CBSE Class 8 Mathematics Proportional Reasoning 1 Short Answer Questions with Answers

Short Answer Questions

Question 1. Reduce each of the following ratios to its simplest form, showing the HCF you divide by:
(i) 84 : 126
(ii) 45 : 75
Answer:
(i) HCF of 84 and 126 is 42. 84 ÷ 42 = 2 and 126 ÷ 42 = 3, so 84 : 126 = 2 : 3.
(ii) HCF of 45 and 75 is 15. 45 ÷ 15 = 3 and 75 ÷ 15 = 5, so 45 : 75 = 3 : 5.
Dividing both terms by the same number does not change the ratio; it only writes it in its simplest form.
In simple words: Divide both numbers by their HCF to get 2 : 3 and 3 : 5.

Teacher's Note:
1. Find the HCF by prime factors if it is not obvious.
2. A single division by the HCF finishes the job.

Question 2. A photograph is 45 mm wide and 30 mm high. It is printed again as a picture 72 mm wide and 48 mm high. By comparing the simplest forms of the two width-to-height ratios, decide whether the two pictures look similar.
Answer:
Original: 45 : 30. HCF is 15, so the simplest form is 3 : 2.
New print: 72 : 48. HCF is 24, so the simplest form is 3 : 2.
Both simplest forms are equal, so 45 : 30 :: 72 : 48. Width and height are both multiplied by 1.6, so the pictures look similar.
In simple words: Both pictures reduce to 3 : 2, so they have the same shape.

Teacher's Note:
1. Equal simplest forms mean the ratios are proportional.
2. 72 ÷ 45 = 48 ÷ 30 = 1.6 confirms the common factor.

Question 3. A class has 18 boys and 24 girls.
(i) Write the ratio of boys to girls in its simplest form.
(ii) Explain, in the "for every … there are …" language of ratios, what this simplest form tells you.
Answer:
(i) Boys : girls = 18 : 24. HCF is 6, so the simplest form is 3 : 4.
(ii) For every 3 boys there are 4 girls in the class. The class is made of 6 such groups: 3 × 6 = 18 boys and 4 × 6 = 24 girls.
In simple words: The class has 3 boys for every 4 girls.

Teacher's Note:
1. A ratio describes groups, not the actual numbers.
2. The HCF tells you how many such groups there are.

Question 4. Seven identical notebooks cost ₹147. Using the unitary method, find the cost of 12 such notebooks. Show both steps.
Answer:
Step 1: Cost of 1 notebook = 147 ÷ 7 = ₹21.
Step 2: Cost of 12 notebooks = 21 × 12 = ₹252.
Check: 7 : 147 :: 12 : 252, since 7 × 252 = 1764 = 147 × 12. So 12 notebooks cost ₹252.
In simple words: One notebook costs ₹21, so 12 cost ₹252.

Teacher's Note:
1. Unitary method: first find the value of one, then of many.
2. Cross multiplication is a quick check.

Question 5. Divide ₹3,600 between Ira and Jay in the ratio 5 : 4. Write the size of one part and each person's share.
Answer:
Number of equal parts = 5 + 4 = 9.
One part = 3600 ÷ 9 = ₹400.
Ira's share = 5 × 400 = ₹2,000; Jay's share = 4 × 400 = ₹1,600.
Check: 2000 + 1600 = 3600 and 2000 : 1600 = 5 : 4.
In simple words: Make 9 parts of ₹400; Ira gets 5 parts and Jay gets 4.

Teacher's Note:
1. Total parts = sum of the ratio terms.
2. Always check both the total and the ratio.

Question 6. Convert, using the conversion facts given in the chapter:
(i) 15 m into feet (take 1 m = 3.281 ft)
(ii) 2.5 L into cubic centimetres
Answer:
(i) 15 m = 15 × 3.281 = 49.215 ft.
(ii) 1 L = 1000 mL and 1 mL = 1 cc, so 2.5 L = 2500 mL = 2500 cc.
In each case the quantity and its converted value are proportional, because every unit is replaced by the same fixed factor.
In simple words: Multiply by the conversion factor: 15 m ≈ 49.215 ft and 2.5 L = 2500 cc.

Teacher's Note:
1. A conversion is just multiplication by a fixed factor.
2. Write the unit with every answer.

Question 7. An oven dial is marked in Celsius but a recipe is written in Fahrenheit.
(i) Convert 35°C into °F.
(ii) Convert 212°F into °C.
Answer:
(i) F = \( \frac{9}{5} \) × 35 + 32 = 63 + 32 = 95, so 35°C = 95°F.
(ii) C = \( \frac{5}{9} \) × (212 − 32) = \( \frac{5}{9} \) × 180 = 100, so 212°F = 100°C.
In simple words: 35°C is 95°F, and 212°F is 100°C, the boiling point of water.

Teacher's Note:
1. °C to °F: multiply by \( \frac{9}{5} \) then add 32.
2. °F to °C: subtract 32 then multiply by \( \frac{5}{9} \).

Question 8. Check whether 15 : 25 and 24 : 40 are proportional. Verify your answer a second time by cross multiplication.
Answer:
Simplest forms: 15 : 25 = 3 : 5 (HCF 5) and 24 : 40 = 3 : 5 (HCF 8).
Both reduce to 3 : 5, so 15 : 25 :: 24 : 40.
Cross multiplication: 15 × 40 = 600 and 25 × 24 = 600. Since ad = bc, the ratios are proportional.
In simple words: Both ratios are 3 : 5, and the cross products are both 600.

Teacher's Note:
1. Two different tests, one result.
2. Use cross multiplication when the numbers are awkward.

Question 9. Find the missing term in each proportion using cross multiplication:
(i) 8 : 14 :: x : 63
(ii) 9 : y :: 24 : 56
Answer:
Use ad = bc.
(i) 8 × 63 = 14 × x, so 14x = 504 and x = 36. Check: 36 : 63 = 4 : 7 = 8 : 14.
(ii) 9 × 56 = y × 24, so 24y = 504 and y = 21. Check: 9 : 21 = 3 : 7 = 24 : 56.
In simple words: Cross multiply and divide: x = 36 and y = 21.

Teacher's Note:
1. Write the product equation first, then divide.
2. Reduce both ratios to check.

Question 10. A rectangular banner is 80 mm wide and 50 mm high. A designer subtracts 20 mm from the width and 20 mm from the height to get a new banner. Show with ratios why the new banner does not look similar to the old one, and state the rule this illustrates.
Answer:
Old banner: 80 : 50 = 8 : 5.
New banner: (80 − 20) : (50 − 20) = 60 : 30 = 2 : 1.
8 : 5 and 2 : 1 are different, so the banners are not similar; the new one looks squashed.
Rule: subtracting (or adding) the same number to both terms changes a ratio. The shape is kept only when both terms are multiplied by the same factor.
In simple words: Taking away the same amount changes the ratio from 8 : 5 to 2 : 1.

Teacher's Note:
1. Equal differences do not give equal ratios.
2. This is why resizing a photo needs a common factor.

Question 11. For 6 people, a payasam recipe needs 750 mL of milk. The same recipe is to be made for 16 people with the same taste. Find the milk needed, stating the quantity for one person on the way.
Answer:
Milk and number of people are in direct proportion.
Milk for 1 person = 750 ÷ 6 = 125 mL.
Milk for 16 people = 125 × 16 = 2000 mL = 2 L.
Check: 6 : 750 :: 16 : 2000, since 6 × 2000 = 12000 = 750 × 16.
In simple words: Each person needs 125 mL, so 16 people need 2 litres.

Teacher's Note:
1. Unitary method: find one person's share first.
2. Change 2000 mL to 2 L for a neat answer.

Question 12. A bottle holds 36 L of sharbat in which syrup and water are mixed in the ratio 2 : 7. Find the volume of syrup and the volume of water in the bottle, and check that your two answers add up correctly.
Answer:
The ratio 2 : 7 makes 2 + 7 = 9 equal parts; one part = 36 ÷ 9 = 4 L.
Syrup = 2 × 4 = 8 L; water = 7 × 4 = 28 L.
Check: 8 + 28 = 36 L, and 8 : 28 = 2 : 7.
In simple words: 9 parts of 4 L give 8 L of syrup and 28 L of water.

Teacher's Note:
1. Split the whole into (2 + 7) equal parts.
2. The two answers must add back to the whole.

Question 13. A tea estate measures 3 hectares.
(i) Express this area in square metres.
(ii) Express it in acres, taking 1 hectare = 2.471 acres.
Answer:
(i) 1 hectare = 10,000 m², so 3 hectares = 3 × 10,000 = 30,000 m².
(ii) 3 hectares = 3 × 2.471 = 7.413 acres.
It is the same land written in different units; only the unit changes.
In simple words: 3 hectares is 30,000 m² or about 7.413 acres.

Teacher's Note:
1. Multiply by the conversion factor for each unit.
2. A hectare is a bit less than 2.5 acres.

Question 14. A train covers 120 km in 90 minutes. Travelling at the same speed, what distance will it cover in 2.5 hours? Explain why the units must be made the same before the proportion is written.
Answer:
Make the units match: 2.5 hours = 150 minutes.
Time : distance gives 90 : 120 :: 150 : x, so 90x = 120 × 150 = 18000 and x = 200.
The train covers 200 km.
Units must match because 90 minutes and 2.5 hours cannot be compared as plain numbers; 2.5 is not 2.5 minutes.
In simple words: In 150 minutes the train goes 200 km.

Teacher's Note:
1. Convert first, then form the proportion.
2. Check: 80 km per hour × 2.5 h = 200 km.

Question 15. Ravi is 4 years old and his father is 36 years old.
(i) Write the ratio of their ages today in its simplest form.
(ii) Write the ratio of their ages 6 years from now in its simplest form, and say what this shows about adding the same number to both terms of a ratio.
Answer:
(i) 4 : 36. HCF is 4, so the simplest form is 1 : 9.
(ii) After 6 years the ages are 10 and 42, so the ratio is 10 : 42 = 5 : 21.
1 : 9 and 5 : 21 are different, so adding the same number to both terms changes the ratio; the two ratios are not proportional.
In simple words: The age ratio changes from 1 : 9 to 5 : 21 because both ages increase by 6.

Teacher's Note:
1. Ages always increase by equal amounts, never by a common factor.
2. This is a classic example of a ratio that changes over time.

Question 16. Meera and Suhail started a tiffin service. Meera put in ₹48,000 and Suhail put in ₹32,000. At the end of the year they shared a profit of ₹25,000 in the ratio of their investments. Find each one's share.
Answer:
Ratio of investments = 48000 : 32000 = 3 : 2 (dividing by 16,000).
Number of parts = 3 + 2 = 5; one part = 25000 ÷ 5 = ₹5,000.
Meera's share = 3 × 5000 = ₹15,000; Suhail's share = 2 × 5000 = ₹10,000.
Check: 15000 + 10000 = 25000 and 15000 : 10000 = 3 : 2.
In simple words: The profit is split 3 : 2, giving ₹15,000 and ₹10,000.

Teacher's Note:
1. Simplify the investment ratio first.
2. Partners who invest more get a larger share.

Question 17. A 54 kg mixture contains sand and cement in the ratio 7 : 2. How much cement must be added, without adding any sand, so that the ratio of sand to cement becomes 7 : 3?
Answer:
7 : 2 gives 9 parts; one part = 54 ÷ 9 = 6 kg. Sand = 7 × 6 = 42 kg; cement = 2 × 6 = 12 kg.
Sand stays 42 kg. For 7 : 3 :: 42 : x, 7x = 3 × 42 = 126, so x = 18 kg.
Cement to be added = 18 − 12 = 6 kg.
Check: 42 : 18 = 7 : 3, and the new total is 60 kg.
In simple words: Sand stays 42 kg, so cement must rise from 12 kg to 18 kg: add 6 kg.

Teacher's Note:
1. Keep the unchanged quantity fixed.
2. The answer is the difference between the new and old cement.

Question 18. Riya rides to her aunt's town at 40 km/h and the journey takes 3 hours. A friend says the time at 60 km/h can be found from 40 : 3 :: 60 : ?.
(i) Explain why this model is wrong.
(ii) Find the correct time taken at 60 km/h.
Answer:
(i) The model assumes speed and time grow by the same factor. They do not: when the speed increases, the time decreases, so they are not in direct proportion and the Rule of Three cannot be used.
(ii) The distance stays fixed: 40 × 3 = 120 km.
Time at 60 km/h = 120 ÷ 60 = 2 hours, which is less, as expected.
In simple words: Faster riding means less time; the distance is 120 km, so the time is 2 hours.

Teacher's Note:
1. Find what stays fixed (the distance) and work from it.
2. The wrong model would have given 4.5 hours, which makes no sense.

Question 19. One acre of land in a township costs ₹24,00,000. Using 1 acre = 43,560 square feet, find the cost of a plot of 3,630 square feet of the same land.
Answer:
43560 ÷ 3630 = 12, so 3,630 ft² is exactly \( \frac{1}{12} \) of an acre.
Cost = 2400000 ÷ 12 = ₹2,00,000.
Check: 43560 : 2400000 :: 3630 : 200000, since 43560 × 200000 = 3630 × 2400000.
In simple words: The plot is one-twelfth of an acre, so it costs ₹2,00,000.

Teacher's Note:
1. Turn the plot into a fraction of an acre first.
2. Large numbers become easy when a neat fraction appears.

Question 20. The ratio of the length to the width of a rectangular playground is 9 : 5 and its perimeter is 448 m. Working backwards from the perimeter, find the length and the width, and verify both the perimeter and the ratio.
Answer:
Perimeter = 2 × (length + width), so length + width = 448 ÷ 2 = 224 m.
224 m is shared in 9 : 5, which has 14 parts; one part = 224 ÷ 14 = 16 m.
Length = 9 × 16 = 144 m; width = 5 × 16 = 80 m.
Check: 2 × (144 + 80) = 448 m, and 144 : 80 = 9 : 5 (HCF 16).
In simple words: Half the perimeter, 224 m, split 9 : 5 gives 144 m and 80 m.

Teacher's Note:
1. Halve the perimeter to get length + width.
2. Then share that sum in the given ratio.