GSEB Class 6 Maths Solutions Chapter 12 Ratio and Proportion InText Questions

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Question 1. In class, there are 20 boys and 40 girls. What is the ratio of the number of boys to the number of girls?
Answer:
Number of boys = 20
Number of girls = 40
The ratio of the number of boys to the number of girls is calculated as:
\( \frac{ \text{Number of boys} }{ \text{Number of girls} } = \frac{ 20 }{ 40 } \)
\( \frac{ 20 \div 20 }{ 40 \div 20 } = \frac{ 1 }{ 2 } \)
\( \implies \) The ratio is \( 1:2 \).
In simple words: First, count the boys and girls. Then, divide the number of boys by the number of girls. Simplify this fraction to get the final ratio.

Exam Tip: When simplifying ratios, always find the greatest common divisor (GCD) of both numbers to reduce the fraction to its simplest form.

 

Question 2. Ravi walks 6 km in an hour while Roshan walks 4 km in an hour. What is the ratio of the distance covered by Ravi to the distance covered by Roshan?
Answer:
Distance covered by Ravi = 6 km
Distance covered by Roshan = 4 km
The ratio of distance covered by Ravi to the distance covered by Roshan is:
\( \frac{ 6 \text{ km} }{ 4 \text{ km} } = \frac{ 6 }{ 4 } \)
\( \frac{ 6 \div 2 }{ 4 \div 2 } = \frac{ 3 }{ 2 } \)
\( \implies \) The ratio is \( 3:2 \).
Note:
(i) We show a ratio using the symbol ':'.
(ii) A ratio does not have a unit.
(iii) Two amounts can only be compared if they are in the same unit.
(iv) In a ratio, the sequence of terms is very important. For instance, \( 2:3 \) and \( 3:2 \) are different ratios.
In simple words: Compare how far Ravi walks with how far Roshan walks. Put Ravi's distance first, then Roshan's, and simplify the numbers. Remember, the order matters in ratios.

Exam Tip: Always make sure the quantities you are comparing are in the same units before calculating the ratio. If not, convert them first.

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Question 1. Saurabh takes 15 minutes to reach school from his house and Sachin takes one hour to reach school from his house. Find the ratio of the time taken by Saurabh to the time taken by Sachin.
Answer:
Time taken by Saurabh = 15 minutes
Time taken by Sachin = 1 hour = 60 minutes
Therefore, the ratio of time taken by Saurabh to the time taken by Sachin is:
\( \frac{ \text{Time taken by Saurabh} }{ \text{Time taken by Sachin} } = \frac{ 15 }{ 60 } \)
\( \frac{ 15 \div 15 }{ 60 \div 15 } = \frac{ 1 }{ 4 } \)
\( \implies \) The ratio is \( 1:4 \).
(The HCF of 15 and 60 is 15)
In simple words: Saurabh takes 15 minutes, and Sachin takes one hour. Change Sachin's time to minutes, then compare 15 minutes to 60 minutes. Simplify this comparison to find the ratio.

Exam Tip: Always convert different units (like minutes and hours) into the same unit before calculating a ratio to avoid mistakes.

 

Question 2. The cost of the toffee is 50 paise and the cost of chocolate is Rs 10. Find the ratio of the cost of toffee to the cost of chocolate.
Answer:
Cost of toffee = 50 paise
Cost of chocolate = Rs 10 = 1000 paise
(Since 1 Rs = 100 paise)
Ratio = \( \frac{ \text{Cost of toffee} }{ \text{Cost of chocolate} } = \frac{ 50 }{ 1000 } \)
\( \frac{ 50 \div 50 }{ 1000 \div 50 } = \frac{ 1 }{ 20 } \)
\( \implies \) The ratio is \( 1:20 \).
(The HCF of 50 and 1000 is 50)
In simple words: First, change the cost of chocolate from rupees to paise. Then, compare the cost of the toffee to the cost of the chocolate, both in paise. Simplify the numbers to get the final ratio.

Exam Tip: It is crucial to convert all monetary values to the smallest common unit (e.g., paise) before calculating the ratio to ensure accuracy.

 

Question 3. In school, there were 73 holidays in one year. What is the ratio of the number of holidays to the number of days in one year?
Answer:
Number of holidays = 73 days
Number of days in one year = 365 days
Ratio = \( \frac{ \text{Number of holidays} }{ \text{Number of days in a year} } = \frac{ 73 }{ 365 } \)
\( \frac{ 73 \div 73 }{ 365 \div 73 } = \frac{ 1 }{ 5 } \)
\( \implies \) The ratio is \( 1:5 \).
(The HCF of 73 and 365 is 73)
In simple words: Compare the number of holidays (73) to the total days in a year (365). Divide both numbers by their greatest common factor to find the simplest ratio.

Exam Tip: Remember that a standard year has 365 days (a leap year has 366). For such questions, assume 365 days unless specified otherwise.

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Question 1. Find the ratio of the number of notebooks to the number of books in your bag.
Answer: To find this ratio, first count how many notebooks you have in your bag. Then, count the total number of books in your bag. Finally, write the count of notebooks and the count of books as a ratio, like "notebooks : books". This simple activity helps you understand ratios using your own school supplies.
In simple words: Count your notebooks and books. Then, write these two numbers as a ratio.

Exam Tip: Clearly label what each number in your ratio represents (e.g., notebooks to books) to make it easy to understand.

 

Question 2. Find the ratio of the number of desks and chairs in your classroom.
Answer: To calculate this ratio, begin by counting all the desks present in your classroom. Next, count the total number of chairs in the classroom. Once you have both numbers, express them as a ratio, such as "number of desks : number of chairs". This helps you see how many desks there are compared to chairs in your learning area.
In simple words: Count all the desks, then count all the chairs. Write these two counts as a ratio.

Exam Tip: When forming ratios, always list the items in the order they are asked in the question (e.g., "desks to chairs," not "chairs to desks").

 

Question 3. Find the number of students above twelve years of age in your class. Then find the ratio of the number of students with age above twelve years and the remaining students.
Answer: To do this, you first need to identify and count all students in your class who are older than twelve years. After that, find the number of remaining students, which includes those twelve years old or younger. Finally, calculate the ratio by comparing the number of students above twelve to the count of the other students. This exercise helps you group and compare different age groups within your class.
In simple words: Count students older than twelve. Count the other students. Then, make a ratio from these two groups.

Exam Tip: Make sure your two groups (above twelve and remaining) correctly cover all students in the class without any overlap or omissions.

 

Question 4. Find the ratio of the number of doors and the number of windows in your classroom.
Answer: To determine this ratio, start by counting all the doors in your classroom. Next, count every window present in the classroom. Once both numbers are known, write them as a ratio in the form "number of doors : number of windows". This quick count allows you to compare the quantity of doors to windows in your learning space.
In simple words: Count the doors. Count the windows. Write the ratio of doors to windows.

Exam Tip: Be careful to count all doors and windows, even if some are partly hidden or are not standard sizes.

 

Question 5. Draw any rectangle and find the ratio of its length to its breadth.
Answer: To complete this, first draw any rectangle you like on a piece of paper. Then, carefully measure the length of one of its longer sides. After that, measure the length of one of its shorter sides, which is its breadth. Finally, write down these two measurements as a ratio, showing the length compared to the breadth. This practical exercise helps you explore how different dimensions of a rectangle relate to each other.
In simple words: Draw a rectangle. Measure its length and breadth. Then, show the ratio of the length to the breadth.

Exam Tip: Use a ruler to get accurate measurements for both length and breadth before calculating the ratio.

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Question 1. Check whether the given ratios are equal, i.e. they are in proportion. If yes, then write them in the proper form.
1. \( 1:5 \) and \( 3:15 \)
2. \( 2:9 \) and \( 18:81 \)
3. \( 15:45 \) and \( 5:25 \)
4. \( 4:12 \) and \( 9:27 \)
5. Rs 10 to Rs 15 and 4 to 6
Answer:
1. \( 1:5 \) and \( 3:15 \)
We have, \( 3:15 = \frac{ 3 }{ 15 } \)
\( \frac{ 3 \div 3 }{ 15 \div 3 } = \frac{ 1 }{ 5 } \)
\( \implies 1:5 \)
(HCF of 3 and 15 is 3)
i.e., \( 1:5 \) and \( 3:15 \) are in proportion.
Thus, the proper form is \( 1:5::3:15 \).
2. \( 2:9 \) and \( 18:81 \)
We have \( 18:81 = \frac{ 18 }{ 81 } \)
\( \frac{ 18 \div 9 }{ 81 \div 9 } = \frac{ 2 }{ 9 } \)
\( \implies 2:9 \)
(HCF of 18 and 81 is 9)
i.e., \( 2:9 \) and \( 18:81 \) are in proportion.
Thus, the proper form is \( 2:9::18:81 \).
3. \( 15:45 \) and \( 5:25 \)
We have, \( 15:45 = \frac{ 15 }{ 45 } \)
\( \frac{ 15 \div 15 }{ 45 \div 15 } = \frac{ 1 }{ 3 } \)
\( \implies 1:3 \)
(HCF of 15 and 45 is 15)
Also, we have \( 5:25 = \frac{ 5 }{ 25 } \)
\( \frac{ 5 \div 5 }{ 25 \div 5 } = \frac{ 1 }{ 5 } \)
\( \implies 1:5 \)
(HCF of 5 and 25 is 5)
Here, \( 1:3 \neq 1:5 \).
Therefore, \( 15:45 \) and \( 5:25 \) are not in proportion.
4. \( 4:12 \) and \( 9:27 \)
We have, \( 4:12 = \frac{ 4 }{ 12 } \)
\( \frac{ 4 \div 4 }{ 12 \div 4 } = \frac{ 1 }{ 3 } \)
\( \implies 1:3 \)
(HCF of 4 and 12 is 4)
Again we have,
We have, \( 9:27 = \frac{ 9 }{ 27 } \)
\( \frac{ 9 \div 9 }{ 27 \div 9 } = \frac{ 1 }{ 3 } \)
\( \implies 1:3 \)
(HCF of 9 and 27 is 9)
Therefore, \( 4:12 = 9:27 \)
or \( 4:12 \) and \( 9:27 \) are in proportion.
i.e., \( 4:12::9:27 \) is the correct form.
5. Rs 10 to Rs 15 and 4 to 6
We have, \( 10:15 = \frac{ 10 }{ 15 } \)
\( \frac{ 10 \div 5 }{ 15 \div 5 } = \frac{ 2 }{ 3 } \)
\( \implies 2:3 \)
(HCF of 10 and 15 is 5)
Again, we have, \( 4:6 = \frac{ 4 }{ 6 } \)
\( \frac{ 4 \div 2 }{ 6 \div 2 } = \frac{ 2 }{ 3 } \)
\( \implies 2:3 \)
(HCF of 4 and 6 is 2)
Thus, \( 10:15 = 4:6 \).
or Rs 10, Rs 15, 4 and 6 are in proportion.
So, the correct form is Rs \( 10 : \text{Rs }15 :: 4:6 \).
In simple words: To check if two ratios are in proportion, simplify each ratio to its simplest form. If both simplified ratios are the same, then they are in proportion. Otherwise, they are not.

Exam Tip: Always simplify each ratio individually before comparing them. Ensure you write the final proportion using the '::' symbol if they are equal.

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Question 1. Prepare five similar problems (on the unitary method) and ask your friends to solve them.
Answer: To prepare these problems, you should first understand the unitary method, which involves finding the value of a single unit to then find the value of many units. Think of scenarios like finding the cost of one item if you know the cost of five, or how long it takes to do one task if you know how long it takes to do ten. Create five unique questions based on this method, making sure they can be solved by your friends. This task helps you apply and teach the unitary method practically.
In simple words: Understand the unitary method. Create five math problems using it. Ask your friends to solve them.

Exam Tip: When creating problems, ensure the numbers are easy to work with and that the concept of finding a "single unit" is clear in each problem.

 

Question 2. Read the table and fill in the boxes.

Answer:
Distance travelled by Kriti in 2 hours = 6 km
\( \implies \) Distance travelled by Kriti in 1 hour = \( \frac{ 6 }{ 2 } \) km = 3 km
Distance travelled by Karan in 1 hour = 4 km
\( \implies \) Distance travelled by Karan in 4 hours = \( 4 \times 4 \) km = 16 km
Distance travelled by Kriti in 1 hour = 3 km
\( \implies \) Distance travelled by Kriti in 4 hours = \( 3 \times 4 \) km = 12 km
Thus, the table is written as given below:In simple words: First, find out how far each person travels in one hour using the given two-hour data. Then, use those one-hour speeds to calculate the distances they travel in four hours. Fill these new distances into the table.

Exam Tip: For problems involving speed, distance, and time, always determine the rate (distance per unit time) for one unit first, as it simplifies further calculations.