Proportional Reasoning-1

7.1 Observing Similarity in Change

The chapter begins with resized tiger images. Image A measures 60×40 mm; C 30×20 mm; D 90×60 mm. Their width:height ratio is 3:2, so they keep their proportions. Image B 40×20 mm has ratio 2:1, while E 60×60 mm has ratio 1:1. Equal absolute changes do not guarantee similar shapes; equal multiplicative scaling does.

7.2 Ratios

A ratio a:b compares quantities in a specified order. For same-kind quantities, convert to matching units before simplifying. For example, 2 m:50 cm =200 cm:50 cm=4:1. A ratio can also compare different units, such as km per hour, but the units and interpretation must then be explicit.

7.3 Ratios in Their Simplest Form

Divide both terms by their highest common factor: 60:40=3:2 and 90:60=3:2. Equivalent ratios have the same value a/b, provided b≠0. Multiplying both terms by a nonzero factor preserves the ratio; adding the same number generally does not.

7.4 Problem Solving with Proportional Reasoning

Direct proportion means y=kx for a constant k. If a:b::c:d, cross-multiplication gives ad=bc. Example: 6 glasses of lemonade need 10 spoons sugar; 18 glasses need 10×18/6=30 spoons. For 120 students requiring 15 kg rice, 80 students need 15×80/120=10 kg. Historical Indian texts describe the rule of three (Trairāśika).

Direct versus inverse variation

At constant speed, distance increases in direct proportion to time. For a fixed distance, travel time varies inversely with speed. Example: 50 km/h for 2 h means 100 km; at 75 km/h time=100/75 h=4/3 h=80 minutes. Do not use a direct proportion between speed and time for fixed distance.

7.5 Sharing, but Not Equally!

To split total T in ratio m:n, calculate mT/(m+n) and nT/(m+n). Example: 42 counters in 4:3 gives 24 and 18. A ₹4000 profit split 3:1 gives ₹3000 and ₹1000. A 40 kg sand:cement mixture at 3:1 contains 30 kg sand and 10 kg cement; to reach 5:2 by adding only cement, final cement=12 kg, so add 2 kg.

7.6 Unit Conversions

Always convert before comparing or cross-multiplying. One litre=1000 mL; one acre=43,560 ft²; one hectare=10,000 m²; one metre≈3.281 feet; one m²≈10.764 ft². Temperature is not directly proportional between Celsius and Fahrenheit because F=(9/5)C+32 includes an offset.

Optimized formula and method table

Worked textbook activities

Filter coffee: regular 15:35=3:7, strong 20:30=2:3, light 10:40=1:4. Comparing decoction share, 2/5>3/10>1/5, so the first is stronger and the last lighter. Sample table: 300:600=1:2 strong; 150:500=3:10 regular in decoction share (about 23.1%)? No: this is lighter than regular 3:7 (30%); classify as light; 200:400=1:2 strong; 24:56=3:7 regular; 100:300=1:3 light. Classroom height and body-measurement activities require actual student measurements; record units and preserve scaling. Shampoo package prices are not proportional to volumes in the textbook sample.

Common mistakes, MCQs and revision

Mistake: treating 50 km/h→75 km/h as direct proportion with time for a fixed trip. Mistake: comparing 1 kg with 200 g without converting. Mistake: adding 2 to both terms of a ratio and assuming equivalence. Original MCQ: Which equals 3:5? (A)6:10 (B)9:10 (C)12:15; answer A. HOTS: A 4:3 ratio totals 56; shares are 32 and 24. Flashcard: Cross-products equal in a proportion. Flashcard: Ratios depend on order. Flashcard: Inverse variation has constant product.

NCERT Figure it Out — page 165

Questions are paraphrased from the official 2026–27 textbook. Diagram- and measurement-dependent answers are identified rather than invented.

1. Question 1. Which of these six proportions 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?

   Step-by-step solution: Compare cross-products for each pair: (i) 4×21=7×12=84, true; (ii) 8×6=48, 3×24=72, false; (iii) 7×7=49, 12×12=144, false; (iv) 21×10=6×35=210, true; (v) 12×12=144, 18×28=504, false; (vi) 24×3=8×9=72, true.

   Final answer: (i), (iv), (vi).

2. Question 2. Give three ratios equivalent to 4:9.

   Step-by-step solution: Multiply both terms by 2, 3 and 4 respectively: (4×2):(9×2), (4×3):(9×3), (4×4):(9×4).

   Final answer: 8:18, 12:27, 16:36.

3. Question 3. Fill 3:__, 12:__, 20:__, 27:__ so that each is proportional to 18:24.

   Step-by-step solution: Reduce 18:24 to 3:4. Multiply the first term by 4/3 to get its matching second term: 3→4, 12→16, 20→80/3, 27→36.

   Final answer: 4; 16; 80/3; 36.

4. Question 4. Which rectangles A–E in the textbook illustration are similar?

   Step-by-step solution: Measure each rectangle and divide width by height. Rectangles with identical width-to-height ratios are similar; labels cannot safely be assigned from text extraction alone.

   Final answer: Diagram-dependent; compare measured width:height for A–E.

5. Question 5. Draw smaller and larger rectangles similar to the printed rectangle.

   Step-by-step solution: Measure the original width w and height h. Draw one rectangle with width w/2 and height h/2, and another with width 2w and height 2h. Both retain ratio w:h.

   Final answer: Any common positive scale factors preserve the shape.

6. Question 6. For each of the two brick-wall patterns, find grey-to-coloured brick ratios.

   Step-by-step solution: Count one complete repeating block in each pattern: (a) 9 grey and 6 coloured; (b) 16 grey and 12 coloured. Divide both terms by their HCF.

   Final answer: (a) 3:2; (b) 4:3.

7. Question 7. Measure a friend's head, torso, arms and legs and draw a proportional figure.

   Step-by-step solution: Record actual measurements H,T,A,L in the same unit. Form H:T, T:A and T:L. Choose a scale factor k and draw lengths kH,kT,kA,kL; each ratio stays unchanged.

   Final answer: Answers depend on measurements; every drawn length must use the same scale factor.

NCERT Figure it Out — page 170

Questions are paraphrased from the official 2026–27 textbook. Diagram- and measurement-dependent answers are identified rather than invented.

1. Question 1. If Earth travels about 940 million km in a year, estimate its travel in one week.

   Step-by-step solution: Assuming 52 equal weeks, divide 940,000,000 by 52 to obtain about 18,076,923 km. This is a textbook approximation; actual orbital speed varies.

   Final answer: Approximately 18.08 million km.

2. Question 2. A mason uses 1450 bricks per 10 ft of wall. How many bricks are needed for all outer and inner walls in the printed house plan?

   Step-by-step solution: The printed plan's wall segments total 108 ft. Rate=1450÷10=145 bricks/ft. Multiply 108×145.

   Final answer: 15,660 bricks (assuming equal height and thickness).

NCERT Figure it Out — page 175

Questions are paraphrased from the official 2026–27 textbook. Diagram- and measurement-dependent answers are identified rather than invented.

1. Question 1. Divide ₹4500 into two shares in ratio 2:3.

   Step-by-step solution: Total parts=2+3=5. Each part=4500÷5=900. Shares=2×900 and 3×900.

   Final answer: ₹1800 and ₹2700.

2. Question 2. A 240 mL solution has acid:water=1:5. How much of each?

   Step-by-step solution: Total parts=6. One part=240÷6=40 mL. Acid=40 mL and water=5×40=200 mL.

   Final answer: 40 mL acid; 200 mL water.

3. Question 3. Make 40 mL paint with blue:yellow=3:5; then add 20 mL yellow. Find the new ratio.

   Step-by-step solution: Original total parts=8, each=5 mL. Blue=15 mL, yellow=25 mL. After adding 20 mL yellow, yellow=45 mL. Simplify 15:45.

   Final answer: 15 mL blue, 25 mL yellow initially; new ratio 1:3.

4. Question 4. For 6 cups of idli mixture with rice:urad dal=2:1, how much of each?

   Step-by-step solution: Total parts=3; one part=6÷3=2 cups. Rice=2×2=4 cups; dal=1×2=2 cups.

   Final answer: 4 cups rice and 2 cups urad dal.

5. Question 5. One bucket of orange paint has red:yellow=3:5. Add one bucket of yellow. What is the new ratio?

   Step-by-step solution: In the original bucket, red=3/8 bucket and yellow=5/8 bucket. New yellow=5/8+1=13/8 bucket. Ratio=(3/8):(13/8).

   Final answer: 3:13.

NCERT Figure it Out — pages 176–177

Questions are paraphrased from the official 2026–27 textbook. Diagram- and measurement-dependent answers are identified rather than invented.

1. Question 1. Simplify the ratio of 600 mL orange juice to 900 mL apple juice.

   Step-by-step solution: HCF(600,900)=300. Divide both quantities by 300.

   Final answer: 2:3.

2. Question 2. Three full buses carried 162 people. How many buses for 204, and will they all be full?

   Step-by-step solution: Capacity per bus=162÷3=54. Three buses hold 162, which is insufficient. Four buses hold 216; 216−204=12 seats remain.

   Final answer: 4 buses; 12 empty seats.

3. Question 3. Compare crowding using Delhi's area 1484 km² and population 30 million, versus Mumbai's area 550 km² and population 20 million.

   Step-by-step solution: Population density is population÷area. Delhi≈30,000,000÷1484≈20,216 people/km². Mumbai≈20,000,000÷550≈36,364 people/km². These are textbook figures, not current census data.

   Final answer: Mumbai is more densely populated using the given figures.

4. Question 4. A crane's neck:rest-of-body ratio is 4:6. What neck length corresponds to your own height?

   Step-by-step solution: Total parts=4+6=10. Neck length is 4/10 of height H, so neck=0.4H. For an illustrative H=150 cm, neck=60 cm.

   Final answer: 0.4×your measured height.

5. Question 5. A historical Lilavati problem: 2½ palas of saffron cost 3/7 niskas. How much for 9 niskas?

   Step-by-step solution: Unit price relation: quantity=2.5×(9÷(3/7))=2.5×21=52.5 palas.

   Final answer: 52½ palas.

6. Question 6. Harmain is 1 and her brother 5. When will their ages be in ratio 1:2?

   Step-by-step solution: Let t years pass. (1+t)/(5+t)=1/2. Cross-multiply: 2+2t=5+t, so t=3. Harmain's age then is 1+3.

   Final answer: After 3 years; Harmain will be 4.

7. Question 7. Equal volumes of gold and water have mass ratio 37:2. If 1 L water is 1 kg, what is mass of 1 L gold?

   Step-by-step solution: Gold:water=37:2, so gold mass=(37/2)×1 kg.

   Final answer: 18.5 kg.

8. Question 8. Apply 10 tonnes of manure per acre to a 200 ft by 500 ft plot. How much manure?

   Step-by-step solution: Area=200×500=100,000 ft². Convert to acres: 100000÷43560≈2.2957 acres. Multiply by 10 tonnes/acre.

   Final answer: Approximately 22.96 tonnes.

9. Question 9. A tap fills 500 mL in 15 s. How long to fill a 10 L bucket?

   Step-by-step solution: 10 L=10000 mL, which is 20 mugs of 500 mL. Time=20×15=300 s.

   Final answer: 300 seconds, or 5 minutes.

10. Question 10. At ₹15,00,000 per acre, find the price of 2400 ft².

    Step-by-step solution: One acre=43,560 ft². Price=1,500,000×2400÷43560≈82,644.63.

    Final answer: Approximately ₹82,645.

11. Question 11. Oxen plough 1 acre in 6 h. A tractor is four times as fast. How long for 20 acres?

    Step-by-step solution: Oxen time=20×6=120 h. Tractor time=120÷4=30 h, assuming constant work rates.

    Final answer: 120 h by oxen; 30 h by tractor.

12. Question 12. A 7.74 g coin contains copper:nickel=3:1. Copper costs ₹906/kg and nickel ₹1341/kg. Find the metal costs.

    Step-by-step solution: Copper mass=7.74×3/4=5.805 g=0.005805 kg; cost=0.005805×906=₹5.25933. Nickel mass=1.935 g=0.001935 kg; cost=0.001935×1341=₹2.594835.

    Final answer: Copper ≈₹5.26; nickel ≈₹2.59; combined ≈₹7.85.

Learning checklist and official source

Check: Can you simplify a ratio, compare two proportions, calculate the fourth proportional, distinguish inverse speed–time variation, split a total in a ratio, convert units and justify a result? This chapter follows NCERT Ganita Prakash Grade 8 Part I Chapter 7, Reprint 2026–27. All 26 numbered Figure it Out questions are represented across four sets. The textbook image-comparison, brick-wall and body-measurement activities require the original figures or real measurements. The optional Binairo puzzle is a separate diagram-based activity and is not claimed solved here.