CBSE Class 8 Mathematics Proportional Reasoning 2 Short Answer Questions with Answers

Short Answer Questions

Question 1. Two shopkeepers prepare lemon squash. Shabnam mixes syrup and water as 6 : 15 and Rehan mixes them as 8 : 20. Using the cross-multiplication method, check whether the two mixtures are proportional, and say whether the two drinks will taste the same.
Answer:
Cross-multiply 6 : 15 and 8 : 20.
First product = 6 × 20 = 120.
Second product = 15 × 8 = 120.
The products are equal, so the two ratios are proportional.
Both mixtures have the same proportion of syrup to water, so the two drinks will taste the same (if everything else is the same).
In simple words: 6 × 20 = 15 × 8 = 120, so both mixtures are in the same ratio (2 : 5) and taste the same.

Teacher's Note:
1. Both ratios also reduce to 2 : 5.
2. Same ratio means same taste.

Question 2. A tourist map carries the Representative Fraction RF 1 : 25,00,000.
(i) Explain what this ratio tells you about the map.
(ii) Convert 25,00,000 cm into kilometres and hence find the geographical distance between two towns that are 6 cm apart on this map.
Answer:
(i) It means that 1 cm measured on the map stands for 25,00,000 cm on the ground.
(ii) 1 km = 1,00,000 cm, so 25,00,000 cm = \( \frac{25,00,000}{1,00,000} \) km = 25 km.
So 6 cm on the map stands for 6 × 25 = 150 km.
In simple words: Every 1 cm on this map is 25 km on the ground, so 6 cm is 150 km.

Teacher's Note:
1. Write the scale as km per cm first.
2. The answer is a straight-line distance.

Question 3. A multigrain flour is prepared by mixing wheat, rice and millet in the ratio 4 : 3 : 2. A miller uses 12 kg of wheat. Find the mass of rice and of millet he must add, and the total mass of flour obtained.
Answer:
Wheat : rice : millet = 4 : 3 : 2.
Wheat is 4 parts = 12 kg, so 1 part = 12 ÷ 4 = 3 kg.
Rice = 3 × 3 = 9 kg; millet = 2 × 3 = 6 kg.
Total flour = 12 + 9 + 6 = 27 kg.
Check: 12 : 9 : 6 divided by 3 gives 4 : 3 : 2.
In simple words: One part is 3 kg, so he adds 9 kg rice and 6 kg millet and gets 27 kg of flour.

Teacher's Note:
1. Find one part from the known ingredient.
2. Always check by reducing back to the given ratio.

Question 4. Divide ₹540 among three friends in the ratio 2 : 3 : 4. Show the sum of the terms you use and write the share of each friend.
Answer:
Sum of the terms = 2 + 3 + 4 = 9.
One part = 540 ÷ 9 = ₹60.
First share = 2 × 60 = ₹120; second share = 3 × 60 = ₹180; third share = 4 × 60 = ₹240.
Check: ₹120 + ₹180 + ₹240 = ₹540.
In simple words: Split ₹540 into 9 parts of ₹60; the friends get ₹120, ₹180 and ₹240.

Teacher's Note:
1. The shares must add back to the whole.
2. Each share = its term × one part.

Question 5. In a class of 72 students, 18 chose cricket as their favourite game. Find the angle of the slice that will represent cricket in a pie chart of this data, and write the fraction of the circle that this slice covers.
Answer:
The angle of a slice is proportional to the number of students it stands for.
Angle for cricket = \( \frac{18}{72} \) × 360° = 90°.
Fraction of the circle = \( \frac{90}{360} \) = \( \frac{1}{4} \), which is also \( \frac{18}{72} \).
So the cricket slice is one quarter of the pie chart.
In simple words: 18 out of 72 is one quarter, so cricket gets a quarter of the circle, 90°.

Teacher's Note:
1. The fraction of students equals the fraction of the circle.
2. 90° is a right angle at the centre.

Question 6. A pie chart drawn for 240 visitors to a book fair has a sector of 60° for the "comics" section.
(i) How many visitors does this sector represent?
(ii) What percentage of the circle is a 60° sector?
Answer:
(i) Number of visitors = \( \frac{60}{360} \) × 240 = \( \frac{1}{6} \) × 240 = 40 visitors.
(ii) A 60° sector is \( \frac{60}{360} \) = \( \frac{1}{6} \) of the circle, that is \( \frac{1}{6} \) × 100% = \( 16\frac{2}{3} \)% of the whole.
In simple words: 60° is one-sixth of the circle, so it stands for 40 visitors and 16 2/3 % of the whole.

Teacher's Note:
1. Turn the angle into a fraction of 360° first.
2. Multiply that fraction by 100 to get a percentage.

Question 7. State, with one reason each, whether the two quantities in each pair are in inverse proportion.
(i) The number of taps of the same kind filling a tank and the time taken to fill it.
(ii) The length of ribbon bought and the price paid at a fixed rate per metre.
Answer:
(i) Inverse proportion. More taps fill the tank in less time, and (number of taps) × (time) stays constant because the tank holds a fixed amount of water.
(ii) Not inverse; this is direct proportion. More metres of ribbon cost more money, and the price per metre \( \frac{\text{price}}{\text{length}} \) stays constant.
In simple words: Taps and time are inversely proportional; ribbon length and price are directly proportional.

Teacher's Note:
1. Ask: when one goes up, does the other go down by the same factor?
2. A constant product means inverse; a constant ratio means direct.

Question 8. A water tank is filled by 15 pumps of the same kind in 8 hours. Find the time 20 such pumps would take, and state the constant product that you used.
Answer:
Pumps and time are inversely proportional, so their product is constant.
Constant = 15 × 8 = 120 pump-hours.
With 20 pumps, time = \( \frac{120}{20} \) = 6 hours.
Check: 20 × 6 = 120. More pumps, fewer hours, as expected.
In simple words: 15 × 8 = 120 pump-hours, so 20 pumps need 120 ÷ 20 = 6 hours.

Teacher's Note:
1. Name the unit of the constant: pump-hours.
2. The answer must be less than 8 hours.

Question 9. On a city map with RF 1 : 50,000, the straight-line distance between a bus stand and a stadium measures 8 cm.
(i) Find the geographical distance between them in kilometres.
(ii) How many centimetres apart would these two places be on another map whose RF is 1 : 2,00,000?
Answer:
(i) 8 cm stands for 8 × 50,000 = 4,00,000 cm, and 4,00,000 cm = 4 km.
(ii) On the second map 1 cm stands for 2,00,000 cm, so the map distance = \( \frac{4,00,000}{2,00,000} \) = 2 cm.
The ground distance is the same, but the smaller-scale map shows it as a shorter length.
In simple words: The places are 4 km apart, which shows as 2 cm on the 1 : 2,00,000 map.

Teacher's Note:
1. Work in centimetres, then convert to km at the end.
2. The ground distance does not change from map to map.

Question 10. Concrete for a roof is mixed as cement : sand : gravel :: 1 : 1.5 : 3. A mason has 12 bags of gravel and wants to use all of it. Find the number of bags of cement and of sand he must take, and the total number of bags of concrete he can make.
Answer:
Gravel is 3 parts and equals 12 bags, so one part = 12 ÷ 3 = 4 bags.
Cement = 1 × 4 = 4 bags; sand = 1.5 × 4 = 6 bags.
Total concrete = 4 + 6 + 12 = 22 bags.
Check: 4 : 6 : 12 divided by 4 gives 1 : 1.5 : 3.
In simple words: One part is 4 bags, so he needs 4 bags of cement and 6 of sand and makes 22 bags in all.

Teacher's Note:
1. Use the gravel to find one part.
2. A decimal term like 1.5 is fine; just multiply.

Question 11. A necklace is strung with 180 beads in the ratio red : blue : green :: 4 : 3 : 2.
(i) Find the number of beads of each colour.
(ii) How many more red beads are there than green beads?
Answer:
Sum of the terms = 4 + 3 + 2 = 9, so one part = 180 ÷ 9 = 20 beads.
(i) Red = 4 × 20 = 80, blue = 3 × 20 = 60, green = 2 × 20 = 40 beads. Total 80 + 60 + 40 = 180.
(ii) Red beads exceed green beads by 80 − 40 = 40 beads.
In simple words: One part is 20 beads: 80 red, 60 blue, 40 green, and red beats green by 40.

Teacher's Note:
1. The difference is also (4 − 2) parts = 40.
2. Check that the colours add to 180.

Question 12. The three angles of a triangle are in the ratio 3 : 4 : 5. Find the three angles and state whether the triangle is acute-angled, right-angled or obtuse-angled.
Answer:
The angle sum 180° is divided in the ratio 3 : 4 : 5.
Sum of the terms = 12, so one part = 180° ÷ 12 = 15°.
Angles: 3 × 15° = 45°, 4 × 15° = 60°, 5 × 15° = 75°.
Every angle is less than 90°, so the triangle is acute-angled.
In simple words: The angles are 45°, 60° and 75°, all less than 90°, so the triangle is acute-angled.

Teacher's Note:
1. Check: 45° + 60° + 75° = 180°.
2. Look at the largest angle to decide the type.

Question 13. A family's monthly income of ₹36,000 is spent as shown: food ₹12,000, rent ₹9,000, school fees ₹6,000 and savings ₹9,000. Find the central angle of each slice of the pie chart and verify that your four angles add up to 360°.
Answer:
₹36,000 stands for 360°, so every ₹100 stands for 1°.
Food = \( \frac{12000}{36000} \) × 360° = 120°.
Rent = \( \frac{9000}{36000} \) × 360° = 90°.
School fees = \( \frac{6000}{36000} \) × 360° = 60°.
Savings = \( \frac{9000}{36000} \) × 360° = 90°.
Check: 120° + 90° + 60° + 90° = 360°.
In simple words: Every ₹100 is 1°, so the slices are 120°, 90°, 60° and 90°, adding to 360°.

Teacher's Note:
1. Finding the money per degree makes every slice quick.
2. Rent and savings are equal, so their slices are equal.

Question 14. A survey of television viewers gives: serials 40%, sports 25%, news 20% and music 15%. Calculate the angle of each sector of the pie chart and check your total.
Answer:
Angle = \( \frac{\text{percentage}}{100} \) × 360°.
Serials = \( \frac{40}{100} \) × 360° = 144°; sports = \( \frac{25}{100} \) × 360° = 90°.
News = \( \frac{20}{100} \) × 360° = 72°; music = \( \frac{15}{100} \) × 360° = 54°.
Check: 144° + 90° + 72° + 54° = 360°, and 40 + 25 + 20 + 15 = 100%.
In simple words: Each 1% is 3.6°, so the sectors are 144°, 90°, 72° and 54°.

Teacher's Note:
1. 100% of the data fills all 360°.
2. 1% = 3.6° is a useful shortcut.

Question 15. A hostel has food provisions for 60 students for 24 days. Twenty more students join the hostel on the first day. For how many days will the same provisions now last? Name the kind of proportion used.
Answer:
More students means the food lasts fewer days, so this is inverse proportion and the product is constant.
Constant = 60 × 24 = 1440 student-days.
New number of students = 60 + 20 = 80.
Number of days = \( \frac{1440}{80} \) = 18 days.
Check: 80 × 18 = 1440.
In simple words: The food is 1440 student-days, so 80 students can eat for 18 days (inverse proportion).

Teacher's Note:
1. Add the new students before dividing.
2. The students join on day one, so no food has been used yet.

Question 16. The quantities x and y are in inverse proportion. When x = 18, y = 20.
(i) Find the constant of the proportion.
(ii) Find y when x = 24, and find x when y = 40.
Answer:
For inverse proportion, xy = k.
(i) k = 18 × 20 = 360.
(ii) When x = 24: y = \( \frac{360}{24} \) = 15. When y = 40: x = \( \frac{360}{40} \) = 9.
Check: 24 × 15 = 360 and 9 × 40 = 360.
In simple words: The product is always 360, so y = 15 when x = 24 and x = 9 when y = 40.

Teacher's Note:
1. Find k once and reuse it.
2. x went up, so y went down, and the reverse.

Question 17. On a trekking map, two villages that lie 5 cm apart represent a geographical distance of 15 km.
(i) Find the Representative Fraction of this map.
(ii) Two other places are 36 km apart on the ground. How far apart will they be drawn on this map?
Answer:
(i) 15 km = 15 × 1,00,000 = 15,00,000 cm, shown by 5 cm. So 1 cm stands for 15,00,000 ÷ 5 = 3,00,000 cm, giving RF = 1 : 3,00,000 (1 cm = 3 km).
(ii) 36 km = 36,00,000 cm, so the map distance = \( \frac{36,00,000}{3,00,000} \) = 12 cm.
In simple words: Each cm is 3 km, so the RF is 1 : 3,00,000 and 36 km is drawn as 12 cm.

Teacher's Note:
1. Bring both distances to centimetres to write the RF.
2. Check: 12 × 3 km = 36 km.

Question 18. In a pie chart showing how children reach a school, the sector for "bus" measures 108° and stands for 54 children.
(i) How many children were surveyed in all?
(ii) Another sector measures 40°. How many children does it represent?
Answer:
The angle of a sector is proportional to the number of children it stands for.
(i) 108° stands for 54 children, so 1° stands for \( \frac{54}{108} \) = 0.5 child. Total = 360 × 0.5 = 180 children.
(ii) Children in the 40° sector = \( \frac{40}{360} \) × 180 = 20 children.
Check: \( \frac{108}{360} \) × 180 = 54, which matches the data.
In simple words: Each degree is half a child, so 360° is 180 children and 40° is 20 children.

Teacher's Note:
1. Find the number for 1° first.
2. Then scale up to 360° or to any other angle.

Question 19. Tap A alone fills a drum in 4 hours and tap B alone fills the same drum in 6 hours. If both taps are opened together, find the time taken to fill the drum. Express the answer in hours and minutes.
Answer:
Take filling the whole drum as 1 unit of work.
In one hour, tap A does \( \frac{1}{4} \) unit and tap B does \( \frac{1}{6} \) unit.
Together in one hour: \( \frac{1}{4} \) + \( \frac{1}{6} \) = \( \frac{3 + 2}{12} \) = \( \frac{5}{12} \) unit.
Time for 1 unit = 1 ÷ \( \frac{5}{12} \) = \( \frac{12}{5} \) = 2.4 hours.
0.4 hour = 0.4 × 60 = 24 minutes, so the drum fills in 2 hours 24 minutes.
In simple words: Together the taps fill 5/12 of the drum each hour, so they need 12/5 hours = 2 hours 24 minutes.

Teacher's Note:
1. Add the rates, then turn the total rate upside down.
2. The answer must be less than 4 hours, the faster tap's time.

Question 20. A prize amount is shared among Anil, Bina and Chetan in the ratio 5 : 4 : 3. Chetan receives ₹1200 less than Anil. Find the total prize amount and each person's share.
Answer:
Let the shares be 5 parts, 4 parts and 3 parts.
Anil's share exceeds Chetan's by 5 − 3 = 2 parts, and this is ₹1200, so 1 part = 1200 ÷ 2 = ₹600.
Total = (5 + 4 + 3) × 600 = 12 × 600 = ₹7200.
Anil = ₹3000, Bina = ₹2400, Chetan = ₹1800.
Check: ₹3000 − ₹1800 = ₹1200, and the shares add to ₹7200.
In simple words: Two parts are ₹1200, so one part is ₹600; the prize is ₹7200, shared as ₹3000, ₹2400 and ₹1800.

Teacher's Note:
1. A difference in money matches a difference in parts.
2. Check both the difference and the total.