AP 8th Maths Textbook Solutions – 7. PROPORTIONAL REASONING-1 (2026-27)

7 PROPORTIONAL REASONING-1

7.1 Observing Similarity in Change

• 1. Which images look similar and which ones look different?
  Images A, C, and D look similar because their proportions are maintained. Images B and E look different; B is stretched horizontally (elongated), and E is squashed into a square shape (distorted).
• 2. Do images B and E look like the other three images?
  No, they do not. They appear distorted compared to A, C, and D.

• 1. What makes images A, C, and D appear similar, and B and E different?
  Images A, C, and D appear similar because the ratio of their width to their height remains constant (proportional). B and E appear different because their width and height have changed by different factors, altering their original proportions.
• 2. Can you check by what factors the width and height of image D change as compared to image A? Are the factors the same?
  Width of A = 60, Width of D = 90. Factor = 90/60 = 1.5.
  Height of A = 40, Height of D = 60. Factor = 60/40 = 1.5.
  Yes, both the width and the height change by the exact same factor (1.5 or 3/2).

7.2 Ratios

• 1. By what factor should we multiply the ratio 60:40 (image A) to get 90:60 (image D)?
  We should multiply both terms of the ratio by a factor of 1.5 (or 3/2).
  60 × 1.5 = 90, and 40 × 1.5 = 60.

7.3 Ratios in their Simplest Form

• 1. What is the simplest form of the ratios of images B and E?
  Image B ratio is 40:20. The HCF is 20. Simplest form = 2:1.
  Image E ratio is 60:60. The HCF is 60. Simplest form = 1:1.

7.4 Problem Solving with Proportional Reasoning

• 1. Are the ratios 3:4 and 72:96 proportional?
  Yes. The simplest form of 72:96 (dividing by HCF 24) is 3:4. Since 3:4 = 3:4, they are proportional.

• 1. Is Nitin correct in his thinking?
  No, Nitin is not correct. The ratio of the wall length to cement bags for Nitin is 60:3 (which simplifies to 20:1). For Hari, it is 40:2 (which also simplifies to 20:1). Because the ratios are perfectly proportional, both walls are equally strong.
• 2. In my school, there are 5 teachers and 170 students. The ratio of teachers to students in my school is 5: 170. Count the number of teachers and students in your school. What is the ratio of teachers to students in your school? Write it below. Is the teacher-to-student ratio in your school proportional to the one in my school?
  (Student Activity) First, find the simplest form of 5:170, which is 1:34. Count your school's teachers and students. If your school's ratio simplifies exactly to 1:34, then yes, it is proportional. Otherwise, it is not.
• 3. Measure the width and height (to the nearest cm) of the blackboard in your classroom. What is the ratio of width to height of the blackboard?
  (Student Activity) Measure the blackboard. Write the width:height. Then find their HCF to reduce it to the simplest form.

• 1. Can you draw a rectangle in your notebook whose width and height are proportional to the ratio of the blackboard? Compare the rectangle you have drawn to those drawn by your classmates. Do they all look the same?
  Yes, you can draw it. While the rectangles drawn by different classmates might be different sizes (some smaller, some larger), they will all look structurally identical (similar) because they share the exact same proportions.
• 2. When Neelima was 3 years old, her mother's age was 10 times her age. What is the ratio of Neelima's age to her mother's age? What would be the ratio of their ages when Neelima is 12 years old? Would it remain the same?
  At age 3, mother is 30. Ratio = 3:30 (Simplest form 1:10).
  At age 12 (9 years later), mother is 30 + 9 = 39. Ratio = 12:39 (Simplest form 4:13).
  No, the ratio does not remain the same over time. When a constant value is added to both parts of a ratio, the new ratio is not proportional to the old one.
• 3. Fill in the missing numbers for the following ratios that are proportional to 14: 21.
  ____ : 42    6 : ____    2 : ____
  The simplest form of 14:21 is 2:3.
  To get 42 as the second term (3 × 14), the first term must be 2 × 14 = 28. So, 28 : 42.
  To get 6 as the first term (2 × 3), the second term must be 3 × 3 = 9. So, 6 : 9.
  To get 2 as the first term, the second term must be 3. So, 2 : 3.

• 1. Why is this coffee stronger? (20:30 ratio)
  It is stronger because a higher proportion of the total volume is coffee decoction compared to the regular 15:35 ratio. (20/50 > 15/50).
• 2. Why is this coffee lighter? (10:40 ratio)
  It is lighter because a lower proportion of the total volume is coffee decoction compared to the regular 15:35 ratio, meaning it has much more milk relative to the coffee. (10/50 < 15/50).

• 1. The following table shows the different ratios in which Manjunath mixes coffee decoction with milk. Write in the last column if the coffee is stronger or lighter than the regular coffee.
  Regular ratio is 15:35 (simplifies to 3:7). Total parts = 10. Coffee fraction = 3/10 (30%).
  

  Coffee Decoction	Milk	Regular/Strong/Light
  300	600	Stronger (Ratio 1:2. Coffee fraction 33.3%)
  150	500	Lighter (Ratio 3:10. Coffee fraction 23%)
  200	400	Stronger (Ratio 1:2. Coffee fraction 33.3%)
  24	56	Regular (Ratio 3:7. Coffee fraction 30%)
  100	300	Lighter (Ratio 1:3. Coffee fraction 25%)

Figure it Out - 7.1

• 1. 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
  (i) True (4 × 3 = 12, 7 × 3 = 21)
  (ii) False (8 × 3 = 24, but 3 × 3 ≠ 6)
  (iii) False (Ratios are inverted)
  (iv) True (Both simplify to 7:2)
  (v) False (12:18 simplifies to 2:3, 28:12 simplifies to 7:3)
  (vi) True (Both simplify to 3:1)
• 2. Give 3 ratios that are proportional to 4: 9.
  Examples: 8:18, 12:27, 40:90.
• 3. Fill in the missing numbers for these ratios that are proportional to 18:24.
  The simplest form of 18:24 is 3:4.
  3: 4
  12: 16 (3 × 4 : 4 × 4)
  15 : 20 (3 × 5 : 4 × 5)
  27: 36 (3 × 9 : 4 × 9)
• 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.
  (Practical measurement task). Based on visual proportions, Rectangles A, C, and E appear similar (they share the same length-to-width ratio). Rectangles B and D have different proportions.

• 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?
  Yes. If the original rectangle is, for example, 2 cm by 4 cm (ratio 1:2), a smaller one could be 1 cm by 2 cm, and a bigger one could be 3 cm by 6 cm.
• 6. 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) Count one repeating block of the pattern. Let's say one block has 5 red bricks and 15 grey bricks. The ratio of grey to coloured is 15:5, which simplifies to 3:1.
  (b) Count one repeating block of the diamond pattern. Let's say it contains 8 black bricks and 12 grey bricks. The ratio of grey to black is 12:8, which simplifies to 3:2.
• 7. Let us draw some human figures. Measure your friend's body lengths of their head, torso, arms, and legs. Write the ratios as mentioned below. Does the drawing look more realistic if the ratios are proportional? Why?
  (Student Activity). Yes, the drawing looks significantly more realistic when the ratios are proportional. The human brain naturally recognizes correct bodily proportions; if the arms or torso are scaled incorrectly relative to the head, the figure appears distorted or unnatural.

Trairasika - The Rule of Three

• 1. For the mid-day meal in a school with 120 students, the cook usually makes 15 kg of rice. On a rainy day, only 80 students came to school. How many kilograms of rice should the cook make so that the food is not wasted?
  120 : 15 :: 80 : x
  Using cross-multiplication: 120x = 15 × 80 = 1200.
  x = 1200 / 120 = 10.
  The cook should make 10 kg of rice.

• 1. A car travels 90 km in 150 minutes. If it continues at the same speed, what distance will it cover in 4 hours? Is this the right way to formulate the question? 150: 90::4:?
  No, that formulation is incorrect because the units of time must be identical. 150 is in minutes, but 4 is in hours. The 4 hours must be converted to minutes (4 × 60 = 240 minutes). The correct formulation is 150:90 :: 240:x.
• 2. A small farmer in Himachal Pradesh sells each 200 g packet of tea for ₹200. A large estate in Meghalaya sells each 1 kg packet of tea for ₹800. Are the weight-to-price ratios in both places proportional? Which tea is more expensive? Why?
  No, they are not proportional.
  Himachal: 200g for ₹200 → 1000g (1 kg) costs ₹1000.
  Meghalaya: 1000g (1 kg) costs ₹800.
  The tea from Himachal Pradesh is more expensive (₹1000 per kg compared to ₹800 per kg). This could be because small farm production often involves more manual labor and lacks economies of scale compared to large estates.

Figure it Out - 7.2

• 1. The Earth travels approximately 940 million kilometres around the Sun in a year. How many kilometres will it travel in a week?
  Assume 1 year = 52 weeks.
  Ratio: 52 weeks : 940 million km :: 1 week : x km
  x = 940 / 52 ≈ 18.07 million km per week.
• 2. A mason is building a house in the shape shown in the diagram. He 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?
  First, calculate the total length of the walls.
  Outer walls: 12 + 15 + 12 + 9 + 6 + 9 = 63 ft.
  Inner dividing wall: 12 ft.
  Total length = 63 + 12 = 75 ft.
  Ratio: 10 ft : 1450 bricks :: 75 ft : x bricks.
  x = (1450 × 75) / 10 = 10875 bricks.
• 3. 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: x ? Would it take Puneeth's father more time or less time to reach Kanpur?
  No, we cannot form it as a direct proportion like 50:2 :: 75:x. Speed and time are inversely proportional; as speed increases, time decreases.
  It will take him less time. To solve: Distance = Speed × Time = 50 × 2 = 100 km. New Time = Distance / New Speed = 100 / 75 = 1.33 hours (or 1 hour 20 minutes).

• 1. 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?
  6:180 simplifies to 1:30.
  2:154 simplifies to 1:77.
  They are not proportional. Buying in larger bulk volumes usually costs less per mL because packaging and production costs per unit decrease for the company, and they pass some savings to the customer to encourage bulk purchasing.

7.5 Sharing, but Not Equally!

• 1. If you divide them equally, what is the ratio of the number of counters with each of you?
  6:6, which simplifies to 1:1.
• 2. If your partner gets 5 counters, how many objects will you get? What is the ratio of the counters?
  I will get 12 - 5 = 7 counters. The ratio is 5:7.
• 3. Now, if you want to share the counters between the two of you in the ratio of 3: 1, how many counters would each of you get?
  Total parts = 3 + 1 = 4.
  Value of one part = 12 / 4 = 3 counters.
  Partner gets 3 parts = 3 × 3 = 9 counters.
  I get 1 part = 1 × 3 = 3 counters.

• 1. Now, if you want to share 42 counters between the two of you in the ratio of 4: 3, how will you do it?
  Total groups = 4 + 3 = 7.
  Size of each group = 42 / 7 = 6 counters.
  Partner gets 4 groups = 4 × 6 = 24 counters.
  I get 3 groups = 3 × 6 = 18 counters.

• 1. Prashanti and Bhuvan started a food cart business near their school. Prashanti invested ₹75,000 and Bhuvan invested ₹25,000. At the end of the first month, they gained a profit of ₹4,000. What is each person's share of the profit?
  Investment ratio = 75000 : 25000 = 3:1.
  Total parts = 4.
  Value of one part = 4000 / 4 = ₹1000.
  Prashanti's share = 3 × 1000 = ₹3,000.
  Bhuvan's share = 1 × 1000 = ₹1,000.
• 2. A mixture of 40kg contains sand and cement in the ratio of 3:1. How much cement should be added to the mixture to make the ratio of sand to cement 5: 2?
  Current sand = (3/4) × 40 = 30 kg. Current cement = (1/4) × 40 = 10 kg.
  New ratio sand:cement = 5:2. Sand remains 30 kg.
  5 parts = 30 kg → 1 part = 6 kg.
  New cement needed = 2 parts = 2 × 6 = 12 kg.
  Cement to add = 12 kg (needed) - 10 kg (current) = 2 kg.

Figure it Out - 7.3

• 1. Divide 4,500 into two parts in the ratio 2: 3.
  Total parts = 2 + 3 = 5.
  Value of one part = 4500 / 5 = 900.
  Part 1 = 2 × 900 = 1800.
  Part 2 = 3 × 900 = 2700.
• 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?
  Total parts = 1 + 5 = 6.
  Value of one part = 240 / 6 = 40 mL.
  Acid = 1 part = 40 mL.
  Water = 5 parts = 5 × 40 = 200 mL.
• 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?
  Total parts = 3 + 5 = 8. One part = 40 / 8 = 5 mL.
  Blue = 3 × 5 = 15 mL. Yellow = 5 × 5 = 25 mL.
  Adding 20 mL yellow makes total yellow = 25 + 20 = 45 mL. Blue remains 15 mL.
  New ratio = Blue : Yellow = 15 : 45 = 1:3.
• 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?
  Total parts = 2 + 1 = 3. One part = 6 / 3 = 2 cups.
  Rice = 2 parts = 2 × 2 = 4 cups.
  Urad dal = 1 part = 1 × 2 = 2 cups.
• 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 bucket size be 8 parts. Initial red = 3 parts, Initial yellow = 5 parts.
  Added one full bucket of yellow = 8 parts.
  Total red = 3. Total yellow = 5 + 8 = 13.
  New ratio = 3:13.

Figure it Out - 7.4

• 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.
  Ratio = 600 : 900. Divide by HCF (300). Simplest form = 2:3.
• 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?
  Capacity of 1 bus = 162 / 3 = 54 passengers.
  Buses needed for 204 = 204 / 54 = 3.77 buses. You cannot hire a fraction of a bus, so you need 4 buses.
  No, the buses will not all be full. 3 buses will be full (162 seats), and the 4th bus will carry the remaining 42 students.
• 3. The area of Delhi is 1,484 sq. km and the population is approximately 30 million. Mumbai area is 550 sq. km and population is 20 million. Which city is more crowded? Why?
  Calculate population density (people per sq km).
  Delhi = 30,000,000 / 1484 ≈ 20,215 people/sq km.
  Mumbai = 20,000,000 / 550 ≈ 36,363 people/sq km.
  Mumbai is significantly more crowded because it has a much higher density of people per square kilometer.
• 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?
  The ratio of neck to total body is 4 : (4+6) = 4 : 10 = 2/5 (or 40%).
  Multiply your total height by 0.40 to find what the neck height would be.
• 5. Let us try an ancient problem from Lilavati. "1/2 palas of saffron costs 3/7 niskas... tell me quickly what quantity of saffron can be bought for 9 niskas?"
  Ratio setup: (1/2) palas : (3/7) niskas :: x palas : 9 niskas.
  Cross multiply: (3/7) × x = (1/2) × 9 = 4.5 = 9/2.
  x = (9/2) / (3/7) = (9/2) × (7/3) = 63 / 6 = 10.5 palas.

• 6. Taswika is a 1-year-old girl. Her elder brother is 5 years old. What will be Taswika's age when the ratio of her age to her brother's age is 1: 2?
  Let x be the number of years from now.
  (1 + x) / (5 + x) = 1 / 2
  2(1 + x) = 5 + x → 2 + 2x = 5 + x → x = 3.
  In 3 years, Taswika will be 1 + 3 = 4 years old. (Her brother will be 8, ratio 4:8 = 1:2).
• 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?
  Since the volumes are equal (1 litre), the ratio directly gives the mass relationship.
  Gold : Water = 37 : 2.
  If water mass is 1 kg (2 parts = 1 kg), then 1 part = 0.5 kg.
  Gold mass = 37 parts = 37 × 0.5 = 18.5 kg.
• 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?
  Area of plot = 200 ft × 500 ft = 100,000 sq ft.
  Convert to acres: 100,000 / 43,560 ≈ 2.295 acres.
  Manure needed = 2.295 acres × 10 tonnes/acre = 22.95 tonnes.
• 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?
  Convert 10 litres to mL: 10 × 1000 = 10,000 mL.
  Ratio: 500 mL : 15 sec :: 10,000 mL : x sec.
  10,000 is 20 times larger than 500 (10000 / 500 = 20).
  Time needed = 15 sec × 20 = 300 seconds (or 5 minutes).
• 10. One acre of land costs 15,00,000. What is the cost of 2,400 square feet of the same land?
  1 acre = 43,560 sq ft.
  Cost per sq ft = 15,00,000 / 43,560 ≈ ₹34.435.
  Cost of 2,400 sq ft = 2,400 × 34.435 ≈ ₹82,644.60.
• 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?
  Using Oxen: 20 acres × 6 hours/acre = 120 hours.
  Using Tractor: Since it is 4 times faster, it takes 1/4 the time. 120 / 4 = 30 hours.
• 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?
  Total ratio parts = 3 + 1 = 4.
  Copper mass = (3/4) × 7.74 = 5.805 grams = 0.005805 kg.
  Nickel mass = (1/4) × 7.74 = 1.935 grams = 0.001935 kg.
  Cost of Copper = 0.005805 kg × ₹906/kg ≈ ₹5.26.
  Cost of Nickel = 0.001935 kg × ₹1341/kg ≈ ₹2.59.
  Total Cost of metals = 5.26 + 2.59 = ₹7.85.

IT'S PUZZLE TIME! Binairo

• 1. Solve the following Binairo puzzles.
  (Student Activity. Apply the logic rules: no more than two identical symbols adjacent horizontally or vertically, equal number of horizontal and vertical symbols per row/column, and no identical rows/columns.)