Proportional Reasoning-1
Why does one photo look "stretched" while another looks perfectly normal when resized? It's all about ratios and proportion. Try the live rectangle-similarity tool and ratio-divider below!
Similarity & ratios
| Image | Width (mm) | Height (mm) | Looks normal? |
|---|---|---|---|
| A | 60 | 40 | โ Yes |
| B | 40 | 20 | โ Stretched |
| C | 30 | 20 | โ Yes |
| D | 90 | 60 | โ Yes |
| E | 60 | 60 | โ Squished |
Image AโC: both width AND height are multiplied by the SAME factor (½) โ looks normal! Image AโB: both width and height dropped by the SAME amount (20mm subtracted) โ but that's NOT the same as a matching factor, so it looks wrong. Proportional means multiplying (not subtracting) by the same factor.
A ratio like a : b means: for every a units of the first thing, there are b units of the second. Reduce to simplest form by dividing both terms by their HCF โ if two ratios reduce to the SAME simplest form, they're proportional (written with "::")!
3:4 is already in its simplest form. To find the simplest form of 72:96, we need to divide both terms by their HCF.
Nitin and Hari were building a compound wall around their house. Nitin built the longer side, 60 ft long, using 3 bags of cement. Hari built the shorter side, 40 ft long, using only 2 bags of cement. Nitin was worried that Hari's wall wouldn't be as strong, since she used less cement. Is Nitin correct?
To check, compare the ratio of wall length to bags of cement for each of them โ if the ratios are proportional, both walls used cement at the same rate, so they're equally strong.
Example 4: The book's own school has 5 teachers and 170 students โ a teacher:student ratio of 5:170 (simplest form 1:34). Count the teachers and students in YOUR school and write your own ratio. Is your school's ratio proportional to 5:170?
Example 5: Measure the width and height of your classroom's blackboard to the nearest cm and write the ratio width:height. Then try drawing a rectangle in your notebook that's proportional to that ratio โ does everyone's rectangle look the same size? (They shouldn't โ only the SHAPE has to match, not the size!)
When Neelima was 3, her mother's age was 10 times hers. Ratio of Neelima's age to her mother's age: 3:30, simplest form 1:10.
9 years later, Neelima is 12 and her mother is 39 (30+9). New ratio: 12:39, simplest form 4:13.
๐ฎ Try it: build proportional rectangles
a:b :: c:d is TRUE exactly when a × d = b × c (cross-multiply and compare!).
๐ Figure It Out (page 165)
Multiply BOTH terms of 4:9 by the same factor. Any factor works โ here are three:
Check: 8×9=72 and 18×4=72 โ โ any multiple of both terms by the same number stays proportional!
18:24 in simplest form is 3:4 (divide both by their HCF, 6). So every proportional ratio must also simplify to 3:4 โ multiply BOTH terms of 3:4 by whatever factor turns 3 into the given first term.
The 20:__ one gives a fraction (26โ ) because 20 isn't a whole-number multiple of 3 โ that's OK! Ratios can scale by fraction factors too, just like in Example 7 below.
Fill in the missing numbers: __:42, 6:__, 2:__ โ all proportional to 14:21.
The book shows 5 rectangles (A, B, C, D, E) at different sizes and rotations โ grab a ruler and measure each one's width and height, then reduce each width:height ratio to simplest form and compare them.
Rectangles whose simplest-form width:height ratios match are similar โ same idea as the tiger photos in Section 1! (Since the rectangles are printed at slightly different rotations in the book, use your ruler directly on the page or ask your teacher for the measured answer key for your printing.)
Say the given rectangle is 6 cm × 4 cm (ratio 3:2 in simplest form). Multiply BOTH sides by the same factor to shrink or grow it:
As long as you multiply BOTH the width and height by the SAME factor, every rectangle you draw looks like a scaled copy of the original โ that's what "proportional" means!
Count one repeating block of the pattern (both walls repeat the same grey/coloured design over and over), then write grey:coloured in simplest form.
Both walls repeat their block over and over, so the ratio for ONE block is the ratio for the WHOLE wall too โ that's proportional reasoning in bricklaying!
This is a hands-on measuring activity โ grab a measuring tape and a friend, and measure the length of their head, torso, arms, and legs in cm. Then write:
Now try drawing a stick figure using EQUIVALENT ratios (same simplest form) for head, torso, arms and legs โ does it look more realistic than a random drawing? That's because real bodies keep these ratios roughly proportional!
The Rule of Three
When you know 3 out of 4 numbers in a proportion, you can always find the missing one! This is called Trairasika โ the Rule of Three.
A school of 120 students usually gets 15 kg of rice cooked for the mid-day meal. On a rainy day, only 80 students come. How much rice should the cook make so none is wasted?
6 glasses of lemonade need 10 spoons of sugar. How much sugar for 18 MORE glasses (same sweetness)?
A car travels 90 km in 150 MINUTES. To find distance in 4 HOURS, you must convert 4 hours to 240 minutes FIRST โ mixing minutes and hours in the same proportion gives a wrong answer!
Ancient Indian mathematician ฤryabhaแนญa described this exact method over 1,500 years ago! He named the three known quantities pramฤแนa (measure), phala (fruit/result), and ichchhฤ (requisition) โ and the unknown ichchhฤphala (yield): "Multiply the phala by the ichchhฤ and divide by the pramฤแนa."
A farmer in Himachal Pradesh sells 200 g packets of tea for โน200. A large estate in Meghalaya sells 1 kg packets for โน800. Are the weight:price ratios proportional? Which tea is more expensive?
If a car travels faster, the travel TIME goes DOWN, not up โ that's an inverse relationship, and the simple Rule of Three (direct proportion) does NOT apply. Always check: does the 4th quantity grow WITH the others, or shrink as they grow?
Puneeth's father went from Lucknow to Kanpur in 2 hours by riding his motorcycle at 50 km/h. If he drives at 75 km/h instead, how long will it take him to reach Kanpur? Can we set this up as the proportion 50:2 :: 75:?
No! Think about what happens when speed goes UP โ the travel time should go DOWN (he covers the same fixed distance faster). But 50:2::75:? would force the time to go UP as speed goes up, which is backwards. Speed and time (for a fixed distance) are inversely related, so the direct Rule of Three cannot be used here at all โ you'd need a different method (multiply speed × time to get the constant distance, then divide by the new speed): distance = 50×2 = 100 km, so at 75 km/h it takes 100÷75 = 1โ hours (1 hour 20 minutes).
Pick your favourite dish and list its ingredients with quantities for your family. Now imagine inviting 15 guests โ use proportional reasoning (Rule of Three!) to scale EVERY ingredient up by the same factor, so the dish tastes just as good for a bigger crowd.
๐ Figure It Out (page 170)
A year has 52 weeks, and the Earth travels the same amount every week (its speed around the Sun is roughly constant), so just divide the yearly distance by 52:
The Earth travels about 18,076,923 km (โ18.08 million km) in a week โ that's how fast our whole planet is zooming through space!
First, find the bricks-per-foot rate. Then add up EVERY wall segment in the floor plan โ the book's own answer key lists these ten wall pieces, walking around the outside plus the inner dividing wall:
The mason needs 15,660 bricks (this is the book's own official answer). It's easy to miscount here โ the safest way is to list EVERY wall segment shown in the diagram one at a time and add them all up, rather than trying to trace one continuous path.
Go to a shop and note the volume and price of different sizes of the same shampoo (sachet, small/medium/large bottle) and fill in a table like the book's own sample:
| Container | Volume | Price |
|---|---|---|
| Sachet | 6 mL | โน2 |
| Small Bottle | 180 mL | โน154 |
| Medium Bottle | 340 mL | โน276 |
| Large Bottle | 1000 mL | โน540 |
See if the volume of shampoo is proportional to the price โ compare volume:price ratios in simplest form for each size. (Usually they're NOT the same โ bigger bottles work out slightly cheaper per mL, which is why buying in bulk often saves money! Try the same activity with rice or atta at different pack sizes.)
Filter Coffee mixing ratios
His REGULAR filter coffee mixes 15 mL decoction with 35 mL milk โ ratio 15:35, simplest form 3:7.
For a customer who wants it STRONGER, he uses 20 mL decoction with 30 mL milk โ ratio 20:30, simplest form 2:3. More decoction relative to milk = stronger coffee!
For LIGHTER coffee, he uses 10 mL decoction with 40 mL milk โ ratio 10:40, simplest form 1:4. Less decoction relative to milk = lighter coffee!
| Decoction (mL) | Milk (mL) | Simplest ratio | Regular / Strong / Light? |
|---|---|---|---|
| 300 | 600 | 1:2 | Stronger than regular (0.5 > 3/7) |
| 150 | 500 | 3:10 | Lighter than regular (0.3 < 3/7) |
| 200 | 400 | 1:2 | Stronger than regular (0.5 > 3/7) |
| 24 | 56 | 3:7 | Exactly regular! โ |
| 100 | 300 | 1:3 | Lighter than regular (0.33 < 3/7) |
Turn decoction:milk into a decimal (decoction ÷ milk). Regular = 3÷7 โ 0.43. A BIGGER decimal than 0.43 means more decoction per mL of milk โ stronger! A smaller decimal means lighter.
Sharing in a ratio
To split a quantity x in the ratio m:n: divide x by (m+n) to find "one group's size," then multiply by m and n separately.
Grab a partner and 12 small objects (coins, seeds, pebbles). Share them equally first โ you'll each get 6, a ratio of 6:6 = 1:1.
Now share them UNEQUALLY in the ratio 3:1 โ one of you takes 3 at a time, the other takes 1 at a time, repeating until all 12 are gone. You end up with 9 and 3 (ratio 3:1, and 9+3=12 โ) โ this hands-on grouping is exactly the idea behind the "split into equal-size groups" method used below!
Prashanti invested โน75,000, Bhuvan invested โน25,000 (ratio 3:1). They made โน4,000 profit, split in the same ratio.
๐ฎ Try the live ratio divider
๐ Figure It Out (page 175)
2 + 1 = 3 groups. Each group = 6 ÷ 3 = 2 cups.
Tricky wording! "One bucket of orange paint" is the WHOLE mixture, made of 3+5 = 8 parts (3 red, 5 yellow). "Another bucket" means ANOTHER FULL bucket โ the same size as the whole original mixture, so it's worth 8 more parts, and it's ALL yellow.
New ratio of red to yellow = 3 : 13.
Unit conversions
Proportional reasoning problems often need unit conversion first!
| Conversion |
|---|
| 1 metre = 3.281 feet |
| 1 square metre = 10.764 square feet |
| 1 acre = 43,560 square feet |
| 1 hectare = 10,000 sq metres |
| 1 hectare = 2.471 acres |
| 1 millilitre (mL) = 1 cubic centimetre (cc) |
| 1 litre = 1,000 mL = 1,000 cc |
| 0ยฐC = 32ยฐF |
| ยฐF = (9/5) ร ยฐC + 32 |
| ยฐC = (5/9) ร (ยฐF โ 32) |
(9/5) × 25 + 32 = 45 + 32 = 77ยฐF โ that's a comfortable room temperature either way!
๐ Figure It Out (page 176)
Simplest ratio = 2 : 3
Each bus's full capacity: 162 ÷ 3 = 54 seats.
Need 4 buses, and they will NOT all be full โ 12 seats vacant.
First, for the crane itself: 4+6 = 10 groups, each group = 155÷10 = 15.5 cm.
For YOUR height, use the same 4:6 (=2:5 of total height) idea: neck length = (4/10) × your height. E.g. for a 150 cm tall student, neck = 0.4 × 150 = 60 cm โ plug in your own height to get your answer!
This is a Rule of Three: 2ยฝ palas costs 3/7 niskas โ find the palas for 9 niskas.
9 niskas buys 52.5 palas of saffron โ ฤryabhaแนญa's method still works over 1,500 years later!
1 litre of gold has a mass of 18.5 kg โ that's why gold jewellery feels so heavy for its size!
Plot area = 200 × 500 = 100,000 sq ft. Convert to acres using 1 acre = 43,560 sq ft.
The farmer should buy about 22,957 kg (โ 22.96 tonnes) of manure.
1 acre = 43,560 sq ft = โน15,00,000.
2,400 sq ft costs about โน82,645.
Every question from the book
TRUE proportions: (i), (iv), (vi).
10 litres = 10,000 mL.
Mumbai is more crowded โ even with fewer total people, its much smaller area makes it far denser!
Let x = years from now: (1+x):(5+x) = 1:2
After 3 years: Harmain is 4, brother is 8 โ ratio 4:8 = 1:2 โ
Practice like the real exam
Section A · MCQ (1 mark each)
(b) aรd=bรc
(b) 43,560 sq ft
Section B · short answer (2 marks each)
Section C · longer answer (3 marks each)
If a car travels at 50 km/h for 2 hours, increasing the speed to 75 km/h does NOT increase the travel time proportionally โ it DECREASES it, since covering the same distance faster takes less time. The Rule of Three only works for DIRECT proportions, where both quantities grow or shrink together by the same factor. Speed and time for a FIXED distance move in OPPOSITE directions โ this is an inverse relationship, requiring a different method.
Section D · 4 marks each
Section E · case study (4 marks)
(a) Copper = 3/4 ร 7.74 = 5.805 g. Nickel = 1/4 ร 7.74 = 1.935 g.
(c) Total โ โน5.26 + โน2.59 = โโน7.85 โ the metal in a โน10 coin is worth less than โน8, much less than its face value!
You did it! ๐
๐ Chapter 7 of 7 · Term 1 Maths · Prishita, Class 8 · All Maths chapters complete!