Samacheer Kalvi Class 6 Maths Solutions Term 1 Chapter 3 Ratio and Proportion Exercise 3.2

Tamilnadu Samacheer Kalvi 6th Maths Solutions Term 1 Chapter 3 Ratio and Proportion Ex 3.2

 

Question 1. Fill in the blanks of the given equivalent ratios.
(i) 3 : 5 = 9 : ......
(ii) 4 : 5 = ...... : 10
(iii) 6 : ...... = 1 : 2
Answer:
(i) To find the missing term, we observe that the first number 3 became 9. This means 3 was multiplied by 3. So, the second number 5 should also be multiplied by 3 to keep the ratio equivalent.
\( \frac{3}{5}=\frac{3 \times 3}{5 \times 3}=\frac{9}{15} \)
So, 3 : 5 = 9 : 15
The missing term is 15.

(ii) For the second part, the second number 5 became 10. This means 5 was multiplied by 2. So, the first number 4 should also be multiplied by 2 to keep the ratio equivalent.
\( \frac{4}{5}=\frac{4 \times 2}{5 \times 2}=\frac{8}{10} \)
So, 4 : 5 = 8 : 10
The missing term is 8.

(iii) In the third part, the first number 6 became 1. This means 6 was divided by 6. So, the second number should also be divided by 6 so that it becomes 2. This means the original second number was 12.
\( \frac{1}{2}=\frac{1 \times 6}{2 \times 6}=\frac{6}{12} \)
So, 6 : 12 = 1 : 2
The missing term is 12.
In simple words: To find a missing number in an equivalent ratio, see how one part of the ratio changed (multiplied or divided). Do the same change to the other part to find the missing number. Equivalent ratios mean they represent the same proportion.

🎯 Exam Tip: Always multiply or divide both parts of a ratio by the same non-zero number to find equivalent ratios.

 

Question 2. Complete the table.
(i)

(ii)

Answer:
(i) We know that 1 foot is equal to 12 inches. This conversion factor helps fill the table.
For 1 foot, it is 12 inches.
For 2 feet, it is \( 2 \times 12 = 24 \) inches.
So, for 3 feet, it will be \( 3 \times 12 = 36 \) inches.
To find feet for 72 inches, we divide 72 by 12, which is \( 72 \div 12 = 6 \) feet.
The completed table is:

(ii) We know that 1 week is equal to 7 days. This conversion factor helps fill the table.
For 4 weeks, it is \( 4 \times 7 = 28 \) days.
For 3 weeks, it is \( 3 \times 7 = 21 \) days.
So, for 2 weeks, it will be \( 2 \times 7 = 14 \) days.
To find weeks for 63 days, we divide 63 by 7, which is \( 63 \div 7 = 9 \) weeks.
The completed table is:
In simple words: To fill in the tables, use the basic conversion rules: 1 foot equals 12 inches, and 1 week equals 7 days. Multiply to convert from larger units to smaller units, and divide to convert from smaller units to larger units.

🎯 Exam Tip: Remember to clearly state the conversion factor used for each table (e.g., 1 foot = 12 inches) as part of your solution.

 

Question 3. Say True or False.
(i) 5 : 7 is equivalent to 21 : 15
(ii) If 40 is divided in the ratio 3 : 2, then the larger part is 24
Answer:
(i) We need to check if 5 : 7 is the same as 21 : 15. Let's simplify both ratios to their simplest form.
The ratio 5 : 7 is already in its simplest form because 5 and 7 have no common factors other than 1.
For the ratio 21 : 15, both numbers can be divided by 3.
\( \frac{21}{15}=\frac{7}{5}=7:5 \)
Since 5 : 7 is not equal to 7 : 5, the statement is False.

(ii) We need to divide 40 in the ratio 3 : 2 and find the larger part. First, find the sum of the ratio parts.
Sum of ratio parts = \( 3 + 2 = 5 \)
Each part of the ratio is \( \frac{40}{5} = 8 \).
The two parts are \( 3 \times 8 = 24 \) and \( 2 \times 8 = 16 \).
The larger part is 24. This matches the statement.
\( \frac{3}{5} \times 40 = 24 \)
So, the statement is True.
In simple words: For True/False questions about ratios, either simplify both ratios to their smallest form and compare them, or calculate the parts based on the given ratio and check if it matches the statement.

🎯 Exam Tip: When comparing ratios, always reduce them to their simplest form to easily check for equivalence. For dividing numbers in a ratio, find the total parts first.

 

Question 4. Give two equivalent ratios for each of the following.
(i) 3:2
(ii) 1 : 6
(iii) 5:4
Answer:
(i) To find equivalent ratios for 3 : 2, we can multiply both numbers by the same whole number. Let's multiply by 2 and then by 3.
Multiplying by 2: \( 3 \times 2 = 6 \) and \( 2 \times 2 = 4 \). So, 6 : 4 is an equivalent ratio.
\( \frac{3}{2}=\frac{3 \times 2}{2 \times 2}=\frac{6}{4} \)
Multiplying by 3: \( 3 \times 3 = 9 \) and \( 2 \times 3 = 6 \). So, 9 : 6 is another equivalent ratio.
\( \frac{3}{2}=\frac{3 \times 3}{2 \times 3}=\frac{9}{6} \)
Thus, two equivalent ratios for 3 : 2 are 6 : 4 and 9 : 6.

(ii) To find equivalent ratios for 1 : 6, we can multiply both numbers by the same whole number. Let's multiply by 2 and then by 3.
Multiplying by 2: \( 1 \times 2 = 2 \) and \( 6 \times 2 = 12 \). So, 2 : 12 is an equivalent ratio.
\( \frac{1}{6}=\frac{1 \times 2}{6 \times 2}=\frac{2}{12} \)
Multiplying by 3: \( 1 \times 3 = 3 \) and \( 6 \times 3 = 18 \). So, 3 : 18 is another equivalent ratio.
\( \frac{1}{6}=\frac{1 \times 3}{6 \times 3}=\frac{3}{18} \)
Thus, two equivalent ratios for 1 : 6 are 2 : 12 and 3 : 18.

(iii) To find equivalent ratios for 5 : 4, we can multiply both numbers by the same whole number. Let's multiply by 2 and then by 3.
Multiplying by 2: \( 5 \times 2 = 10 \) and \( 4 \times 2 = 8 \). So, 10 : 8 is an equivalent ratio.
\( \frac{5}{4}=\frac{5 \times 2}{4 \times 2}=\frac{10}{8} \)
Multiplying by 3: \( 5 \times 3 = 15 \) and \( 4 \times 3 = 12 \). So, 15 : 12 is another equivalent ratio.
\( \frac{5}{4}=\frac{5 \times 3}{4 \times 3}=\frac{15}{12} \)
Thus, two equivalent ratios for 5 : 4 are 10 : 8 and 15 : 12.
In simple words: To get an equivalent ratio, just multiply both numbers in the ratio by the same whole number. For example, if you have 3:2, multiplying both by 2 gives 6:4, which is the same as 3:2.

🎯 Exam Tip: Always show the multiplication steps clearly when finding equivalent ratios to avoid errors and earn full marks.

 

Question 5. Which of the two ratios is larger?
(i) 4 : 5 or 8: 15
(ii) 3: 4 or 7:8
(iii) 1: 2 or 2:1
Answer:
(i) To compare 4 : 5 and 8 : 15, we write them as fractions and find a common denominator.
\( 4:5 = \frac{4}{5} \)
\( 8:15 = \frac{8}{15} \)
The least common multiple (LCM) of 5 and 15 is 15. So, we convert \( \frac{4}{5} \) to a fraction with a denominator of 15.
\( \frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15} \)
Now we compare \( \frac{12}{15} \) and \( \frac{8}{15} \).
Since \( 12 > 8 \), it means \( \frac{12}{15} > \frac{8}{15} \).
Therefore, 4 : 5 is larger than 8 : 15. Comparing fractions with a common denominator helps to identify the larger ratio.

(ii) To compare 3 : 4 and 7 : 8, we write them as fractions and find a common denominator.
\( 3:4 = \frac{3}{4} \)
\( 7:8 = \frac{7}{8} \)
The LCM of 4 and 8 is 8. So, we convert \( \frac{3}{4} \) to a fraction with a denominator of 8.
\( \frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8} \)
Now we compare \( \frac{6}{8} \) and \( \frac{7}{8} \).
Since \( 7 > 6 \), it means \( \frac{7}{8} > \frac{6}{8} \).
Therefore, 7 : 8 is larger than 3 : 4.

(iii) To compare 1 : 2 and 2 : 1, we write them as fractions.
\( 1:2 = \frac{1}{2} \)
\( 2:1 = \frac{2}{1} \)
We compare \( \frac{1}{2} \) and \( \frac{2}{1} \). We know that \( \frac{2}{1} = 2 \) and \( \frac{1}{2} \) is less than 1.
Clearly, \( \frac{2}{1} > \frac{1}{2} \).
Therefore, 2 : 1 is larger than 1 : 2.
In simple words: To compare two ratios, turn them into fractions. Then, make the bottom numbers (denominators) the same. Once the bottom numbers are the same, just look at the top numbers (numerators) – the fraction with the bigger top number is the larger one.

🎯 Exam Tip: Always find the least common multiple (LCM) of the denominators to efficiently compare fractions and determine the larger ratio.

 

Question 6. Divide the numbers given below in the required ratio.
(i) 20 in the ratio 3 : 2
(ii) 27 in the ratio 4: 5
(iii) 40 in the ratio 6 : 14.
Answer:
(i) To divide 20 in the ratio 3 : 2, first find the total number of parts.
Sum of the ratio parts = \( 3 + 2 = 5 \).
Now, find the value of one part: \( \frac{20}{5} = 4 \).
The first part is \( 3 \times 4 = 12 \).
The second part is \( 2 \times 4 = 8 \).
So, 20 divided in the ratio 3 : 2 results in 12 and 8.

(ii) To divide 27 in the ratio 4 : 5, first find the total number of parts.
Sum of the ratio parts = \( 4 + 5 = 9 \).
Now, find the value of one part: \( \frac{27}{9} = 3 \).
The first part is \( 4 \times 3 = 12 \).
The second part is \( 5 \times 3 = 15 \).
So, 27 divided in the ratio 4 : 5 results in 12 and 15.

(iii) To divide 40 in the ratio 6 : 14, first find the total number of parts.
Sum of the ratio parts = \( 6 + 14 = 20 \).
Now, find the value of one part: \( \frac{40}{20} = 2 \).
The first part is \( 6 \times 2 = 12 \).
The second part is \( 14 \times 2 = 28 \).
So, 40 divided in the ratio 6 : 14 results in 12 and 28.
In simple words: To divide a number using a ratio, first add up all the parts of the ratio. Then, divide the total number by this sum to find out how much each "part" is worth. Finally, multiply each number in the ratio by this "part" value to get your divided amounts.

🎯 Exam Tip: Double-check your answer by adding the divided parts; their sum should always equal the original number.

 

Question 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 Rs 4000, then what will be the amount spent for
(i) Provisions and
(ii) Vegetables?
Answer:
The total amount allotted is Rs 4000.
The ratio of spending on Provisions to Vegetables is 3 : 2.
First, find the total number of parts in the ratio: \( 3 + 2 = 5 \) parts.
Next, find the value of one part from the total amount: \( \text{1 part} = \text{Rs } \frac{4000}{5} = \text{Rs } 800 \). This represents the cost of one unit in the ratio.
(i) Amount spent for Provisions: Since Provisions correspond to 3 parts of the ratio,
\( \text{Amount for Provisions} = 3 \times \text{Rs } 800 = \text{Rs } 2400 \).
(ii) Amount spent for Vegetables: Since Vegetables correspond to 2 parts of the ratio,
\( \text{Amount for Vegetables} = 2 \times \text{Rs } 800 = \text{Rs } 1600 \).
In simple words: First, add the numbers in the ratio to get the total number of shares. Then, divide the total money by this sum to find the value of one share. Finally, multiply the value of one share by each number in the ratio to find out how much was spent on Provisions and Vegetables.

🎯 Exam Tip: Clearly label each calculated amount (e.g., "Amount for Provisions") to ensure your answer is easy to understand and follow.

 

Question 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.
Answer:
The total length of the line segment is 63 cm.
The line segment is divided in the ratio 3 : 4.
First, find the total number of parts in the ratio: Sum of the ratio = \( 3 + 4 = 7 \) parts.
Next, find the length of one part from the total length: \( \text{1 part} = \frac{63 \text{ cm}}{7} = 9 \text{ cm} \). This represents the length of one unit in the ratio.
Length of the first part: Since the first part corresponds to 3 parts of the ratio,
\( \text{Length of first part} = 3 \times 9 \text{ cm} = 27 \text{ cm} \).
Length of the second part: Since the second part corresponds to 4 parts of the ratio,
\( \text{Length of second part} = 4 \times 9 \text{ cm} = 36 \text{ cm} \).
So, the 63 cm line segment is divided into two parts with lengths 27 cm and 36 cm. When added together, these parts correctly sum up to 63 cm.
In simple words: Add the numbers in the ratio to find the total shares. Divide the total length by this sum to get the length of one share. Then, multiply the length of one share by each number in the ratio to find the lengths of the two parts of the line segment.

🎯 Exam Tip: Always include the units (e.g., cm) in your final answer for measurements to avoid losing marks.

Objective Type Questions

 

Question 9. If 2 : 3 and 4: ...... are equivalent ratios, then the missing term is
(a) 6
(b) 2
(c) 4
(d) 3
Answer: (a) 6
In simple words: Since 2 multiplied by 2 gives 4, then 3 multiplied by 2 will give 6. So the missing number is 6.

🎯 Exam Tip: For equivalent ratios, look for the factor by which one part of the ratio has been multiplied or divided, and apply the same factor to the other part.

 

Question 10. An equivalent ratio of 4 : 7 is
(a) 1:2
(b) 6:15
(c) 14:8
(d) 12:21
Answer: (d) 12:21
In simple words: An equivalent ratio means that both numbers have been multiplied or divided by the same factor. Here, 4 times 3 is 12, and 7 times 3 is 21, making 12:21 an equivalent ratio.

🎯 Exam Tip: Test each option by checking if both numbers in the option can be divided by the same factor to get the original ratio, or if the original ratio can be multiplied by a factor to get the option.

 

Question 11. Which is not an equivalent ratio of \( \frac{16}{24} \)?
(a) \( \frac{6}{9} \)
(b) \( \frac{12}{18} \)
(c) \( \frac{10}{15} \)
(d) \( \frac{20}{28} \)
Answer: (d) \( \frac{20}{28} \)
In simple words: To find the ratio that is NOT equivalent, simplify each option. The original ratio \( \frac{16}{24} \) simplifies to \( \frac{2}{3} \). Options (a), (b), and (c) also simplify to \( \frac{2}{3} \), but option (d) simplifies to \( \frac{5}{7} \), which is different.

🎯 Exam Tip: Always simplify the given ratio and each option to their lowest terms to easily identify non-equivalent ratios.

 

Question 12. If Rs 1600 is divided
(a) Rs 480
(b) Rs 800
(c) Rs 1000
(d) Rs 200
Answer: (c) Rs 1000
In simple words: This question implies dividing Rs 1600 into parts. Without a specific ratio, it's hard to explain the process, but based on the provided answer, Rs 1000 would be one of the resulting parts.

🎯 Exam Tip: In ratio division problems, ensure the problem statement specifies the ratio or conditions for division clearly. When missing, rely on the given solution as a direct fact.