Figure it Out
Q1. Circle the following statements of proportion that are true. (i) 4 : 7 :: 12 : 21 (ii) 8 : 3 :: 24 : 6 (iii) 7 : 12 :: 12 : 7 (iv) 21 : 6 :: 35 : 10 (v) 12 : 18 :: 28 : 12 (vi) 24 : 8 :: 9 : 3
Answer:
For two ratios a:b and c:d to be proportional, cross multiplication gives a*d = b*c.
(i) 4*21 = 84, 7*12 = 84, true.
(ii) 8*6 = 48, 3*24 = 72, false.
(iii) 7*7 = 49, 12*12 = 144, false.
(iv) 21*10 = 210, 6*35 = 210, true.
(v) 12*12 = 144, 18*28 = 504, false.
(vi) 24*3 = 72, 8*9 = 72, true.
True statements: (i), (iv), (vi).
Q2. Give 3 ratios that are proportional to 4 : 9.
Answer:
Any ratio obtained by multiplying both terms of 4:9 by the same non-zero number gives a proportional ratio. Examples: 8:18 (multiply by 2), 12:27 (multiply by 3), 16:36 (multiply by 4).
Q3. Fill in the missing numbers for these ratios that are proportional to 18 : 24. 3 : ______? 12 : ______? 20 : ______? 27 : ______
Answer:
Simplify 18:24 to 3:4 (divide by 6). So all proportional ratios are equivalent to 3:4.
For 3 : ?, multiply both terms of 3:4 by 1 gives 3:4.
For 12 : ?, multiply 3 by 4 to get 12, so 4*4=16 gives 12:16.
For 20 : ?, multiply 3 by 20/3 = 20/3, but easier: 20/3 = ?/4 gives ? = (20*4)/3 = 80/3 ≈ 26.67, not integer. Alternatively, note 20 = 5*4, so first term is 5 times 4? Actually, 3:4 = 20:? implies 3*? = 4*20 => ? = 80/3, so it is 20:80/3. Typically we expect whole numbers, but the problem allows fractions: 20 : 80/3. However, if we want integer, we can multiply by 2: 6:8; but 20 is not a multiple of 3.
For 27 : ?, 3*9=27, so 4*9=36 gives 27:36.
So answers: 3:4, 12:16, 20:80/3, 27:36.
Q4. 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.
Answer:
Measure the width and height of each rectangle (A, B, C, D, E) with a ruler. Compute the ratio width:height for each. Rectangles with the same simplest form ratio are similar. For example, if A has ratio 3:2, C has 3:2, D has 3:2, then A, C, D are similar; B and E may have different ratios. Draw the rectangles as labeled in the text.
Q5. 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?
Answer:
Measure the width and height of the given rectangle. Let its ratio be w:h (simplified). To draw a smaller rectangle, choose a scale factor less than 1 (e.g., 1/2) and multiply both dimensions. For a bigger rectangle, choose a factor greater than 1. The drawings will have the same shape but different sizes. Classmates may choose different scale factors, so rectangles will differ in size but are all correct as long as the ratio matches.
Q6. 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.
Answer:
Count the number of grey bricks and coloured bricks in the given pattern (from the figure in the text). Suppose there are 4 grey and 6 coloured bricks; then ratio grey:coloured = 4:6 = 2:3 in simplest form. Adjust based on actual count from the diagram.
Q7. 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— head : torso ______ : ______ torso : arms ______ : ______ torso : legs ______ : ______ Now, draw a figure with head, torso, arms, and legs with equivalent ratios as above. Does the drawing look more realistic if the ratios are proportional? Why? Why not?
Answer:
Measure a friend: head length (e.g., 20 cm), torso (e.g., 50 cm), arms (e.g., 60 cm), legs (e.g., 70 cm). Compute ratios: head:torso = 20:50 = 2:5; torso:arms = 50:60 = 5:6; torso:legs = 50:70 = 5:7. Draw a figure using these ratios (e.g., head 2 cm, torso 5 cm, arms 6 cm, legs 7 cm). The drawing looks more realistic if ratios match real proportions because the human body has consistent proportions.
Q1. The Earth travels approximately 940 million kilometres around the Sun in a year. How many kilometres will it travel in a week?
Answer:
First, find the distance traveled in one day: 940,000,000 km / 365 days ≈ 2,575,342.47 km per day. Then multiply by 7 days: 2,575,342.47 × 7 ≈ 18,027,397.26 km. So, the Earth travels approximately 18.03 million km in a week.
Q2. 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. (Diagram: 12 ft, 15 ft, 9 ft, 9 ft, 6 ft)
Answer:
We need to calculate the total length of all walls. From the diagram: outer walls: 12 ft + 15 ft + 9 ft + 6 ft = 42 ft. Inner wall: 9 ft. Total wall length = 42 + 9 = 51 ft. For 10 ft wall, 1450 bricks are needed. For 1 ft, 1450/10 = 145 bricks. For 51 ft, 145 × 51 = 7395 bricks. So, he needs approximately 7395 bricks.
Q3. Puneeth's father went from Lucknow to Kanpur in 2 hours by riding his motorcycle at a speed of 50 km/h. If he drives at 75 km/h, how long will it take him to reach Kanpur? Can we form this problem as a proportion?— 50 : 2 :: 75 : __. Would it take Puneeth's father more time or less time to reach Kanpur? Think about it. Even though this problem looks similar to the previous problems, it cannot be solved using the Rule of Three!
Answer:
Distance = speed × time = 50 × 2 = 100 km. At 75 km/h, time = distance / speed = 100 / 75 = 4/3 hours = 1 hour 20 minutes. So, it takes less time. This is an inverse proportion (as speed increases, time decreases), not a direct proportion, so the Rule of Three (direct proportion) cannot be applied. The correct proportion would be 50 × 2 = 75 × time, i.e., product is constant.
Q1. Divide ₹4,500 into two parts in the ratio 2:3.
Answer:
Total parts = 2+3 = 5. Value of each part = 4500/5 = 900. First part = 2 × 900 = ₹1,800. Second part = 3 × 900 = ₹2,700.
Q2. 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?
Answer:
Total parts = 1+5 = 6. Each part = 240/6 = 40 mL. Acid = 1 × 40 = 40 mL. Water = 5 × 40 = 200 mL.
Q3. 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?
Answer:
Total parts = 3+5 = 8. Each part = 40/8 = 5 mL. Blue = 3×5 = 15 mL. Yellow = 5×5 = 25 mL. After adding 20 mL yellow: new yellow = 25+20 = 45 mL, blue remains 15 mL. New ratio blue:yellow = 15:45 = 1:3.
Q4. 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?
Answer:
Total parts = 2+1 = 3. Each part = 6/3 = 2 cups. Rice = 2×2 = 4 cups. Urad dal = 1×2 = 2 cups.
Q5. 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?
Answer:
Original mixture: total parts = 3+5 = 8. Let each part = 1 unit. Red = 3 units, Yellow = 5 units. Added another bucket of yellow paint (assuming same volume as original bucket = 8 units). New yellow = 5 + 8 = 13 units. Red remains 3 units. New ratio red:yellow = 3:13.
Q1. 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.
Answer:
Ratio = 600:900 = 600/300 : 900/300 = 2:3.
Q2. 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?
Answer:
Last year: 162 people in 3 buses, so each bus capacity = 162/3 = 54 people. This year: 204 students. Buses needed = 204/54 = 3.78, so we need 4 buses. 4 buses can carry 4×54 = 216 people, so 204 will fill 3 full buses (162) and the 4th bus will have 204-162 = 42 people, not full.
Q3. 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?
Answer:
Population density of Delhi = 30,000,000 / 1484 ≈ 20,215 people per sq km. Population density of Mumbai = 20,000,000 / 550 ≈ 36,364 people per sq km. Mumbai is more crowded because it has higher population density.
Q4. 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?
Answer:
Total parts = 4+6 = 10. Neck height = (4/10) × 155 = 62 cm. For a person of height H, neck height = (4/10) × H = 0.4H. So if student's height is, say, 150 cm, neck would be 60 cm. (General formula: neck = 0.4 × height)
Q5. 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 2½ palas of saffron costs 3/7 niskas, O expert businessman! tell me quickly what quantity of saffron can be bought for 9 niskas?'
Answer:
By direct proportion: (2.5 palas) / (3/7 niskas) = x palas / 9 niskas. Cross multiply: 2.5 × 9 = (3/7) × x => 22.5 = (3/7)x => x = 22.5 × 7/3 = 22.5 × 7 ÷ 3 = 157.5/3 = 52.5 palas. So, 52.5 palas of saffron can be bought for 9 niskas.
Q6. 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?
Answer:
Let Harmain's age = x. Brother's age = x + 4 (since brother is 4 years older). Ratio x : (x+4) = 1:2. So x/(x+4) = 1/2 => 2x = x+4 => x = 4. So Harmain will be 4 years old and brother will be 8 years old (ratio 4:8 = 1:2).
Q7. 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?
Answer:
Ratio gold:water = 37:2. 1 litre water = 1 kg. So 1 litre gold = (37/2) × 1 kg = 18.5 kg.
Q8. 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?
Answer:
Area of plot = 200 ft × 500 ft = 100,000 sq ft. Convert to acres: 1 acre = 43,560 sq ft. So, area in acres = 100,000 / 43,560 ≈ 2.295 acres. Manure needed = 2.295 × 10 = 22.95 tonnes. So, he should buy approximately 23 tonnes of manure.
Activity 1
Q1. Take your favourite dish. Find out all the ingredients and their respective quantities needed to make the dish for your family. Suppose you are celebrating a festival and you want to invite 15 guests. Find out the quantities of the ingredients required to cook the same dish for them.
Answer:
This is an open-ended activity. Students should list ingredients for their dish for their family (e.g., 4 people), then scale up each ingredient quantity proportionally for 15 guests. For example, if a dish requires 2 cups of rice for 4 people, then for 15 people, required rice = (2/4) × 15 = 7.5 cups. All ingredients should be scaled similarly.
Activity 2
Q1. Go to the market and collect the prices of different sizes of shampoo containers of the same shampoo and create a table like the one given below. See if the volume of shampoo is proportional to the price. Container Volume Price Sachet 6 mL ₹2 Small Bottle 180 mL ₹154 Medium Bottle 340 mL ₹276 Large Bottle 1000 mL ₹540. Let us compare the ratios for the sample table above. The ratio of the volume of a sachet to a small bottle is 6 : 180. The ratio of their prices is 2 : 154. Are these ratios proportional? Why do you think that the ratio of the prices is not proportional to the ratio of the volumes? Discuss the pros and cons of different size bottles for the company and for customers. For reducing ecological footprint, what would you recommend to the company and to the customer? Does the same occur for other products? Make similar tables for other products in the market, capturing different prices for different measures of the same product, e.g., rice or atta (flour). Observe the products for which the prices are proportional to the different measures. Discuss in class the proportionality of prices to measures of the same product.
Answer:
This is an activity-based question. For the sample data: Ratio of volumes 6:180 simplifies to 1:30. Ratio of prices 2:154 simplifies to 1:77. Since 1:30 ≠ 1:77, they are not proportional. Prices are not proportional to volume because larger containers often have lower per-unit cost due to packaging and marketing. For ecological footprint, recommend larger containers to reduce plastic waste. Students should collect real market data and observe if prices are proportional or not and discuss reasons.
Activity 3
Q1. Form a pair. Collect 12 countable objects or counters (it can be coins, seeds, or pebbles). Now, share them between the two of you in different ways. If you divide them equally, what is the ratio of the number of counters with each of you?
Answer:
Each gets 6 counters. Ratio is 6:6 = 1:1.
Q2. Now let us not share equally. If your partner gets 5 counters, how many objects will you get? What is the ratio of the counters?
Answer:
You will get 12 - 5 = 7 counters. Ratio of partner to you is 5:7.
Q3. Now, if you want to share 12 counters between the two of you in the ratio of 3:1, how many counters would each of you get?
Answer:
Total parts = 3+1 = 4. Size of each part = 12/4 = 3. Partner gets 3×3 = 9 counters, you get 1×3 = 3 counters.
Q4. Now, if you want to share 42 counters between the two of you in the ratio of 4:3, how will you do it?
Answer:
Total parts = 4+3 = 7. Size of each part = 42/7 = 6. Partner gets 4×6 = 24 counters, you get 3×6 = 18 counters.