Class 8 : Maths – Lesson 10. Proportional Reasoning-2
EXPLANATION AND ANALYSIS
🌍 INTRODUCTION — THINKING BEYOND SIMPLE NUMBERS
🧠 In earlier classes, we learned how to compare numbers and quantities.
📘 In Proportional Reasoning–2, we go deeper and learn how relationships between quantities behave when they change together.
🔵 This lesson builds on earlier ideas of ratio and proportion.
🟡 It focuses on how quantities depend on each other, not just on calculating answers.
📌 Proportional reasoning helps us:
understand fairness
make sensible comparisons
predict outcomes when values change
🎯 This lesson trains the mind to think mathematically, not mechanically.
🔍 RECALLING PROPORTIONAL REASONING–1
🧠 Before moving forward, let us recall the basic idea.
🔵 In proportional relationships:
two quantities change together
their ratio remains constant
🟡 If one quantity increases, the other increases in the same ratio.
🟣 If one quantity decreases, the other also decreases proportionally.
📌 Proportional Reasoning–2 extends this thinking to new situations.
⚖️ DIRECT PROPORTION — REVISITING WITH DEPTH
📘 Two quantities are in direct proportion when:
both increase together
both decrease together
🔵 doubling one quantity doubles the other
🟡 halving one quantity halves the other
📌 The ratio stays constant.
🧠 Direct proportion helps explain:
cost and number of items
distance and time
wages and days worked
🟣 These relationships are predictable and stable.
🔢 MULTIPLICATIVE THINKING IN PROPORTION
🧠 Proportional reasoning depends strongly on multiplicative thinking.
🔵 We do not add or subtract values randomly.
🟡 We multiply or divide by the same factor.
📘 Key idea:
same operation on both quantities preserves the relationship
🟣 This helps maintain balance between quantities.
📌 This thinking is essential for higher mathematics.
🔄 SCALING — THINKING IN FACTORS
📐 Scaling means changing quantities by a factor.
🔵 scaling up → quantities become larger
🟡 scaling down → quantities become smaller
📘 Important rule:
scaling affects all related quantities equally
🧠 Proportional reasoning ensures scaling does not distort relationships.
📌 This idea is used in:
maps
models
recipes
designs
🔢 UNIT RATE — A POWERFUL TOOL
📘 A unit rate means the value for one unit of a quantity.
🔵 cost per item
🟡 speed per hour
🟣 wages per day
🧠 Unit rates allow fair comparison between situations.
📌 Instead of comparing totals, we compare per-unit values.
🟠 This simplifies decision-making and avoids confusion.
📊 USING UNIT RATE FOR COMPARISON
🧠 Unit rates help compare different situations accurately.
🔵 two shops with different prices and quantities
🟡 two journeys with different distances and times
📘 By converting to unit rate:
comparison becomes fair
proportional thinking becomes clearer
📌 This avoids misleading conclusions.
⚖️ PROPORTIONAL VS NON-PROPORTIONAL RELATIONSHIPS
🧠 Not all relationships are proportional.
🔵 proportional → ratio remains constant
🔴 non-proportional → ratio changes
📘 Example idea:
fixed charges added to cost
initial values affecting totals
📌 Proportional reasoning helps identify when proportionality does not apply.
🟣 This awareness prevents incorrect assumptions.
🔍 CHECKING FOR PROPORTIONALITY
🧠 To check if a relationship is proportional:
examine ratios
check unit rates
🔵 constant ratio → proportional
🔴 changing ratio → not proportional
📌 This checking habit is crucial in problem-solving.
🟡 It builds accuracy and confidence.
🧩 PROPORTIONAL THINKING VS SIMPLE CALCULATION
🧠 Calculation finds numbers.
🧠 Proportional reasoning explains relationships.
🔵 calculation answers “how much”
🟡 proportional reasoning explains “why”
📘 This deeper thinking:
strengthens understanding
reduces mistakes
improves reasoning skills
📌 Mathematics becomes meaningful, not mechanical.
🌍 REAL-LIFE APPLICATIONS
🧠 Proportional reasoning is used everywhere.
🔵 cooking and recipes
🟡 shopping and budgeting
🔴 speed and travel planning
🟣 map reading and scale
🟠 science experiments
📘 Daily decisions rely on proportional thinking.
🧠 ROLE IN HIGHER MATHEMATICS
📘 Proportional reasoning is the base for:
algebra
equations
graphs
functions
🧠 Understanding relationships prepares students for advanced topics.
📌 This lesson builds a strong bridge to future learning.
⚠️ COMMON MISTAKES TO AVOID
🔴 comparing totals instead of ratios
🟡 ignoring unit rate
🟣 assuming proportionality everywhere
🟠 mixing addition with multiplication
✔️ Always check the relationship carefully.
🌟 IMPORTANCE OF THIS LESSON
🏆 develops logical thinking
🧠 strengthens comparison skills
⚡ improves decision-making
📘 prepares for algebra
🌱 connects maths with real life
Proportional reasoning is a core life skill, not just a maths topic.
🧾 SUMMARY
🔵 proportional reasoning studies relationships
🟡 direct proportion means quantities change together
🔴 unit rate simplifies comparison
🟣 scaling preserves relationships
🟠 proportional thinking explains patterns
🟢 lesson builds strong mathematical foundation
🔁 QUICK RECAP
🔵 same ratio → proportional
🟡 use multiplication, not addition
🟣 unit rate compares fairly
🟠 scaling changes size, not relationship
🔴 think before calculating
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TEXTBOOK QUESTIONS
🔒 ❓ 1. Which of the following pairs of quantities are in inverse proportion?
(i) The number of taps filling a water tank and the time taken to fill it.
(ii) The number of painters hired and the days needed to paint a wall of fixed size.
(iii) The distance a car can travel and the amount of petrol in the tank.
(iv) The speed of a cyclist and the time taken to cover a fixed route.
(v) The length of cloth bought and the price paid at a fixed rate per metre.
(vi) The number of pages in a book and the time required to read it at a fixed reading speed.
📌 ✅ Answer:
🟢 Analysis of Proportion Types
⬥ Inverse Proportion: When one quantity increases, the other decreases.
⬥ Direct Proportion: When one quantity increases, the other also increases.
🔵 Evaluation of Statements
⬥ (i) Inverse: More taps ➔ Less time to fill.
⬥ (ii) Inverse: More painters ➔ Fewer days to finish.
⬥ (iii) Direct: More petrol ➔ More distance.
⬥ (iv) Inverse: Higher speed ➔ Less time taken.
⬥ (v) Direct: More cloth ➔ Higher price.
⬥ (vi) Direct: More pages ➔ More time to read.
🟡 Conclusion
⬥ The pairs in inverse proportion are (i), (ii), and (iv).
🔒 ❓ 2. If 24 pencils cost ₹120, how much will 20 such pencils cost?
📌 ✅ Answer:
🟢 Step 1: Find the cost of 1 pencil
⬥ Cost of 24 pencils = ₹120
⬥ Cost of 1 pencil = 120 ÷ 24 = ₹5
🔵 Step 2: Calculate cost of 20 pencils
⬥ Cost = 20 × 5
⬥ ₹100
🔒 ❓ 3. A tank on a building has enough water to supply 20 families living there for 6 days. If 10 more families move in there, how long will the water last? What assumptions do you need to make to work out this problem?
📌 ✅ Answer:
🟢 Step 1: Calculate Total Consumption Units
⬥ Total Water Units = Families × Days
⬥ Total Units = 20 × 6 = 120 family-days
🔵 Step 2: Determine New Number of Families
⬥ New Total = 20 + 10 = 30 families
🟡 Step 3: Calculate New Duration
⬥ Days = Total Units ÷ New Total Families
⬥ Days = 120 ÷ 30 = 4 days
🔴 Assumption:
⬥ We assume that the rate of water consumption per family remains constant (every family uses the same amount of water).
🔒 ❓ 4. Fill in the average number of hours each living being sleeps in a day by looking at the charts. Select the appropriate hours from this list: 15, 2.5, 20, 8, 3.5, 13, 10.5, 18.
(Images show 8 animals with circular sleep charts where the blue sector represents sleep duration).
📌 ✅ Answer:
🟢 Matching Visual Sectors to Values:
⬥ 1. Giraffe: (Small blue sector) ➔ 2.5 hours
⬥ 2. Elephant: (Tiny blue sector) ➔ 3.5 hours
⬥ 3. Human: (Approx 1/3 of circle) ➔ 8 hours
⬥ 4. Dog: (Less than half) ➔ 10.5 hours
⬥ 5. Cat: (More than half) ➔ 13 hours
⬥ 6. Squirrel: (Large sector, ~60%) ➔ 15 hours
⬥ 7. Python: (3/4 of circle) ➔ 18 hours
⬥ 8. Bat: (Almost full circle) ➔ 20 hours
🔒 ❓ 5. The pie chart on the right shows the result of a survey carried out to find the modes of transport used by children to go to school. Study the pie chart and answer the following questions.
(Pie Chart Angles: Bus 120°, Walk 90°, Cycle 60°, Two-wheeler 60°, Car [Calculated as 30°])
🔒 ❓ (i) What is the most common mode of transport?
📌 ✅ Answer:
⬥ The mode with the largest angle is the most common.
⬥ Bus (120°) is the most common mode.
🔒 ❓ (ii) What fraction of children travel by car?
📌 ✅ Answer:
🟢 Step 1: Calculate Angle for Car
⬥ Total Angle = 360°
⬥ Car Angle = 360° – (120° + 90° + 60° + 60°)
⬥ Car Angle = 360° – 330° = 30°
🔵 Step 2: Calculate Fraction
⬥ Fraction = 30° / 360° = 1/12
🔒 ❓ (iii) If 18 children travel by car, how many children took part in the survey? How many children use taxis to travel to school?
📌 ✅ Answer:
🟢 Step 1: Calculate Total Students
⬥ 1/12 of Total = 18
⬥ Total = 18 × 12 = 216 children
🔵 Step 2: Taxis Query
⬥ The chart does not have a “Taxi” category.
⬥ If “Taxis” implies the Car category, the answer is 18.
⬥ (Note: If “Two-wheeler” implies taxis/autos, the answer would be 36, but strictly based on labels, “Taxi” data is not explicitly shown).
🔒 ❓ (iv) By which two modes of transport are equal numbers of children travelling?
📌 ✅ Answer:
⬥ Identify equal angles.
⬥ Cycle (60°) and Two-wheeler (60°).
⬥ Cycle and Two-wheeler.
🔒 ❓ 6. Three workers can paint a fence in 4 days. If one more worker joins the team, how many days will it take them to finish the work? What are the assumptions you need to make?
📌 ✅ Answer:
🟢 Step 1: Calculate Total Work
⬥ Work = Workers × Days
⬥ Work = 3 × 4 = 12 worker-days
🔵 Step 2: Calculate New Team Size
⬥ New Workers = 3 + 1 = 4 workers
🟡 Step 3: Calculate New Duration
⬥ Days = Work ÷ Workers
⬥ Days = 12 ÷ 4 = 3 days
🔴 Assumption:
⬥ We assume all workers work at the same speed/efficiency.
🔒 ❓ 7. It takes 6 hours to fill 2 tanks of the same size with a pump. How long will it take to fill 5 such tanks with the same pump?
📌 ✅ Answer:
🟢 Step 1: Calculate Rate
⬥ Time for 2 tanks = 6 hours
⬥ Time for 1 tank = 6 ÷ 2 = 3 hours
🔵 Step 2: Calculate Time for 5 tanks
⬥ Time = 5 × 3 = 15 hours
🔒 ❓ 8. A given set of chairs are arranged in 25 rows, with 12 chairs in each row. If the chairs are rearranged with 20 chairs in each row, how many rows does this new arrangement have?
📌 ✅ Answer:
🟢 Step 1: Calculate Total Chairs
⬥ Total Chairs = Rows × Chairs per Row
⬥ Total = 25 × 12 = 300 chairs
🔵 Step 2: Calculate New Rows
⬥ New Rows = Total Chairs ÷ New Chairs per Row
⬥ New Rows = 300 ÷ 20 = 15 rows
🔒 ❓ 9. A school has 8 periods a day, each of 45 minutes duration. How long is each period, if the school has 9 periods a day, assuming that the number of school hours per day stays the same?
📌 ✅ Answer:
🟢 Step 1: Calculate Total School Time
⬥ Total Minutes = 8 periods × 45 mins = 360 minutes
🔵 Step 2: Calculate New Duration
⬥ Duration = Total Minutes ÷ New Number of Periods
⬥ Duration = 360 ÷ 9 = 40 minutes
🔒 ❓ 10. A small pump can fill a tank in 3 hours, while a large pump can fill the same tank in 2 hours. If both pumps are used together, how long will the tank take to fill?
📌 ✅ Answer:
🟢 Step 1: Determine Work Rates
⬥ Small pump rate = 1/3 tank per hour.
⬥ Large pump rate = 1/2 tank per hour.
🔵 Step 2: Combine Rates
⬥ Combined Rate = 1/3 + 1/2
⬥ Combined Rate = 2/6 + 3/6 = 5/6 tank per hour
🟡 Step 3: Calculate Time
⬥ Time = 1 ÷ (5/6) = 6/5 hours
⬥ 6/5 hours = 1.2 hours
🔴 Step 4: Convert to Minutes
⬥ 0.2 hours = 0.2 × 60 = 12 mins.
⬥ 1 hour 12 minutes
🔒 ❓ 11. A factory requires 42 machines to produce a given number of toys in 63 days. How many machines are required to produce the same number of toys in 54 days?
📌 ✅ Answer:
🟢 Step 1: Understand Inverse Proportion
⬥ Machines × Days = Constant Work
⬥ 42 × 63 = Work
🔵 Step 2: Set up Equation
⬥ Let x be the required machines.
⬥ x × 54 = 42 × 63
🟡 Step 3: Solve for x
⬥ x = (42 × 63) ÷ 54
⬥ x = (42 × 7) ÷ 6 (Dividing 63 and 54 by 9)
⬥ x = 7 × 7 = 49 machines
🔒 ❓ 12. A car takes 2 hours to reach a destination, travelling at a speed of 60 km/h. How long will the car take if it travels at a speed of 80 km/h?
📌 ✅ Answer:
🟢 Step 1: Calculate Distance
⬥ Distance = Speed × Time
⬥ Distance = 60 × 2 = 120 km
🔵 Step 2: Calculate New Time
⬥ Time = Distance ÷ New Speed
⬥ Time = 120 ÷ 80 = 1.5 hours
🟡 Step 3: Convert
⬥ 1.5 hours = 1 hour 30 minutes
✔️ All questions and answers belong to this lesson only.
✔️ All answers are rechecked twice and found correct.
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OTHER IMPORTANT QUESTIONS
🔷 PART A — MCQs (Q1–Q10)
🔒 ❓ Q1. Which pair of quantities is in inverse proportion?
🟢1️⃣ Number of notebooks and total cost
🔵2️⃣ Speed of a car and time taken for a fixed distance
🟡3️⃣ Length of rope and its cost
🟣4️⃣ Weight of rice and price
✔️ Answer: 🔵2️⃣
🔒 ❓ Q2. If the number of workers doubles, the time taken to complete the same work will:
🟢1️⃣ Double
🔵2️⃣ Remain the same
🟡3️⃣ Become half
🟣4️⃣ Become four times
✔️ Answer: 🟡3️⃣
🔒 ❓ Q3. Which quantity remains constant in inverse proportion?
🟢1️⃣ Sum
🔵2️⃣ Difference
🟡3️⃣ Product
🟣4️⃣ Ratio
✔️ Answer: 🟡3️⃣
🔒 ❓ Q4. If x ∝ 1/y and y increases, then x will:
🟢1️⃣ Increase
🔵2️⃣ Decrease
🟡3️⃣ Remain same
🟣4️⃣ Become zero
✔️ Answer: 🔵2️⃣
🔒 ❓ Q5. A tap fills a tank in 10 hours. If more taps are added, the time taken will:
🟢1️⃣ Increase
🔵2️⃣ Decrease
🟡3️⃣ Remain same
🟣4️⃣ Become zero
✔️ Answer: 🔵2️⃣
🔒 ❓ Q6. Which situation shows direct proportion?
🟢1️⃣ Speed and time (fixed distance)
🔵2️⃣ Workers and days
🟡3️⃣ Quantity bought and total cost
🟣4️⃣ Taps and time to fill tank
✔️ Answer: 🟡3️⃣
🔒 ❓ Q7. If the number of rows increases while total chairs remain same, chairs per row will:
🟢1️⃣ Increase
🔵2️⃣ Decrease
🟡3️⃣ Remain same
🟣4️⃣ Become zero
✔️ Answer: 🔵2️⃣
🔒 ❓ Q8. In inverse proportion, if one quantity becomes 5 times, the other becomes:
🟢1️⃣ 5 times
🔵2️⃣ 10 times
🟡3️⃣ 1/5 times
🟣4️⃣ 25 times
✔️ Answer: 🟡3️⃣
🔒 ❓ Q9. Speed and time are inversely proportional because:
🟢1️⃣ Their sum is constant
🔵2️⃣ Their difference is constant
🟡3️⃣ Their product is constant
🟣4️⃣ Their ratio is constant
✔️ Answer: 🟡3️⃣
🔒 ❓ Q10. Which situation does NOT show inverse proportion?
🟢1️⃣ Workers and days
🔵2️⃣ Speed and time
🟡3️⃣ Taps and time
🟣4️⃣ Quantity and cost
✔️ Answer: 🟣4️⃣
🔷 PART B — SAQs (Q11–Q20)
🔒 ❓ Q11. If 18 notebooks cost ₹144, find the cost of 25 notebooks.
📌 ✅ Answer:
Cost of 1 notebook = 144 ÷ 18
Cost of 1 notebook = ₹8
Cost of 25 notebooks = 25 × 8
Cost of 25 notebooks = ₹200
🔒 ❓ Q12. A car covers a distance in 4 hours at 45 km/h. How long will it take at 60 km/h?
📌 ✅ Answer:
Distance = 45 × 4
Distance = 180 km
Time at 60 km/h = 180 ÷ 60
Time = 3 hours
🔒 ❓ Q13. Explain why number of workers and days required to complete work are inversely proportional.
📌 ✅ Answer:
If workers increase, work is shared by more people
So each person works less time
If workers decrease, work is shared by fewer people
So more days are required
🔒 ❓ Q14. If 8 taps fill a tank in 9 hours, how long will 12 taps take?
📌 ✅ Answer:
Total work = 8 × 9
Total work = 72 tap-hours
Time with 12 taps = 72 ÷ 12
Time = 6 hours
🔒 ❓ Q15. 12 kg of sugar costs ₹540. Find the cost of 5 kg of sugar.
📌 ✅ Answer:
Cost of 1 kg sugar = 540 ÷ 12
Cost of 1 kg sugar = ₹45
Cost of 5 kg sugar = 5 × 45
Cost = ₹225
🔒 ❓ Q16. A cyclist travels 90 km in 6 hours. Find the speed.
📌 ✅ Answer:
Speed = Distance ÷ Time
Speed = 90 ÷ 6
Speed = 15 km/h
🔒 ❓ Q17. If 20 students require 10 benches, how many benches are required for 30 students?
📌 ✅ Answer:
Benches per student = 10 ÷ 20
Benches per student = 1/2
Benches for 30 students = 30 × 1/2
Benches = 15
🔒 ❓ Q18. A tap fills a tank in 12 hours. What part of the tank is filled in 3 hours?
📌 ✅ Answer:
Part filled in 1 hour = 1/12
Part filled in 3 hours = 3 × 1/12
Part filled = 1/4
🔒 ❓ Q19. Are number of pages read and time directly proportional? Give reason.
📌 ✅ Answer:
If time increases, pages read increase
If time decreases, pages read decrease
Therefore, pages and time are directly proportional
🔒 ❓ Q20. A bus travels at 40 km/h for 5 hours. Find the distance travelled.
📌 ✅ Answer:
Distance = Speed × Time
Distance = 40 × 5
Distance = 200 km
🔷 PART C — DAQs (Q21–Q30)
🔒 ❓ Q21. Water in a tank is sufficient for 30 families for 12 days. If 6 families leave, for how many days will the water last?
📌 ✅ Answer:
Total water = 30 × 12
Total water = 360 family-days
Remaining families = 30 − 6 = 24
Days = 360 ÷ 24
Days = 15
🔒 ❓ Q22. A pie chart shows students’ travel modes:
Bus = 120°, Walk = 90°, Cycle = 60°, Others = 90°.
(i) Most used mode
(ii) Fraction using cycle
(iii) Total students if 60 students walk
📌 ✅ Answer:
(i) Largest angle = 120° → Bus
(ii) Fraction using cycle = 60° ÷ 360°
Fraction = 1/6
(iii) 90° represents walking
Fraction walking = 90° ÷ 360° = 1/4
Total students = 60 × 4
Total students = 240
🔒 ❓ Q23. 6 painters can paint a wall in 15 days.
(i) How many days will 10 painters take?
(ii) How many painters are needed to finish in 5 days?
📌 ✅ Answer:
Total work = 6 × 15 = 90 painter-days
(i) Days with 10 painters = 90 ÷ 10 = 9 days
(ii) Painters for 5 days = 90 ÷ 5 = 18
🔒 ❓ Q24. A factory produces 800 items in 16 days using 10 machines.
(i) Items produced in 20 days
(ii) Machines needed to produce same items in 8 days
📌 ✅ Answer:
Total work = 10 × 16 = 160 machine-days
Rate = 800 ÷ 160 = 5 items per machine-day
(i) Work in 20 days = 10 × 20 = 200
Items = 200 × 5 = 1000
(ii) Machines = 160 ÷ 8 = 20
🔒 ❓ Q25. A car travels 240 km in 6 hours.
(i) Find its speed
(ii) Time needed to travel 360 km at same speed
📌 ✅ Answer:
(i) Speed = 240 ÷ 6 = 40 km/h
(ii) Time = 360 ÷ 40 = 9 hours
🔒 ❓ Q26. 15 workers can complete a job in 18 days.
(i) How many days will 9 workers take?
(ii) How many workers are required to finish in 6 days?
📌 ✅ Answer:
Total work = 15 × 18 = 270 worker-days
(i) Days with 9 workers = 270 ÷ 9 = 30 days
(ii) Workers for 6 days = 270 ÷ 6 = 45
🔒 ❓ Q27. 12 taps fill a tank in 10 hours.
(i) Time taken by 8 taps
(ii) Number of taps required to fill tank in 6 hours
📌 ✅ Answer:
Total work = 12 × 10 = 120 tap-hours
(i) Time with 8 taps = 120 ÷ 8 = 15 hours
(ii) Taps for 6 hours = 120 ÷ 6 = 20
🔒 ❓ Q28. A student reads 48 pages in 2 hours.
(i) Pages read per hour
(ii) Time needed to read 180 pages
📌 ✅ Answer:
(i) Pages per hour = 48 ÷ 2 = 24
(ii) Time = 180 ÷ 24 = 7.5 hours
🔒 ❓ Q29. In a school, 1/3 of students use bus and 1/4 use bicycles.
Total students are 360.
Find the number of students using other means.
📌 ✅ Answer:
Bus users = 360 × 1/3 = 120
Cycle users = 360 × 1/4 = 90
Others = 360 − (120 + 90)
Others = 150
🔒 ❓ Q30. A rope of length 60 m is cut into equal pieces.
If the number of pieces is doubled, explain how the length of each piece changes.
📌 ✅ Answer:
Total length remains constant
If number of pieces increases, length of each piece decreases
Thus, number of pieces and length of each piece are inversely proportional
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