
The extreme terms are 25 cm and Rs. 160.
In simple words: We changed meters to centimeters to compare correctly. Both ratios became 1:4, so they are in proportion. The two numbers in the middle are "middle terms," and the two numbers at the ends are "extreme terms."

Exam Tip: Always make sure units are consistent within a ratio before simplifying. For proportions, if the ratios are equal, clearly identify the middle (means) and extreme terms.

Question 4. Determine whether the given ratios are in proportion or not. If the ratios are in proportion, write their middle terms and extreme terms.
(b) 39 litres: 65 litres and 6 bottles: 10 bottles
Answer:
Ratio of 39 litres to 65 litres \( = \frac{39 \text{ litres}}{65 \text{ litres}} = \frac{39 \div 13}{65 \div 13} \) (HCF of 39 and 65 is 13)
\( = \frac{3}{5} = 3:5 \)
Ratio of 6 bottles to 10 bottles \( = \frac{6 \text{ bottles}}{10 \text{ bottles}} = \frac{6 \div 2}{10 \div 2} \) (HCF of 6 and 10 is 2)
\( = \frac{3}{5} = 3:5 \)
Here, both ratios are equal.
Thus, 39 litres : 65 litres and 6 bottles : 10 bottles are in proportion.
The middle terms are 65 litres and 6 bottles.
The extreme terms are 39 litres and 10 bottles.
In simple words: Both the ratio of litres and the ratio of bottles simplify to 3:5, showing they are in proportion. The numbers in the middle are the middle terms, and the numbers at the ends are the extreme terms.

Exam Tip: Remember the property of proportion: the product of the extreme terms equals the product of the middle terms. This can be used as a quick check for your answers.

Question 4. Determine whether the given ratios are in proportion or not. If the ratios are in proportion, write their middle terms and extreme terms.
(c) 2 kg : 80 kg and 25 g : 625 g
Answer:
Ratio of 2 kg to 80 kg \( = \frac{2 \text{ kg}}{80 \text{ kg}} = \frac{2 \div 2}{80 \div 2} \) (HCF of 2 and 80 is 2)
\( = \frac{1}{40} = 1:40 \)
Ratio of 25 g to 625 g \( = \frac{25 \text{ g}}{625 \text{ g}} = \frac{25 \div 25}{625 \div 25} \) (HCF of 25 and 625 is 25)
\( = \frac{1}{25} = 1:25 \)
Here, \( 1 : 40 \neq 1 : 25 \)
Thus, 2 kg : 80 kg and 25 g : 625 g are not in proportion.
In simple words: The ratio of kilograms simplifies to 1:40, but the ratio of grams simplifies to 1:25. Since these simplified ratios are different, the given quantities are not in proportion.

Exam Tip: Even if the quantities are of the same type (e.g., mass), they must simplify to the same numerical ratio to form a proportion. Always state clearly if they are not in proportion.

Question 4. Determine whether the given ratios are in proportion or not. If the ratios are in proportion, write their middle terms and extreme terms.
(d) 200 ml : 2.5 litres and Rs. 4 : Rs. 50
Answer:
Ratio of 200 ml to 2.5 litres:
First, convert units: 2.5 litres = \( 2.5 \times 1000 \) ml = 2500 ml.
So, 200 ml : 2500 ml \( = \frac{200 \text{ ml}}{2500 \text{ ml}} = \frac{200 \div 100}{2500 \div 100} = \frac{2}{25} = 2:25 \)
Ratio of Rs. 4 to Rs. 50:
\( = \frac{4}{50} = \frac{4 \div 2}{50 \div 2} \) (HCF of 4 and 50 is 2)
\( = \frac{2}{25} = 2:25 \)
Here, both ratios are equal.
Thus, 200 ml : 2.5 litres and Rs. 4 : Rs. 50 are in proportion.
The middle terms are 2.5 litres and Rs. 4.
The extreme terms are 200 ml and Rs. 50.
In simple words: We converted litres to milliliters for the first ratio, which then simplified to 2:25. The second ratio for money also simplified to 2:25. Since both match, the quantities are proportional. The middle two numbers are the middle terms, and the outside two are the extreme terms.

Exam Tip: When converting between units like millilitres and litres, or grams and kilograms, ensure you use the correct conversion factor to avoid calculation errors. A litre is 1000 millilitres.