Samacheer Kalvi Class 6 Maths Solutions Term 1 Chapter 3 Ratio and Proportion Exercise 3.3

Tamilnadu Samacheer Kalvi 6th Maths Solutions Term 1 Chapter 3 Ratio and Proportion Ex 3.3

Question 1. Fill in the blanks.
(i) 3 : 5 :: ......... : 20
(ii) ......... : 24 :: 3 : 8
(iii) 5 : ......... :: 10 : 8 :: 15 : .........
(iv) 12 : ......... :: .......... : 4 = 8 : 16
Answer:
(i) 3 : 5 :: 12 : 20
(ii) 9 : 24 :: 3 : 8
(iii) 5 : 4 :: 10 : 8 :: 15 : 12
(iv) 12 : 24 :: 2 : 4 = 8 : 16
In simple words: When two ratios are in proportion, the product of their outer terms (extremes) is equal to the product of their inner terms (means). This rule helps us find any missing value in the proportion.

🎯 Exam Tip: Remember the "product of extremes equals product of means" rule for solving proportion problems. This is the main concept for finding missing values.

Question 2. Say True or False.
(i) 2 : 7 :: 14 : 4
(ii) 7 Persons are to 49 Persons as 11 kg is to 88 kg
(iii) 10 books are to 15 books as 3 books are to 5 books.
Answer:
(i) False
Hint: For 2 : 7 :: 14 : 4 to be true, the product of means must equal the product of extremes.
\( 7 \times 14 = 98 \)
\( 2 \times 4 = 8 \)
Since \( 98 \ne 8 \), the statement is False.
(ii) False
Hint: For 7 : 49 :: 11 : 48 to be true, the product of means must equal the product of extremes.
\( 49 \times 11 = 539 \)
\( 7 \times 48 = 336 \)
Since \( 539 \ne 336 \), the statement is False.
(iii) False
Hint: For 10 : 15 :: 3 : 5 to be true, the ratios must be equal when simplified.
\( \frac{10}{15} = \frac{5 \times 2}{5 \times 3} = \frac{2}{3} \)
\( \frac{3}{5} \)
Since \( \frac{2}{3} \ne \frac{3}{5} \), the statement is False.
In simple words: For two ratios to be in proportion, the multiplication of the two middle numbers must be the same as the multiplication of the two outside numbers. If they are not the same, then the statement is false.

🎯 Exam Tip: Always verify proportionality by checking if the product of the means equals the product of the extremes, or by simplifying both ratios to their simplest form and comparing them. Both methods work and lead to the same conclusion.

Question 3. Using the numbers 3, 9, 4, 12 write two ratios that are in proportion.
Answer:
(i) Using 3, 9, 4, 12 in order:
Product of extremes \( = 3 \times 12 = 36 \)
Product of means \( = 9 \times 4 = 36 \)
Since the products are equal, the ratios are in proportion: 3 : 9 :: 4 : 12.
(ii) Using 9, 3, 12, 4 in order:
Product of extremes \( = 9 \times 4 = 36 \)
Product of means \( = 3 \times 12 = 36 \)
Since the products are equal, these ratios are also in proportion: 9 : 3 :: 12 : 4.
In simple words: To make a proportion, arrange four numbers so that when you multiply the first and last numbers, you get the same result as when you multiply the two middle numbers. We can arrange the given numbers in different ways to form valid proportions.

🎯 Exam Tip: When forming proportions from a set of numbers, remember that the order matters. Always check if the product of extremes equals the product of means for each arrangement.

Question 4. Find whether 12, 24,18, 36 are in order that can be expressed as two ratios that are in proportion.
Answer: Yes, 12, 24, 18, 36 are in proportion.
To check if they are in proportion, we arrange them as 12 : 24 :: 18 : 36.
Product of extremes \( = 12 \times 36 = 432 \)
Product of means \( = 24 \times 18 = 432 \)
Since the product of extremes equals the product of means (432 = 432), the numbers are in proportion.
In simple words: We check if these four numbers can form a proportion. They can, because multiplying the first and last numbers gives the same result as multiplying the two middle numbers. This shows they are balanced.

🎯 Exam Tip: Always show your work when verifying proportionality by calculating both the product of extremes and the product of means separately before comparing them.

Question 5. Write the mean and extreme terms in the following ratios and check whether they are in proportion.
(i) 78 liters is to 130 liters and 12 bottles are to 20 bottles
(ii) 400 gm is to 50 gm and 25 rupees is to 625 rupees
Answer:
(i) For the ratios 78 : 130 and 12 : 20:
The proportion is 78 : 130 :: 12 : 20.
Extreme terms are 78 and 20.
Mean terms are 130 and 12.
Product of Extremes \( = 78 \times 20 = 1560 \)
Product of Means \( = 130 \times 12 = 1560 \)
Since the Product of Extremes = Product of Means, these ratios are in proportion.
(ii) For the ratios 400 : 50 and 25 : 625:
The proportion is 400 : 50 :: 25 : 625.
Extreme terms are 400 and 625.
Mean terms are 50 and 25.
Product of Extremes \( = 400 \times 625 = 2,50,000 \)
Product of Means \( = 50 \times 25 = 1250 \)
Since the product of extremes (2,50,000) is not equal to the product of means (1250), these ratios are not in proportion. It's important to keep units consistent when comparing quantities.
In simple words: First, list the numbers that are at the ends (extremes) and in the middle (means) of the proportion. Then, multiply the extreme numbers together and the mean numbers together. If both multiplications give the same answer, the ratios are in proportion; otherwise, they are not.

🎯 Exam Tip: Always identify the extreme and mean terms correctly before calculating their products. Misidentifying them is a common error that leads to incorrect conclusions about proportionality.

Question 6. America's famous Golden Gate bridge is 6480 ft long with 756 ft tall towers. A model of this bridge exhibited in a fair is 60 ft long with 7 ft tall towers. Is the model in proportion to the original bridge?
Answer: Yes, the model is in proportion to the original bridge.
For the original bridge, the ratio of length to tower height is 6480 : 756.
For the model, the ratio of length to tower height is 60 : 7.
To check if these ratios are in proportion, we write: 6480 : 756 :: 60 : 7.
Product of the means \( = 756 \times 60 = 45360 \)
Product of the extremes \( = 6480 \times 7 = 45360 \)
Since the product of the means equals the product of the extremes \( (45360 = 45360) \), the model is indeed in proportion to the original bridge. This means all parts of the model scale down consistently.
In simple words: We check if the big bridge and its small model are perfectly scaled. We do this by comparing how the length relates to the tower height for both. If these relations are the same, then the model is a perfect, smaller version of the real bridge.

🎯 Exam Tip: When dealing with real-world proportion problems, set up the ratios consistently (e.g., always length:height) for both objects before applying the product of means and extremes rule.

Objective Type Questions

Question 7. Which of the following ratios are in proportion?
(a) 3:5, 6:11
(b) 2:3, 9:6
(c) 2 : 5, 10:25
(d) 3:1, 1:3
Answer: (c) 2 : 5, 10:25
For the option (c) 2 : 5 :: 10 : 25:
Product of extremes \( = 2 \times 25 = 50 \)
Product of means \( = 5 \times 10 = 50 \)
Since \( 50 = 50 \), this option shows ratios that are in proportion. This balance is key in many mathematical concepts.
In simple words: We look for the option where the multiplication of the two outer numbers gives the same answer as the multiplication of the two inner numbers. Option (c) is the only one where this is true.

🎯 Exam Tip: For MCQ questions testing proportionality, quickly test each option by multiplying the extremes and the means. The option where these products match is the correct answer.

Question 8. If the ratios formed using the numbers 2, 5, x, 20 in the same order are in proportion, then 'x' is
(a) 50
(b) 4
(c) 10
(d) 8
Answer: (d) 8
If 2, 5, x, 20 are in proportion, then 2 : 5 :: x : 20.
Using the rule: Product of extremes = Product of means
\( 2 \times 20 = 5 \times x \)
\( 40 = 5x \)
\( x = \frac{40}{5} \)
\( x = 8 \)
So, the missing number 'x' is 8. Understanding this relationship helps solve many ratio-based problems.
In simple words: We have four numbers that are in proportion. To find the missing number 'x', we multiply the first and last numbers, and then we multiply the two middle numbers. Since these results must be equal, we can solve to find 'x'.

🎯 Exam Tip: Remember the order of numbers is crucial in a proportion. Always set up the equation (product of extremes = product of means) carefully according to the given sequence.

Question 9. If 7 : 5 is in proportion to x : 25, then 'x' is
(a) 27
(b) 49
(c) 35
(d) 14
Answer: (c) 35
If 7 : 5 :: x : 25 is a proportion,
Using the rule: Product of extremes = Product of means
\( 7 \times 25 = 5 \times x \)
\( 175 = 5x \)
\( x = \frac{175}{5} \)
\( x = 35 \)
Thus, the value of 'x' is 35. This method is fundamental for solving any unknown in a proportion.
In simple words: We are given that two ratios are equal. We multiply the outer numbers (7 and 25) and set it equal to the multiplication of the inner numbers (5 and x). Then we solve this simple equation to find what 'x' is.

🎯 Exam Tip: Write down the proportion clearly as "a : b :: c : d" before applying the cross-multiplication rule \( (a \times d = b \times c) \) to avoid errors when solving for the unknown.