RBSE Solutions Class 6 Maths Chapter 13 Ratio and Proportion Exercise 13.2

Question 1. Determine if the following are in proportion.
(i) 8, 6, 48, 36
(ii) 12, 18, 20, 30
(iii) 14, 20, 26, 32
(iv) 26, 65, 32, 60
Answer:

For a set of four numbers to be in proportion, the product of the first and last numbers (extreme terms) must be equal to the product of the second and third numbers (middle terms). The table shows whether this condition is met for each set of numbers, determining if they are in proportion. A proportion indicates that two ratios are equivalent.
In simple words: We check if the multiplication of the first and last number is the same as the multiplication of the two middle numbers. If they are equal, the numbers are in proportion.

🎯 Exam Tip: Always remember the fundamental rule of proportion: "product of extremes = product of means". This is the fastest way to verify if numbers are in proportion.

Question 2. Write True or False against each of the following statements.
(i) 20 : 60 :: 15 : 45
(ii) 20 : 22 :: 32 : 42
(iii) 0.9 : 0.36 :: 10 : 4
(iv) 5.2 : 26 :: 1.8 : 0.9
Answer:

To check if a statement about proportion is true or false, we multiply the first number by the fourth number, and the second number by the third number. If these two products are equal, the statement is true; otherwise, it is false. This method helps us quickly verify the balance between the two ratios.
In simple words: For each statement, we check if the product of the outside numbers equals the product of the inside numbers. If they match, it's True. If not, it's False.

🎯 Exam Tip: When dealing with decimals in proportion, be careful with multiplication. A small error can lead to a wrong True/False conclusion.

Question 3. Are the following statements True?
(i) 30 sec : 1 min :: 16 m : 32 m
(ii) 2.5l : 5l :: 25 m : 50 m
(iii) 44 Rs : 20 Rs :: 66l : 30l
(iv) 12 m : 15 m :: 48 kg : 64 kg
Answer:
(i) We need to make sure the units are the same for comparison. We know that 1 minute is equal to 60 seconds.
\( 30 \text{ sec} : 1 \text{ min} :: 16 \text{ m} : 32 \text{ m} \)
\( \implies 30 : 60 :: 16 : 32 \) (Since \( 1 \text{ min} = 60 \text{ sec} \))
\( \implies 30 \times 32 = 60 \times 16 \)
\( \implies 960 = 960 \)
Thus, This statement is true.

(ii) For this proportion, we just compare the numbers as the units are consistent (liters with meters).
\( 2.5\text{l} : 5\text{l} :: 25 \text{ m} : 50 \text{ m} \)
\( \implies 2.5 : 5 :: 25 : 50 \)
\( \implies 2.5 \times 50 = 5 \times 25 \)
\( \implies 125 = 125 \)
Thus, This statement is true.

(iii) For this proportion, the units (Rs and liters) are different but we only compare the numerical values for the check.
\( 44 \text{ Rs} : 20 \text{ Rs} :: 66\text{l} : 30\text{l} \)
\( \implies 44 : 20 :: 66 : 30 \)
\( \implies 44 \times 30 = 20 \times 66 \)
\( \implies 1320 = 1320 \)
Thus, This statement is true.

(iv) For this proportion, we compare the numerical values of the given units (meters and kilograms).
\( 12 \text{ m} : 15 \text{ m} :: 48 \text{ kg} : 64 \text{ kg} \)
\( \implies 12 : 15 :: 48 : 64 \)
\( \implies 12 \times 64 = 15 \times 48 \)
\( \implies 768 = 720 \)
Thus, This statement is false.
In simple words: To check if statements are true proportions, we multiply the first and last numbers, and then the two middle numbers. If the products are the same, it's true. Remember to convert units to be the same if needed, like minutes to seconds.

🎯 Exam Tip: Always make sure to use consistent units when comparing ratios. For example, convert minutes to seconds before checking proportion, or ensure both sides of the ratio use the same unit if possible, for accurate comparison.

Question 4. Determine if the following ratios form a proportion. Also, write the middle terms and extreme terms where the ratios from a proportion.
(i) 200 cm : 2.5 m and Rs 40 : Rs 5
(ii) 1 kg : 250 gm and Rs 40 : Rs 10
(iii) 4 kg : 16 gm and 20 gm : 400 gm
(iv) 39 Persons : 65 Persons and 91 : 151
Answer:
(i) We first convert all measurements to a common unit. We know that 1 meter is equal to 100 centimeters. So, 200 cm becomes 2 m.
\( 200 \text{ cm} : 2.5 \text{ m} \text{ and } \text{Rs } 40 : \text{Rs } 5 \)
\( \implies 2 \text{ m} : 2.5 \text{ m} :: \text{Rs } 40 : \text{Rs } 5 \) (Since \( 200 \text{ cm} = 2 \text{ m} \))
\( \implies 2 \times 5 = 2.5 \times 40 \)
\( \implies 10 = 100 \)
This statement is not in a proportion.

(ii) We convert kilograms to grams to have consistent units. We know that 1 kg is 1000 gm.
\( 1 \text{ kg} : 250 \text{ gm} \text{ and } \text{Rs } 40 : \text{Rs } 10 \)
\( \implies 1000 \text{ gm} : 250 \text{ gm} :: \text{Rs } 40 : \text{Rs } 10 \) (Since \( 1 \text{ kg} = 1000 \text{ gm} \))
\( \implies 1000 \times 10 = 250 \times 40 \)
\( \implies 10000 = 10000 \)
This statement makes a proportion. Its middle terms are 250 gm and Rs 40, and the extreme terms are 1 kg and Rs 10.
An easy way to find middle terms is to look at the two numbers in the middle, and extreme terms are the numbers at the ends.

(iii) For this statement, we convert all kilograms to grams. We know 1 kg is 1000 gm, so 4 kg is 4000 gm.
\( 4 \text{ kg} : 16 \text{ kg} \text{ and } 20 \text{ gm} : 400 \text{ gm} \)
\( \implies 4 : 16 :: 20 : 400 \)
\( \implies 4 \times 400 = 16 \times 20 \)
\( \implies 1600 = 320 \)
This statement is not in a proportion.

(iv) This involves a count of persons and a simple ratio. The units are consistent.
\( 39 \text{ Persons} : 65 \text{ Persons} \text{ and } 91 : 151 \)
\( \implies 39 : 65 :: 9 : 15 \)
\( \implies 39 \times 15 = 65 \times 9 \)
\( \implies 585 = 585 \)
This statement forms a proportion. Its middle terms are 65 Persons and 91, and the extreme terms are 39 Persons and 151.
In simple words: To see if ratios form a proportion, first make sure all units are the same (like cm to m, or kg to gm). Then, multiply the first number by the last, and the two middle numbers together. If the answers are the same, it's a proportion. The first and last numbers are "extreme terms," and the two in the middle are "middle terms."

🎯 Exam Tip: Always pay close attention to units given in the question. Converting all quantities to a common unit before checking for proportion is a crucial step to avoid errors.