Proportional Reasoning-1 - Chapter 7 Class 8 (Ganita Prakash)
Master Proportional Reasoning-1 - Chapter 7 Class 8 (Ganita Prakash) with comprehensive NCERT Solutions, Practice Questions, MCQs, Sample Papers, Case Based Questions, and Video lessons.
NCERT Solutions
Proportional Reasoning-1 - Chapter 7 Class 8 (Ganita Prakash) – NCERT Solutions
Each question below opens its complete step-by-step Teachoo solution.
Figure it out - Page 165-167
7 questionsQuestion 1
(i) Circle the following statements of proportion that are true. (i) 4 : 7 :: 12 : 21To check proportional, we convert them into fractions
4/7=12/21
Question 2
Give 3 ratios that are proportional to 4 : 9. ______ : ______ ______ : ______ ______ : ______ To find ratios that are proportional, we just multiply both sides by any number
View solutionQuestion 3
Fill in the missing numbers for these ratios that are proportional to 18 : 24. 3 : ______ 12 : ______ 20 : ______ 27 : ______We can write this as
View solutionQuestion 4
Look at the following rectangles. Which rectangles are similar to each other? You can verify this by measuring the width and height using a scale and comparing their ratios.Rectangles C, D, and E are similar.
View solutionQuestion 5
Look at the following rectangle. Can you draw a smaller rectangle and a bigger rectangle with the same width to height ratio in your notebooks? Compare your rectangles with your classmates’ drawings. Are all of them the same? If they are different from yours, can you think why? Are they wrong?Draw a rectangle. Let's say 4 cm × 2cm
Draw a bigger one:
Multiply sides by 2→ 8 cm × 4 cm
Draw a smaller one:
Divide sides by 2→ 2 cm × 1 cm
Are they wrong?
If your classmate draws a rectangle that is 4 cm × 3 cm, it is "wrong" because the ratio changed (it got fatter).
Question 6
(a) The following figure shows a small portion of a long brick wall with patterns made using coloured bricks. Each wall continues this pattern throughout the wall. What is the ratio of grey bricks to coloured bricks? Try to give the ratios in their simplest form. (a)We calculate from one repeatable block
Repeatable block is marked in the photo
Question 7
Let us draw some human figures. Measure your friend’s body—the lengths of their head, torso, arms, and legs. Write the ratios as mentioned below— Now, draw a figure with head, torso, arms, and legs with equivalent ratios as above.Okay let’s first measure our friend
Measurements of my friend's body:
Head
Figure it out - Page 170-171
2 questionsQuestion 1
The Earth travels approximately 940 million kilometres around the Sun in a year. How many kilometres will it travel in a week?In one year, Earth travels 940 million kilometers
Let’s convert year into weeks
1 year = 52 weeks
Question 2
A mason is building a house in the shape shown in the diagram. He needs to construct both the outer walls and the inner wall that separates two rooms. To build a wall of 10-feet, he requires approximately 1450 bricks. How many bricks would he need to build the house? Assume all walls are of the same height and thickness.
First, we calculate the Total Length of wall required
And, then we find Number of bricks
Total Length of wall required
Figure it out - Page 175
5 questionsQuestion 1
Divide ₹4,500 into two parts in the ratio 2 : 3.Let’s divide it into two parts – Part A & Part B
View solutionQuestion 2
In a science lab, acid and water are mixed in the ratio of 1 : 5 to make a solution. In a bottle that has 240 mL of the solution, how much acid and water does the solution contain?Given that acid and water are mixed in the ratio of 1 : 5
So, we can write
Amount of Acid : Amount of Water = 1 : 5
Question 3
Blue and yellow paints are mixed in the ratio of 3 : 5 to produce green paint. To produce 40 mL of green paint, how much of these two colours are needed? To make the paint a lighter shade of green, I added 20 mL of yellow to the mixture. What is the new ratio of blue and yellow in the paint?First let us answer the first part of the question
To produce 40 mL of green paint, how much of these two colours are needed?
Question 4
To make soft idlis, you need to mix rice and urad dal in the ratio of 2 : 1. If you need 6 cups of this mixture to make idlis tomorrow morning, how many cups of rice and urad dal will you need?Given that rice and urad dal are mixed in the ratio of 2 : 1
So, we can write
Cups of Rice : Cups of Urad Dal = 2 : 1
Question 5
I have one bucket of orange paint that I made by mixing red and yellow paints in the ratio of 3 : 5. I added another bucket of yellow paint to this mixture. What is the ratio of red paint to yellow paint in the new mixture?Let's assume "One Bucket" holds 8 Litres
(we chose 8 because 3+5=8, making the math easy).
Figure it out - Page 176, 177
12 questionsQuestion 1
Anagh mixes 600 mL of orange juice with 900 mL of apple juice to make a fruit drink. Write the ratio of orange juice to apple juice in its simplest form.Since Anagh mixes 600 mL of orange juice with 900 mL of apple juice
The ratio of orange juice to apple juice is
600 : 90
Question 2
Last year, we hired 3 buses for the school trip. We had a total of 162 students and teachers who went on that trip and all the buses were full. This year we have 204 students. How many buses will we need? Will all the buses be full?Let’s find the Number of people that can fit in 1 bus
Since 162 students and teachers went on 3 buses and the bus was full
Thus,
Number of people that can fit in 1 bus = 𝟏𝟔𝟐/𝟑
= 54
Question 3
The area of Delhi is 1,484 sq. km and the area of Mumbai is 550 sq. km. The population of Delhi is approximately 30 million and that of Mumbai is 20 million people. Which city is more crowded? Why do you say so?To find which city is more crowded, we find Population Density
And, Population density is defined as
Population Density = 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛/𝐴𝑟𝑒𝑎
Question 4
A crane of height 155 cm has its neck and the rest of its body in the ratio 4 : 6. For your height, if your neck and the rest of the body also had this ratio, how tall would your neck be?Given ratio of Neck to Rest of Body is 4 : 6
View solutionQuestion 5
Let us try an ancient problem from Lilavati. At that time weights were measured in a unit named palas and niskas was a unit of money. “If 2 1/2 palas of saffron costs 3/7 niskas, O expert businessman! tell me quickly what quantity of saffron can be bought for 9 niskas?” First, we convert mixed fractions to fractions
𝟐 𝟏/𝟐 palas =2+1/2 palas
=(2 × 2 + 1)/2 palas
=(4 + 1)/2 palas
=𝟓/𝟐 palas
Question 6
Harmain is a 1-year-old girl. Her elder brother is 5 years old. What will be Harmain’s age when the ratio of her age to her brother’s age is 1 : 2?Now,
Harmain’s age = 1 year
Harmain’s brothers age = 5 years
Question 7
The mass of equal volumes of gold and water are in the ratio 37 : 2. If 1 litre of water is 1 kg in mass, what is the mass of 1 litre of gold?Given that the mass of equal volumes of gold and water are in the ratio 37 : 2
So, we can say
Mass of Gold : Mass of Water = 37 : 2
Question 8
It is good farming practice to apply 10 tonnes of cow manure for 1 acre of land. A farmer is planning to grow tomatoes in a plot of size 200 ft by 500 ft. How much manure should he buy? (Please refer to the section on Unit Conversions earlier in this chapter).We follow these steps
We find area of plot in square feet
We convert area into acres as manure is in terms of acres
We find amount of Manure
Question 9
A tap takes 15 seconds to fill a mug of water. The volume of the mug is 500 mL. How much time does the same tap take to fill a bucket of water if the bucket has a 10-litre capacity?First, we convert units so they match.
Bucket capacity =10 Litres =𝟏𝟎,𝟎𝟎𝟎" " 𝐦𝐋.
Question 10
One acre of land costs ₹15,00,000. What is the cost of 2,400 square feet of the same land?Since we need to find of square feet of land,
We find cost per square foot.
Question 11
A tractor can plough the same area of a field 4 times faster than a pair of oxen. A farmer wants to plough his 20-acre field. A pair of oxen takes 6 hours to plough an acre of land. How much time would it take if the farmer used a pair of oxen to plough the field? How much time would it take him if he decides to use a tractor instead?Given that
A pair of oxen takes 6 hours to plough an acre of land
Since farmer wants to plough his 20-acre field
Time taken by pair of oxen = 20 acres × 6 hours/acre
= 120 hours
Question 12
The ₹10 coin is an alloy of copper and nickel called ‘cupro-nickel’. Copper and nickel are mixed in a 3 : 1 ratio to get this alloy. The mass of the coin is 7.74 grams. If the cost of copper is ₹906 per kg and the cost of nickel is ₹1,341 per kg, what is the cost of these metals in a ₹10 coin?Let’s Find the mass of each metal.
View solutionWhy Learn This With Teachoo?
Proportional Reasoning–1 is Chapter 7 of NCERT Class 8 Ganita Prakash Part 1. It teaches students to recognise situations in which two quantities change in the same ratio, solve proportional problems, apply the traditional Trairāśika or Rule of Three, divide quantities in unequal shares and convert units. Teachoo provides concept explanations and complete solutions to the chapter’s activities and Figure it out questions.
What is proportional reasoning?
Two quantities are directly proportional when multiplying one by a factor multiplies the other by the same factor. Their ratio remains constant. If twice as many identical items cost twice as much, the cost and number of items show direct proportion.
Observing Similarity in Change asks students to compare how quantities vary rather than looking at isolated numbers. Ratio, rate and unit rate supply the language for this comparison. A proportion states that two ratios are equal.
Problem-solving methods include scaling up or down, finding the value for one unit and using equivalent ratios. A reliable method should preserve units and show the relation between the quantities. Cross multiplication may be efficient, but it should represent an understood equality of ratios.
Trairāśika, sharing and conversion
Trairāśika, the Rule of Three, is an Indian mathematical procedure for finding a fourth value from three related values. It is closely connected with the unitary method and proportional scaling. Studying it gives both a practical technique and historical perspective.
“Sharing, but Not Equally!” divides a total in a given ratio. If a quantity is shared in the ratio a, there are a + b total parts. The value of one part is found first, after which each share is calculated.
Unit conversion is another proportional situation. Converting kilometres to metres or hours to minutes requires a known scale factor. Units must change together with numerical values, and dimensional meaning should remain consistent.
Topics covered on Teachoo
Teachoo includes:
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observing similarity in change;
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definitions of ratio, rate and proportion;
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proportional problem solving;
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Figure it out solutions for pages 165–167;
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Trairāśika or the Rule of Three;
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Figure it out solutions for pages 170–171;
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the activity and question on page 171;
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unequal sharing in a ratio;
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Figure it out solutions for page 175;
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unit conversion; and
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Figure it out solutions for pages 176–177.
Learning outcomes
Students should be able to decide whether a relationship is proportional, find a constant ratio or unit rate and solve through scaling, equivalent ratios or the Rule of Three. They should divide a whole in a stated ratio, convert units accurately and explain what each ratio compares. They should also reject a proportional method when an additive rather than multiplicative relationship is present.
Why is this chapter important?
Proportional reasoning is used in recipes, maps, prices, speed, scale drawings, mixtures, finance and science. It links fractions, decimals and percentages with algebra. Many later formulas describe proportional relationships, so understanding the structure is more valuable than memorising one procedure.
How Teachoo helps
Teachoo organises the chapter by concept and application. For each problem, write the two quantities with units in consistent columns. Ask whether doubling one should double the other. Find a unit rate or equivalent ratio, estimate the direction of change and then calculate.
When sharing, add the ratio parts before finding one part. When converting, write the conversion factor as a fraction whose units cancel appropriately. Compare your work with Teachoo’s explanation only after making an independent attempt.
Common mistakes to avoid
Not every pair of increasing quantities is proportional. A fixed starting charge plus a variable charge is not a direct proportion. Keep ratio order consistent and convert quantities to compatible units before comparing. In unequal sharing, do not divide by only one term of the ratio; divide the total by the sum of all parts.
Quick revision checklist
Test several tables to decide whether their ratios are constant. Solve one question each by scaling, unit rate and the Rule of Three, then compare the methods. Divide a total in a two-part and a three-part ratio and verify that the shares add to the original amount. Finish with two unit conversions in which the numerical value changes in opposite directions, explaining why the quantity itself remains unchanged.
Deeper reasoning and concept connections
In Proportional Reasoning–1 (Ganita Prakash), fluency means more than repeating a procedure. Students should be able to recognise the underlying structure when the numbers, diagram, wording or orientation changes. A useful routine is: identify the mathematical objects, list the known and unknown quantities, state the governing property, carry out the steps and verify that every condition has been used.
Look for connections within the chapter as well. A definition usually leads to a representation; the representation reveals a pattern; and the pattern supports a rule or calculation. Explaining this chain improves retention and helps with case-based questions. It also prevents the common mistake of selecting a formula simply because its symbols resemble the numbers in the question.
How to solve unfamiliar and competency-based questions
Use a five-step response: interpret, represent, select, solve and verify. Interpret the wording; represent the information; select a definition, property or formula; solve without skipping the logical step; and verify through substitution, estimation, measurement or an alternative representation. This routine works for direct exercises as well as case-based questions.
If information appears unnecessary, ask whether it establishes a hidden condition. If information is missing, state what cannot be determined instead of inventing a value. In written answers, name the rule being used. Clear reasoning helps a teacher award method marks and also makes the page easier for a student—or an AI answer system—to retrieve for the precise doubt being asked.
What complete mastery looks like
For Proportional Reasoning–1 (Ganita Prakash), a student should be able to define the central ideas in simple language, recognise them in different representations, solve routine questions accurately and explain the method used. They should also be able to correct a flawed solution, create an example satisfying given conditions and combine two ideas from the chapter in one problem. A reliable mastery test is to solve one direct question, one application question and one reasoning question without looking at notes, then explain all three aloud or in writing.
Keep a compact error log with four labels: concept, interpretation, calculation and presentation. Reattempt each error after a gap instead of rereading the answer immediately. Improvement comes from correcting the decision that caused the mistake, not from repeating questions whose method is already known.
Additional frequently asked questions
What should a student know before starting Proportional Reasoning–1 (Ganita Prakash)?
Revise the definitions, number operations, diagrams or notation used at the beginning of the chapter. The prerequisite list should be short: if an earlier skill blocks progress, repair that skill with two or three focused questions and return to the chapter.
How can a student check an answer in Proportional Reasoning–1 (Ganita Prakash)?
Use an independent check whenever possible: substitute the result, reverse the operation, estimate its size, compare it with the figure, test a simpler case or solve using another representation. A check should examine the mathematical condition, not merely repeat the same arithmetic.
How many questions are enough for strong preparation?
There is no fixed number. Stop counting questions and track coverage: every concept, every standard method, at least one mixed problem, one competency-based problem and every previously incorrect type should be solved independently. Ten varied, analysed questions are more valuable than fifty copied solutions.
How should Teachoo solutions be used without becoming dependent on them?
Attempt the question first and mark the exact step where progress stops. Read only enough of the solution to repair that step, close it and restart the question. Finally, solve a similar problem without help. This turns a solution into feedback rather than a substitute for thinking.
Frequently asked questions
What is direct proportion?
It is a relationship in which two quantities change by the same multiplicative factor and maintain a constant ratio.
What is the Rule of Three?
It is a method for finding a fourth proportional value from three known related values.
How is a quantity divided in a ratio?
Add the ratio terms, find the value of one part and multiply that value by each ratio term.
Always identify the relationship before choosing the method. Correct proportional reasoning begins with structure, not cross multiplication.