- Cross-products test whether two ratios are proportional.
- Representative Fraction map scales use the same distance units.
- Multi-term ratios preserve every ingredient's relative share.
- To divide total T in a ratio, multiply T by each part divided by the sum of parts.
- Pie-chart sector angle=360°×category/total.
- Direct proportion has a constant quotient; inverse proportion has a constant product.
- For fixed work, workers×days is constant under equal productivity.
- For independent pumps, add their rates rather than their times.
Use these notes alongside your current textbook. Check important formulas, worked answers and syllabus coverage with your teacher or official learning material.
Proportional Reasoning–2 — Class 8 NCERT Ganita Prakash Chapter 10
Direct answer: Proportional Reasoning–2 builds on ratios through map scales, multi-term mixtures, splitting totals, pie charts and inverse proportion. Direct proportion has a constant ratio; inverse proportion has a constant product.
Why does doubling the number of equally efficient pumps halve the time to fill a tank, while doubling the number of paint cans doubles the amount of paint? Understanding the difference helps solve practical problems without choosing the wrong formula.
Official textbook: Part II, Chapter 3, printed pages 55–69.
3.1 Proportionality — A Quick Recap
Two ratios a:b and c:d are proportional when ad=bc (with nonzero second terms). Example: 6:3 and 4:2 are proportional because 6×2=3×4=12. The idli recipe 2 cups rice to 1 cup urad dal scales consistently: 6:3 and 4:2 preserve flavour proportions if all other ingredients scale equally.
3.2 Ratios in Maps
Representative Fraction (RF) compares a map distance with the actual straight-line geographical distance using the same units. At RF 1:60,00,000, 1 cm on the map represents 60,00,000 cm=60 km on the ground. At RF 1:50, 1 cm in a classroom plan represents 50 cm in reality. A map scale does not automatically give road-travel distance. Measure printed maps with a ruler; different print scaling may affect measurements.
3.3 Ratios with More than 2 Terms
A mixture 8:4:2:1 represents coriander, chillies, toor dal and fenugreek in order. Halving every quantity gives 4:2:1:0.5, an equivalent ratio. For red:blue:white=2:3:5, if white is 10 L, one part is 2 L, so red=4 L and blue=6 L. In construction, cement:sand:gravel=1:1.5:3; 3 bags of cement require 4.5 bags sand and 9 bags gravel, 16.5 bags total. Such simplified examples assume compatible units and additive volumes.
3.4 Dividing a Whole in a Given Ratio
For a total T split in ratio a:b:c, first compute parts sum S=a+b+c. Shares are Ta/S, Tb/S, Tc/S. Example: 110 units of concrete in 1:1.5:3 have 5.5 parts; one part=20 units, so cement 20, sand 30, gravel 60. A 50 mL paint mix in 2:3:5 has red 10 mL, blue 15 mL and white 25 mL. Triangle angles in ratio 1:3:5 total 180°, giving 20°,60°,100°.
3.5 A Slice of the Pie
A pie chart represents each category by a central angle. For count n out of total N, angle=(n/N)×360°. Example: 40 students have grades A,B,C,D,E counts 12,10,8,6,4. Angles are 108°,90°,72°,54°,36°, summing to 360°. Draw a circle, mark one radius, use a protractor to measure each sector successively, and label each slice with its category and count.
3.6 Inverse Proportions
In direct proportion y=kx and y/x is constant. In inverse proportion xy=k, so when x doubles y halves. For a fixed 90 km route, speed/time pairs (5 km/h,18 h), (15,6), (30,3), (60,1.5) all give product 90 km. Under constant individual productivity, workers×days stays constant for fixed work. For two independent pumps, add filling rates, not their completion times.
Direct and inverse comparison table
| Feature | Direct proportion | Inverse proportion |
|---|---|---|
| Relationship | y=kx | xy=k |
| Invariant | y/x | x×y |
| If x doubles | y doubles | y halves |
| Example | Items bought and cost at fixed price | Speed and time for fixed distance |
| Calculation | y₂=y₁(x₂/x₁) | y₂=y₁(x₁/x₂) |
Map, multi-part ratio and pie-chart formula table
| Task | Formula | Illustration |
|---|---|---|
| Ground distance | Map distance×scale denominator | 2 cm at 1:100000→2 km |
| Share of total T | T×part/sum of parts | 150 in 3:4:3:5→30,40,30,50 |
| Pie-sector angle | 360°×category/total | 25%→90° |
| Fixed work | Workers×days=constant | 3×4=4×3 |
| Combined work rate | 1/T=1/T₁+1/T₂ | 3h and 2h→1.2h |
Math Talk, activities and visual understanding
Map-making activity: measure your classroom, choose RF 1:50, draw the room boundary and place desks and doors at scaled positions. Multi-part ratio activity: mix a mock recipe and compare equivalent ingredient amounts. Pie-chart activity: survey classmates about one favourite subject each, count responses and calculate sector angles. Inverse-proportion activity: record speed and time for the same journey and test constant products. Do not invent survey results or map distances; use actual observations.
Common mistakes, HOTS, MCQs and revision
Common errors: treating map distance as road distance; changing only one term of a multi-part ratio; using 180° instead of 360° for pie charts; using direct proportion for fixed-work workers and days; adding pump completion times rather than rates. Original MCQ: A category represents 25% of a pie chart. Angle? 90°. HOTS: Two taps fill in 4h and 6h; together time=1/(1/4+1/6)=12/5h=2h24m. Competency: A scale 1:25000 and a 4 cm map segment represent 100000 cm=1 km. Flashcards: direct→constant quotient; inverse→constant product; pie chart→360° total.
NCERT Figure it Out — page 60
Each official question is paraphrased with its own solution and final answer. Diagram-dependent results are marked rather than guessed.
Question 1. A cricket coach divides 150 minutes into warm-up/cool-down, batting, bowling and fielding in ratio 3:4:3:5. Find the time for each activity.
Step-by-step solution: Add ratio parts: 3+4+3+5=15. One part=150/15=10 min. Multiply by each part: 30,40,30,50.
Final answer: Warm-up/cool-down 30 min; batting 40 min; bowling 30 min; fielding 50 min.
Question 2. A library has Odia:Hindi:English books in ratio 3:2:1 and 288 Odia books. Find Hindi and English counts.
Step-by-step solution: Three parts represent 288, so one part=288/3=96 books. Hindi=2×96; English=1×96.
Final answer: 192 Hindi books and 96 English books.
Question 3. A collection has 100 coins of ₹10, ₹5, ₹2 and ₹1 in number ratio 4:3:2:1. Find the total value.
Step-by-step solution: Parts sum=10, so each part is 10 coins. Counts=40,30,20,10. Total value=40×10+30×5+20×2+10×1=400+150+40+10.
Final answer: ₹600.
Question 4. Construct a triangle with sides in ratio 3:4:5. Are all such triangles congruent?
Step-by-step solution: Choose scale k>0; side lengths are 3k,4k,5k, which satisfy triangle inequality. For k=1 construct 3,4,5 with a ruler and compass. For k=2 construct 6,8,10. Corresponding angles are equal, but sizes differ.
Final answer: Construction is possible; triangles with different scale factors are similar, not congruent.
Question 5. Can a triangle be constructed with sides in ratio 1:3:5? Explain.
Step-by-step solution: Let sides k,3k,5k with k>0. Triangle inequality requires k+3k>5k, but 4k<5k.
Final answer: No; the sum of the two shorter sides is less than the longest side.
NCERT Figure it Out — pages 62–63
Each official question is paraphrased with its own solution and final answer. Diagram-dependent results are marked rather than guessed.
Question 1. Of 360 people, 90 prefer summer, 120 rainy and the remainder winter. Construct a pie chart.
Step-by-step solution: Winter count=360−90−120=150. Since the total count is 360, each count corresponds to the same number of degrees. Draw sectors 90°,120°,150° and label them.
Final answer: Summer 90°; rainy 120°; winter 150°.
Question 2. Construct a pie chart for favourite TV channels: entertainment 50%, sports 25%, news 15%, information 10%.
Step-by-step solution: Multiply each percentage by 360°: 0.50×360=180; 0.25×360=90; 0.15×360=54; 0.10×360=36. Sum=360.
Final answer: Entertainment 180°; sports 90°; news 54°; information 36°.
Question 3. Survey classmates about their favourite subjects and draw a pie chart from the counts.
Step-by-step solution: Each student chooses one subject. Count nᵢ students in each category, sum N. Compute sector angle 360nᵢ/N for each category. Check angles sum to 360°, then draw and label the chart.
Final answer: Student-specific results; a correct chart uses the actual survey counts, not invented numbers.
NCERT Figure it Out — page 65
Each official question is paraphrased with its own solution and final answer. Diagram-dependent results are marked rather than guessed.
Question 1. Which of the three given x–y tables represent inverse proportion?
Step-by-step solution: (i) Products 40×20=800,80×10=800,25×32=800,16×50=800: constant. (ii) Products 800,800,312.5,128: not constant. (iii) Products 30×15=450,90×5=450,150×3=450,10×45=450: constant.
Final answer: Tables (i) and (iii) are inverse proportions; (ii) is not.
Question 2. Fill the missing entries in the inverse-proportion table with x values 16, 12 and 36 and given y values 9 and 48.
Step-by-step solution: The PDF text extraction separates the table labels from their column positions. Use xy=k for a complete known column and fill each blank with k/x or k/y; the positions of 9 and 48 must be verified from the printed grid.
Final answer: Method verified; numerical blanks await a visual check of the table alignment.
NCERT Figure it Out — pages 67–68
Each official question is paraphrased with its own solution and final answer. Diagram-dependent results are marked rather than guessed.
Question 1. Identify which pairs are inversely proportional: taps/time, painters/days, distance/petrol, speed/time, cloth/price, pages/reading time.
Step-by-step solution: At equal flow rates, more taps reduce time; equal painters reduce days for fixed wall; for a fixed route higher speed reduces time. More fuel allows more distance (direct); longer cloth costs more at fixed price (direct); more pages require more time at fixed speed (direct).
Final answer: Inverse: (i) taps/time, (ii) painters/days, (iv) speed/time. Direct: (iii), (v), (vi), under stated assumptions.
Question 2. If 24 pencils cost ₹120, find the price of 20 identical pencils.
Step-by-step solution: Unit price=120/24=₹5. Multiply by 20.
Final answer: ₹100.
Question 3. Water supplies 20 families for 6 days. If 10 more families arrive, how long will it last? State assumptions.
Step-by-step solution: Total family-days=20×6=120. Families=30. Days=120/30=4. Assumptions: equal daily consumption per family, fixed usable water volume, no replenishment or losses.
Final answer: 4 days under those assumptions.
Question 4. Match eight living beings to their daily sleeping hours using the printed charts and choices 15, 2.5, 20, 8, 3.5, 13, 10.5, 18.
Step-by-step solution: The chart's bars and animal labels must be read together; the plain text PDF does not reliably preserve this pairing. Measure each bar against the axis and match to the listed values, checking all eight choices are used once.
Final answer: Requires the original sleep-chart image; no fabricated animal-to-hour mapping.
Question 5. Interpret the textbook pie chart of transport to school: most common mode, car fraction, survey size if 18 travel by car, taxi count, and equal modes.
Step-by-step solution: The chart's transport labels and sector angles are partly separated in text extraction. Read each printed sector label and its angle. Car fraction=car angle/360; if 18 students are in the car sector, total=18×360/(car angle). Taxi count=total×(taxi angle)/360. Equal modes have equal angles.
Final answer: Diagram-dependent; exact mode assignments and totals require verifying the printed pie chart.
Question 6. Three workers paint a fence in four days. How long with four equally productive workers?
Step-by-step solution: Fixed work=3×4=12 worker-days. With 4 workers, time=12/4.
Final answer: 3 days.
Question 7. A pump fills two identical tanks in six hours. How long for five tanks?
Step-by-step solution: Time per tank=6/2=3 h, at constant pump rate. Five tanks take 5×3.
Final answer: 15 hours.
Question 8. Chairs occupy 25 rows of 12 chairs each. If rearranged into 20 chairs per row, how many rows?
Step-by-step solution: Total chairs=25×12=300. Number of rows=300/20.
Final answer: 15 rows.
Question 9. A school has eight 45-minute periods. What is each period's length if there are nine periods in the same school day?
Step-by-step solution: Total class time=8×45=360 min. New period length=360/9.
Final answer: 40 minutes.
Question 10. One pump fills a tank in three hours and another in two hours. How long together?
Step-by-step solution: Combined hourly rate=1/3+1/2=5/6 tank per hour. Time=1/(5/6)=6/5 h=72 min.
Final answer: 1 hour 12 minutes.
Question 11. A factory needs 42 machines for 63 days to produce fixed output. How many machines for 54 days?
Step-by-step solution: Machine-days=42×63=2646. Machines=2646/54=49, assuming equal production rates.
Final answer: 49 machines.
Question 12. A car travels at 60 km/h and takes two hours. How long at 80 km/h for the same distance?
Step-by-step solution: Distance=60×2=120 km. Time=120/80=1.5 h.
Final answer: 1 hour 30 minutes.
Learning checklist and source
Check that you can read an RF map scale, scale multi-ingredient ratios, divide a total, construct a pie chart, distinguish direct and inverse relationships, solve constant-work problems, and add independent work rates. Original NCERT source: Ganita Prakash Part II, Chapter 3. This page covers all 22 numbered Figure it Out questions in four sets, but the sleep chart, transport pie chart and inverse table need image-level verification for complete numerical answers. The classroom survey requires student-collected data.
Frequently asked questions
What is Proportional Reasoning-2 about?
Map scales, multi-term ratios, proportional division, pie charts and inverse proportion.
What does RF 1:6000000 mean?
1 cm on the map represents 6000000 cm or 60 km of geographical distance.
How do you split a quantity in ratio 2:3:5?
Divide the quantity into ten equal parts and allocate two, three and five parts.
How do you calculate a pie-chart angle?
Multiply the category fraction of the whole by 360°.
What is inverse proportion?
Two quantities x and y are inversely proportional if xy remains constant.
What is direct proportion?
Two quantities are directly proportional when their ratio remains constant.
Are speed and time inversely proportional?
Yes, for the same fixed distance at constant speeds.
How long do four workers take if three workers finish in four days?
Three days if each worker has the same constant productivity.
How long do pumps taking 3h and 2h individually take together?
6/5 hours, or 1 hour 12 minutes.
How many Figure it Out questions are in this chapter?
22 numbered questions across four sets: 5, 3, 2 and 12.
Is map distance the same as road distance?
No. A map's RF scale represents geographical distance, which can differ from road routes.
Are every diagram and survey result fully solved?
No. Some printed charts need visual verification and classroom survey results depend on observations.
