Complete the Ganita Prakash Class 8 Worksheet and NCERT Class 8 Maths Chapter 7 Proportional Reasoning 1 Worksheet with Answers before your unit tests for better preparation.
Proportional Reasoning 1 Worksheet Class 8
Class 8 Maths Proportional Reasoning 1 Worksheet
Proportional Reasoning 1 Class 8 Ganita Prakash Worksheet
Observing Similarity in Change
The Magic of Paper Sizes
Rohait was helping his father in the stationery shop one afternoon when something caught his eye. He noticed stacks of papers in different sizes – A4, A3, A5,… and many more.
What puzzled him was that, although the papers were bigger or smaller, they all looked the same in shape.
Curious, he asked his father why this was so.
His father smiled and explained, “That is because all A-series papers are designed using the same proportions. No matter how large or small the sheet, the length and width of the paper change by the same factor.”
He showed Rohail a few examples:
• Size of A4 paper = 29.7 cm × 21 cm → \(\frac{29.7}{21}\) ≈ 1.414
• Size of A3 paper = 42 cm × 29.7 cm → \(\frac{42}{29.7}\) ≈ 1. 414
So, whenever a paper size is doubled or halved, its shape stays similar.
That is why photocopies, enlargements, and reductions fit perfectly without any distortion.
Question 1.
Based on the above, answer the following questions:
(a) Calculate scale factors of width and height from A4 to A3, and from A3 to A1. Are they the same?
Answer:
A4 → A3 : Scale: 1.414; A3 → Al : Scale : 2; They are not the same.
(b) Why do photocopiers use A-series paper sizes for enlargements/reductions?
Answer:
√2 aspect ratio allows perfect enlargement
Ratios
Question 2.
(a) If an A4 sheet is 29.7 cm × 21 cm, what is the ratio of length to width?
Answer:
1.4 14
(b) An A3 sheet is double the size of an A4 sheet. Check if the ratio is the same.
Answer:
same
(c) An A5 sheet is half of an A4 sheet. What is its size, and what ratio do you get?
Answer:
21 cm x 14.85 cm, 1.414
Riya was excited to set a new profile picture on WhatsApp. She tried cropping and resizing her photo in different ways. The first version looked just right, but the second one appeared stretched, and the third looked squashed. Confused, she wondered why the same photo looked so different each time.
Her elder brother explained, “That happens because of the ratio. When you change the width and height in the same proportion, the.picture keeps its natural shape. But if the proportions are changed unevenly, the image gets distorted either stretched or squashed.”. Riya smiled, now understanding the secret behind a perfect profile picture: keeping the right ratio!
Question 3.
Riga’s original photo was 600 pixels wide and 800 pixels high. She resizes it to a width of 300 pixels and to a height of 500 pixels.
(a) Compare the width and height of Riya’s resized photo with the original one.
(b) By what factors have the width and height changed? Are these factors equal, and does the image remain in the same ratio?
Answer:
(a) Width : reduced by half, Height : reduced by 0.625 Factors are not equal, Image doesn’t remain in same ratio.
Question 4.
Find 3 objects at home or ¡n class with a Length: wLdth ratio close to 2:3. Draw or describe them.
Answer:
Rectangular objects such as: Book, paper, Tablets etc:
Question 5.
Ananya was helping her mother bake a cake. Her mother said, “The recipe uses sugar and flour in the ratio 1 : 4.” What does this mean in practical terms?
Answer:
I unit of sugar and 4 units of flour
Question 6.
What type of operations on the terms of a ratio (for example, a : b) will keep the ratio each section of same?
Answer:
Multiplication and Division by the same number
Ratios in Their Simplest Form
Question 7.
The ratio of the number of boys to the number of girls in a class is 18 : 12.
(a) Write the ratio in its simplest form.
Answer:
(a) 3 : 2
(b) If there are 30 boys in a section of the class, how many girls should there be so that the ratio remains the same?
Answer:
20 8. Yes; 120 : 84 = 90 : 63 = 10 : 7
Question 8.
A rectangle has sides in the ratio 120 : 84. A second rectangle has sides in the ratio 90 : 63. Are these rectangles similar? Justify your conclusion using ratios in their simplest form.
Answer:
Yes; 120 = 84 = 9: 63 = 10 : 7
Question 9.
Width and height of the adjoining three pictures are as follows:
A: 48 cm, 32 cm.
B: 36 cm, 24 cm.
C: 50 cm, 30 cm
Find the ratios of width: height and write in simplest forms for pictures A, B, and C.
Picture A : 48 : 32 = 3:2;
PictureB:3624 = 3 : 2;
Picture C : 50 : 30 = 5 : 3
Problem Solving with Proportional Reasoning
Question 10.
In a conference meeting, the refreshment team prepared 50 glasses of lemonade using 30 spoons of sugar. But when they heard that the number of guests had doubled, they knew they had to make double the lemonade. To keep the same taste, they used the same ratio of sugar to lemonade.
Based on the above, answer the following questions:
(a) What is the ratio of lemonade to sugar in the original preparation? Express it in simplest form.
Answer:
50 : 30; 5 : 3
(b) If instead double, the number of guests triples, how much lemonade and sugar should the organisers prepare to keep the taste consistent? Show the proportional reasoning used.
Answer:
Lemonade: 150 glasses, Sugar : 90 spoons
(c) If the community decides to prepare 125 glasses of lemonade while keeping the same sweetness ratio, how many spoons of sugar will be required? Show how you calculate it using proportional reasoning.
Answer:
75 spoons
Question 11.
Riya mixes 6 lemons and 18 spoons of sugar for 12 glasses of lemonade.
(a) If she wants 24 glasses, how many lemons and spoons of sugar are needed for the same taste?
Answer:
12 lemons and 36 spoons of sugar
(b) If she only used 12 spoons of sugar for 24 glasses, what happens to the taste?
Answer:
Much less sweet
Question 12.
Write your favourite recipe. Double and triple each ingredient, and explain any surprises (e.g., does oven size or cooking time change?)
Answer:
Do it yourself
Question 13.
Kesar mixes 6 glasses of lemonade with 10 spoons of sugar. For double the glasses of lemonade, what is the new ratio, and how much sugar will be needed?
Answer:
6 : 5; 20 spoons of sugar
Question 14.
Two brothers, Ramesh and Suresh, started saving money. Ramesh saved ₹75 and Suresh saved ₹45 in a week.
(a) What is the ratio of Ramesh’s savings to Suresh’s savings?
Answer:
5 : 3
(b) If Ramesh increases his savings to 50, how much should Suresh save to keep the ratio same?
Answer:
₹ 90
Question 15.
A solution contains acid and water in the ratio 84 : 126. Another solution is made by mixing 60 mL of acid and y mL of water; find the value of y such that both solutions have the same simplest ratio of acid to water.
Answer:
90 mL
Question 16.
A parent divides 120 candies in.the ratio 18:12 between their two children. How many does each child get? What is the simplest form of the ratio, and could this ratio be further simplified if the parent wanted to share 180 candies instead?
Answer:
72, 48; 3 : 2, No
Question 17.
The school is preparing for its annual fair, and the organisers need to buy tea leaf for the refreshment stall. Supplier A offers 200 grams for ₹250, while Supplier B offers 500 grams for ₹600. To make a smart purchase, the students decide to solve the following questions:
(a) Calculate the price per gram of tea leaf for both Supplier A and Supplier B. Which supplier offers better value?
Answer:
Supplier A: 1.25 per gram, supplier B : 1.2O per supplier B
(b) The organisers found a third tea supplier C offering 1 kg packets for ?850. Compare the price per gram among all three suppliers and rank them from cheapest to most expensive.
Answer:
Supplier C : 0.85 per gram, C < B < A
Question 18.
(a) If the gift packs contain items in the ratio 3 : 2, how many of each item are needed to prepare 300 gift packs?
Answer:
180 and 120
(b) To celebrate, the organisers want to add a third item to the gift packs such that the ratio of the three items is 3 : 2 : 4. If the total number of items in 300 gift packs is 2700, how many of each item will there be?
Answer:
900, 600 and 1200
Question 19.
Fill in the missing numbers/or these ratios that are proportional to 15: 25.
(a) 3 : _____
(b) 9 : _____c
(c) _____ : 50
(d) _____ : 10
Answer:
(a) 5
(b) 15
(c) 30
(d) 6
Trairasika – The Rule of Three
Question 20.
Green Valley Farmers’ Cooperative grows organic vegetables. For their local market, they prepare for bulk orders. The farmers apply proportional reasoning (Trairasika – the Rule of Three) to plan harvest, packaging, and sales. To supply 40 families, they usually harvest 120 kg of vegetables in a week.
One carton holds 15 kg of vegetables. The cooperative sells 8 cartons of vegetables for ?4,800 to a supermarket.
A local restaurant wants to place a special order of organic spinach. For every 3 kg spinach, they add 1 kg herbs In the package. The farmers also donate some vegetables: when they provide 25 kg for 10 families, they want to know how much Is needed for 60 families.
Based on the above, answer the following questions:
(a) If 120 kg of vegetables serves 40 families, how many kilograms will be required for 150 families?
Answer:
450 kg
(b) How many cartons are required to pack 240 kg of vegetables?
Answer:
16 cartons
(c) If 8 cartons cost ?4,800, what will be the cost of 15 cartons at the same rate?
Answer:
90O0
(d) For the restaurant’s spinach order, if 12 kg of spinach is used, how many kilograms of herbs should be mixed to maintain the ratio?
Answer:
4 kg
(e) For the donation program, if 25 kg of vegetables are enough for 10 families, how many kilograms are needed for 60 families?
Answer:
150 kg
Question 21.
A car travels 240 km in 3 hours. Applying the Rule of Three, calculate the distance it will cover in 7 hours at the same speed. Identify the pramana, phala, and ichchha in this problem.
Answer:
560 km; Pramana : 3 hours, Phala : 240 km, lchchha : 7 hours
Question 22.
Aryabhata’s rule requires consistent units. If 5 metres of cloth cost ₹375, what is the cost of 200 centimetres of the same cloth? (Use the Rule of Three to find the answer)
Answer:
₹150
Question 23.
A recipe states that 2 \(\frac{1}{2}\) (pramana) palas of rice (ichchhaphala) can be served with 15 palas of rice? (phala × ichchha)/pramana.
Answer:
18 people
Question 24.
Using the Rule of Three, determine which of the following is the better deal:
– Packet A: 400 grams of soap for ₹80
– Packet B: 750 grams of soap for ₹135
(Hurt: Find the cost for a common weight, e.g., 100 grams, for both packets and compare).
Answer:
Packet B : 750 grams for 135
Question 26.
What should the price of the 180 mL bottle be if a 6 mL sachet costs ?2? (consider that price 15 proportional to the valume)
Answer:
₹60
Question 27.
An artist wants to scale up a painting. The original canvas is 30 cm x 20 cm. If the new canvas has a width of 120 cm, what should its height be to maintain the proportions? Use the Rule of Three to solve.
Answer:
80 m
Sharing, but not Equally!
Charity Event Donations
Question 28.
A youth club is organising a charity event. They collected 900 food packets to be distributed among three NGOs: A, B, and C.
- They decided to distribute them in the ratio 2:3:4.
- NGO A requests that 50 extra packets be given to them from the share of C.
- Later, 180 more packets are donated, and they want to divide the new packets among A, B, and C in the same original ratio (2:3: 4).
- Meanwhile, another committee member suggests checking what fraction of the total each NGO actually received after these changes.
- Finally, they wonder if they had instead divided the 720 packets equally among all three NGOs, how different would the shares be.
(a) How many packets did each NGO (A, B, C) originally get when 720 packets were distributed in the ratio 2:3:4?
Answer:
A : 160, B : 240, C : 320
(b) After giving 50 extra packets from C’s share to A, how many packets does each NGO have?
Answer:
A : 210, B : 240, C : 270
(c) When 180 more packets arrive, divide them in the original ratio 2:3:4. Add these to each NGO’s new total.
Answer:
A : 250, B : 300, C : 350
(d) What fraction of the final total packets did each NGO get?
Answer:
A : \(\frac{5}{18}\), B : \(\frac{1}{3}\), C : \(\frac{7}{18}\)
Question 29.
Red and white paints are mixed in the ratio 4 : 7 to make pink paint.
(a) To produce 55 mL of pink paint, how much red and white paint are needed?
Answer:
Red paint : 20 ml white point: 35 mL
(b) To make the pink paint slightly darker, 10 mL of red paint is added to the mixture. What is the new ratio of red paint and white paint in the mixture?
Answer:
(b) 6 : 7
Unit Conversions
Question 30.
Convert the following:
(a) 5 metres into feet
(Use 1 m = 3.281 ft)
(b) 2500 square feet into sq metres
(Use 1 sq m – 10.764 sq ft)
(c) 95°F into degrees Celsius
[UseC= – x (F – 32)]
(d) 19.768 acres into hectares
(Use 1 hectare – 2.471 acres)
Answer:
(a) 16405
(b) 232.25 sq metres
(c) 35 degree colours
(d) 8 hectares
Question 31.
Answer the following:
(a) How many millilitres (mL) are there in 3.5 litres?
Answer:
3500 ml
(b) A bucket has a capacity of 15 litres. What is its capacity in cubic centimetres (cc)?
Answer:
15000cc
Question 32.
(a) A farmer has a plot of land measuring 2 acres. How many square feet is this measurement? (1 acre = 43,560 sq ft)
Answer:
87,120 sq ft
(b) A paint can cover 400 square feet of wall space. How many square metres of wall can it cover? (Round to the nearest whole number).
Answer:
37 square metres
(c) A swimming pool has a volume of 80,000 litres. What is its volume in cubic metres? (Hint: 1 cubic metre = 1000 litres)
Answer:
80 cubic metres
(d) One acre of land costs ?15,00,000. What is the cost of a plot of land that is 60 ft by 120 ft?
(Hint: Find area in sq ft, convert to acres, then find cost, 1 acre = 4046.86 sq m)
Answer:
247,95O (app)
(e) A tractor can plough 2 acres of land in 3 hours. How many square metres of land can it plough per hour?
Answer:
2698 square metres per hour
Question 33.
A baby is prescribed 5 cc of medicine. The dropper is labelled in ml. How many ml. should be administered to the baby?
Answer:
5 ml
34. A coiu is made from aa alloy of copper aad aickel ia tKe ratio 3:2. The mass is 6.2 grams. If the cost of copper is ?850 per kg aad aickel is ? 1,200 per kg, what is the cost of the metal ia a siagle coia? (Waraiag: This is a multi-step problem requiriag mass coaversioa aad ratio from a previous sectiou).
Answer:
₹ 6.14
35. Shampoo is sold in two bottles, A : 500 mL for ₹150 aad B : 1.2 L bottle for ₹330. Calculate the price per litre for each bottle. Which one is the better buy? Is the price proportioual to the volume?
Answer:
A : ₹3OO per litre,
B = ₹275 per litre;
Bottle B is the better buy, Not proportional
36. Arbaaz waots to fill bottles with juice for a party. Each bottle holds 500 millilitres (mL). He has prepared 6 litres of juice aad waats to kaow how maay bottles he caa fill. Arbaaz sets up the calculatiou as follows:
He coaverts 6 litres iato millilitres incorrectly.
6 litres = 600 ml
Thea, he calculates the uumber of bottles as:
\(\frac{600 \mathrm{~mL}}{500 \mathrm{~mL}}\) = 1.2 bottle\
He coacludes that he caa fill 1 whole bottle aad a little more.
Ideatify the errors ia Arbaaz’s calculatioa. Correct the errors and calculate how many bottles he can actually fill with 6 litres of juice.
Answer:
Errors: 6 litres = 600 ml, \(\frac{600 \mathrm{~mL}}{500 \mathrm{~mL}}\) = 1.2 bottles
Worksheet On Proportional Reasoning 1 Class 8
A. Choose the correct option.
1. The ratio of wLdth to length of a rectangle s 60 : 40. The ratio in its simplest form, LS:
(a) 6 : 4
(b) 3 : 2
(c) 12 : 8
(d) 30 : 20
Answer:
(b) 3 : 2
2. Which of the following pairs of ratios is proportional?
(a) 4 : 7 and 12 : 21
(b) 8 : 3 and 24 : 6
(c) 7: 12 and 12 : 7
(d) 5: 10 and 8 : 15
Answer:
(a) 4 : 7 and 12 : 21
3. If 4,500 is divided in the ratio 2 : 3, the two parts are:
(a) 1,500 and 3,000
(b) 2,000 and 2,500
(c) 1,800 and 2,700
(d) 2,200 and 2,300
Answer:
(c) 1,800 and 2,700
4. A car travels qO km in 150 minutes. At the same speed, in 4 hours it will travel:
(a) 120 km
(b) 144 km
(c) 1 35 km
(d) 180 km
Answer:
(b) 144 km
5. The cross multiplication condition, for the two ratios a : b arid c : d being proportional is:
(a) a+b = c + d
(b) ad – bc
(c) ab – cd
(d) a – d = b – c
Answer:
(b) ad – bc
Directions. (For Q.6 — 10): In the following questions, a statement of Assertion (A) is followed
by a statement of Reason (R). Choose the correct option OS:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of
Assertion (A).
(c) Assertion (A) is true but Reason (R) is false.
(d) Assertion (A) is false but Reason (R) is true.
6. Assertton (A): If a : b :: c : d, iheri ad = bc.
Reason. (R): When two ratios are proportional, cross multipLying their terms gives equal products.
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
7. Assertion. (A): A ratio remains unchanged if the same number LS added to both its terms.
Reason. (R): For any ratio a : b, (a + k) : (b + k) LS not equaL to a : b.
Answer:
(d) Assertion (A) is false but Reason (R) is true.
8. AssertLon (A): If prices are proportional to quartttttes, the unit price is constant.
Reason (R): For proportional pairs, price + quantity has the same value for all pairs.
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
9. Assertion. (A): When x is divided in the ratio m : n, the two parts are \(m \times \frac{x}{m+n}\)
and \(n \times \frac{x}{m+n}\)
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Reason (R): The ‘group size’ method gives equaL-group size \(\frac{x}{m+n}\) which, multiplied by m and n, gives the two parts respectively.
10. Assertion (A): The ratios 4 : 6 arid 10 : 15 are proportional.
Reason (R): Both ratios reduce to the same simplest form.
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
B. Fill In the blanks.
1. The terms of a ratio are simplified by dividing them with their _______.
Answer:
Highest common factor
2. If x : y :: 6 : 9, then the value of x when y = 18 is _______.
Answer:
12
3. In a ratio 7 : 3 the first part of a total 100 is _______ and the second part is
Answer:
70,30
4. In a proportion 6:10-18: 30, the factor of multiplication is _______.
Answer:
3
5. If a coin contains copper and zink in the ratio 7 : 5 the fraction of copper in the coin is _______.
Answer:
\(\frac{7}{12}\)
C. State whether the following statements are True (T) or False (F).
1. When we add the same number to both the terms of a ratio, the ratio always remains proportional.
Answer:
False
2. The sum of two ratios is always a ratio.
Answer:
False
3. The Rule of Three was known and used in ancient India.
Answer:
True
4. In a proportion 6 : 10 – 18 : 30, the factor of multiplication is _______.
Answer:
True
5. If two quantities are proportional, multiplying both by the same factor keeps them proportional.
Answer:
False
D. Solve the following.
1. Divide ₹6,000 into two parts in the ratio 5 : 7.
Answer:
₹2500, ₹3500
2. A fruit drink is prepared by mixing 600 ml. of orange juice with 900 mL of apple juice. Write the ratio in the simplest form.
Answer:
2 : 3
3. In a school, 150 students require 18 kg of rice for mid-day meals. How much rice is required for 90 students?
Answer:
10.8 kg
4. When Sanya is 4 years old, the ratio of her age to her mother’s age is 1 : 8. When Sanya is 10 years old, what does the ratio become?
Answer:
5 : 19
5. A mixture of 60 kg has sand and cement in the ratio 4 : 3. How much sand should be added to the mixture to make the ratio 2:1?
Answer:
17.13 kg