Proportional Reasoning – 1
Why This Matters
You change the size of pictures all the time.
You drag the corner of a photo on a phone to make it bigger. You print a passport photo. You stretch a sticker to fit a wall. Sometimes the picture still looks right. But sometimes the face looks too thin and tall, or too fat and squashed. Why?
Here is a small story from the Ganita Prakash textbook. There are five pictures of a tiger, all of different sizes. Three of them — call them A, C and D — look perfectly fine, just bigger or smaller copies of each other. But two of them, B and E, look wrong. In B the tiger looks stretched and thin. In E it looks squashed and fat. They are all just rectangles holding a tiger, so what is different?
The answer is one of the most useful ideas in all of mathematics: proportional reasoning. It tells you when a change keeps the same shape and when it spoils it. The same idea decides how much sugar to add when you make more lemonade, how to share money fairly between two friends, and how a map can shrink a whole country onto one page without lying about its shape.
By the end of this chapter you will know exactly why some changes keep a shape and others ruin it. You will read and simplify ratios, test if two ratios match, and split any amount in any ratio. Let’s begin.
The Big Idea
A change keeps a shape only when every length changes by the same factor — the same multiplication, not the same addition. Making both sides of a rectangle twice as long keeps it similar; taking 20 off both sides does not. We capture this “same factor” idea with a ratio, written a : b. Two ratios that describe the same shape are called proportional. Almost this whole chapter — similar figures, scale factors, mixing, sharing, the Rule of Three — is just this one idea seen from different sides: equal factors, not equal differences.
Let’s Break It Down
We will go step by step. First, see why some images look similar and others look distorted. Then meet the scale factor. Then see the surprising way area grows. Then learn ratios and their simplest form, the test for proportional ratios, and finally how to share an amount in a ratio.
Before we start, let us quickly remember what a ratio’s “factor” and HCF mean, because everything leans on them.
Why some images look similar and others look distorted
Look again at the tiger pictures. Here are their measurements from the textbook.
| Image | Width (mm) | Height (mm) |
|---|---|---|
| A | 60 | 40 |
| B | 40 | 20 |
| C | 30 | 20 |
| D | 90 | 60 |
| E | 60 | 60 |
Compare A and C. The width of C (30) is half the width of A (60). The height of C (20) is also half the height of A (40). Both the width and the height changed by the same factor, ½. Because both shrank by the same factor, C is just a smaller copy of A. They look similar.
Now compare A and B. The width drops from 60 to 40, that is 20 less. The height drops from 40 to 20, that is also 20 less. The difference is the same (20 each), so it feels like a fair change. But look at the factor. The height went from 40 to 20, a factor of ½. The width went from 60 to 40, a factor of 2/3, not ½. The two sides did not change by the same factor. So the shape is spoiled — B looks stretched.
This is the heart of the chapter. The picture below puts a similar pair and a distorted one side by side, so you can see it.
So the rule is: two figures are similar when their matching lengths change by the same factor. We say their lengths are proportional. For rectangles, that means the width-to-height ratio stays the same.
Let’s test image D against A to be sure we have it.
Image A is 60 by 40. Image D is 90 by 60. By what factor does the width change? By what factor does the height change? Is D similar to A?
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Find the width factor. Width goes from 60 to 90. The factor is 90 ÷ 60 = 3/2 (which is 1.5). So the width is multiplied by 3/2.
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Find the height factor. Height goes from 40 to 60. The factor is 60 ÷ 40 = 3/2 again. So the height is also multiplied by 3/2.
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Compare the two factors. Width factor = 3/2. Height factor = 3/2. They are the same.
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Because the width and height changed by the same factor (3/2), image D is a bigger copy of A with the same shape. So D is similar to A. This is why A, C and D all looked alike — same factor on both sides.
The scale factor
That “same factor” has a name. The scale factor is the single number you multiply every length by to enlarge or shrink a figure while keeping its shape.
- A scale factor greater than 1 makes the figure bigger (an enlargement). Scaling 30 by 20 with a factor of 2 gives 60 by 40.
- A scale factor between 0 and 1 makes it smaller (a reduction). Scaling 60 by 40 with a factor of ½ gives 30 by 20.
The key point: the scale factor must hit every length, not just one. If you multiply only the width, you stretch the shape and break it.
The picture below shows a small rectangle enlarged by a scale factor of 2.
Notice that the scale factor only changes the size, never the ratio. Both 30 : 20 and 60 : 40 simplify to 3 : 2. That is exactly why scaling keeps a shape.
A photo is 12 cm wide and 8 cm tall. You enlarge it with a scale factor of 3. What are its new width and height, and is it still similar to the original?
Multiply every length by the scale factor 3. New width = 12 × 3 = 36 cm. New height = 8 × 3 = 24 cm. Because both sides were multiplied by the same factor (3), the new photo is similar to the original. (Check: 12 : 8 = 3 : 2, and 36 : 24 = 3 : 2 — the same.)
Length scales by k, but area scales by k squared
Here is something that surprises almost everyone. When you double the lengths of a figure, the area does not double. It becomes four times bigger.
Why? Think of a tiny square that is 1 unit on each side. Its area is 1 × 1 = 1. Now scale it by a factor of 2, so each side becomes 2 units. The new area is 2 × 2 = 4. The side doubled, but the area is four times as big.
The reason is simple once you see it. Area is two lengths multiplied together. If each length is multiplied by the scale factor k, then the area is multiplied by k × k = k². So:
length is multiplied by k
area is multiplied by k²
The picture below shows the 2 × 2 square sitting on the small one. You can literally count four small squares inside it.
This matters in real life. If a small pizza and a large pizza have the same shape but the large one is twice as wide, it does not give you twice the pizza — it gives you about four times the pizza. That can change which one is better value.
A square tile is 10 cm on each side. You scale it up by a factor of 3. What is its new side length, and how many times bigger is its area?
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The new side length is the old length times the scale factor: 10 × 3 = 30 cm.
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The old area was 10 × 10 = 100 cm². The new area is 30 × 30 = 900 cm².
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Find how many times bigger: 900 ÷ 100 = 9. So the area is 9 times bigger.
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Notice 9 = 3². The length grew by the scale factor k = 3, but the area grew by k² = 3² = 9. So the side is 3 times longer while the area is 9 times larger. Length scales by k, area by k squared.
Ratios and the terms of a ratio
We have been writing things like 60 : 40 and 3 : 2. This is a ratio. A ratio compares two quantities by how many of one go with how many of the other.
A ratio a : b means: for every a units of the first quantity, there are b units of the second quantity.
The two numbers, a and b, are called the terms of the ratio. For image A, the ratio of width to height is 60 : 40. So for every 60 mm of width there are 40 mm of height.
A ratio is not just for shapes. The ratio of teachers to students might be 5 : 170. The ratio of sugar to glasses of lemonade might be 10 : 6. The ratio of cement bags to wall length might be 3 : 60. Any time two amounts go together, a ratio describes the pairing.
One important thing: a ratio cares about the relationship, not the exact amounts. 60 : 40 and 30 : 20 describe the same relationship, because both say “the first is one and a half times the second.” That is why we need a way to tell when two ratios are really the same — which is the next idea.
Ratios in their simplest form
To compare ratios easily, we shrink each one to its simplest form. We do this by dividing both terms by their HCF.
Take 60 : 40. The HCF of 60 and 40 is 20. Divide both terms by 20:
60 ÷ 20 : 40 ÷ 20
= 3 : 2
So the simplest form of 60 : 40 is 3 : 2. You cannot reduce it any further, because 3 and 2 share no common factor bigger than 1.
Now take image D’s ratio, 90 : 60. The HCF of 90 and 60 is 30. Divide both by 30: 90 ÷ 30 : 60 ÷ 30 = 3 : 2 again. So A and D have the same simplest form. That is the precise reason they look alike.
Once two ratios share a simplest form, we say they are proportional and write them with a double colon :: like this: 60 : 40 :: 90 : 60. Read it “60 is to 40 as 90 is to 60.”
The picture below shows both ratios being reduced and matching.
In contrast, image B is 40 : 20, whose simplest form is 2 : 1, and image E is 60 : 60, whose simplest form is 1 : 1. Neither matches 3 : 2, so B and E are not proportional to A, C and D. That is the real reason they looked wrong.
Are the ratios 3 : 4 and 72 : 96 proportional?
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Put each in simplest form. 3 : 4 is already simplest — 3 and 4 share no common factor bigger than 1.
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Reduce 72 : 96. Find the HCF of 72 and 96. It is 24 (since 72 = 24 × 3 and 96 = 24 × 4).
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Divide both terms by 24: 72 ÷ 24 : 96 ÷ 24 = 3 : 4.
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Both ratios in simplest form are 3 : 4 — the same. So they are proportional: 3 : 4 :: 72 : 96.
There is also a quicker test that avoids finding HCFs. For a : b :: c : d, the ratios are proportional exactly when a × d = b × c. This is called cross multiplication. For example, 3 : 4 and 72 : 96 give 3 × 96 = 288 and 4 × 72 = 288 — equal, so they are proportional. We can use this to find a missing term too.
Kesang adds 10 spoons of sugar to 6 glasses of lemonade. For the same sweetness, how many spoons of sugar does she need for 18 glasses?
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Set up the proportion. Glasses to sugar must stay the same. So 6 : 10 :: 18 : ? Let the unknown be the spoons of sugar for 18 glasses.
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Find the factor of change in the known term. Glasses went from 6 to 18. The factor is 18 ÷ 6 = 3.
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The other term must change by the same factor. Sugar was 10, so new sugar = 10 × 3 = 30.
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So 6 : 10 :: 18 : 30. She needs 30 spoons of sugar for 18 glasses to keep the same sweetness. (Check by cross multiplication: 6 × 30 = 180 and 10 × 18 = 180 — equal, so it is correct.)
When Neelima is 3, her mother is 30. The ratio of their ages is 3 : 30, or 1 : 10. When Neelima turns 12, her mother is 39. Is the new ratio 12 : 39 still proportional to 1 : 10?
No. Simplify 12 : 39. The HCF of 12 and 39 is 3, so 12 : 39 = 4 : 13. That is not the same as 1 : 10. The ratio changed because we added the same number (9 years) to both ages — and adding the same amount does not keep a ratio proportional. Only multiplying both terms by the same factor keeps a ratio.
Sharing an amount in a given ratio
Ratios also let you split something fairly, but not equally. Suppose two partners must share 42 counters in the ratio 4 : 3. How many does each get?
The trick is to think in groups. The ratio 4 : 3 means: for every 4 the first person takes, the second takes 3. So the whole amount is split into 4 + 3 = 7 equal groups. The first person gets 4 of those groups, the second gets 3.
Total groups = 4 + 3 = 7
Size of each group = 42 ÷ 7 = 6
First share = 4 × 6 = 24, second share = 3 × 6 = 18
The picture below lays out the seven groups so you can see the split.
In general, to share a quantity x in the ratio m : n, the two parts are m × (x ÷ (m + n)) and n × (x ÷ (m + n)). You divide by the total groups, then multiply by each share’s number of groups.
Prashanti invested ₹75,000 and Bhuvan invested ₹25,000 in a food cart. They share a profit of ₹4,000 in the same ratio as their investments. What is each person's share?
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Find the ratio of investments: 75000 : 25000. Reduce it — divide both by 25000 to get 3 : 1.
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Find the total groups: 3 + 1 = 4.
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Find the size of one group: ₹4,000 ÷ 4 = ₹1,000.
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Give each their groups. Prashanti = 3 × 1,000 = ₹3,000. Bhuvan = 1 × 1,000 = ₹1,000. (Check: 3,000 + 1,000 = ₹4,000, the whole profit, and 3,000 : 1,000 = 3 : 1 as required.)
Common Mistakes
These are the slip-ups students make most often with ratios and proportion. Spotting them now will save you marks later.
If you subtract the same number from both terms of a ratio, the ratio stays the same. So 60 : 40 and 40 : 20 are the same shape.
Taking away an equal amount from both sides feels perfectly fair and balanced, so it seems like the ratio should not change.
Only multiplying (or dividing) both terms by the same factor keeps a ratio. Subtracting changes it: 60 : 40 is 3 : 2, but 40 : 20 is 2 : 1 — different ratios, so different shapes. That is exactly why image B looked stretched.
If you double the side of a square, you double its area too.
Doubling the side feels like it should double everything about the square, since the only number you changed was the side, and you doubled it.
Area uses two lengths multiplied together, so doubling the side multiplies the area by 2 × 2 = 4, not 2. A 2 by 2 square holds four 1 by 1 squares. In general, length × k makes area × k².
To make a ratio simpler, you can just subtract to make the numbers smaller, so 18 : 24 becomes 0 : 6 by taking 18 off each.
Subtracting really does make the numbers smaller, and 'simplest form' sounds like it just means smaller numbers, so subtracting looks like a shortcut.
Simplest form comes from dividing both terms by their HCF, not subtracting. 18 : 24 has HCF 6, so it reduces to 3 : 4. Dividing keeps the relationship the same; subtracting destroys it.
Quick Check
Try these quick questions. The explanation appears after you answer, so read it either way.
A rectangle is 50 cm by 30 cm. Which of these rectangles is similar to it (same shape)?
The original ratio is 50 : 30 = 5 : 3. Check 25 : 15 — its HCF is 5, giving 5 : 3, the same. So 25 by 15 is similar (it is the original scaled by ½). The others reduce to 2 : 1, 2 : 1 and 2 : 1, which do not match 5 : 3.
A figure is enlarged by a scale factor of 4. By what factor does its area grow?
Length grows by the scale factor k = 4, but area grows by k² = 4² = 16. Area is two lengths multiplied, so each ×4 stacks up to ×16. (Picture a square: 4 times wider and 4 times taller holds 16 copies of the original.)
Which statement of proportion is TRUE?
Use cross multiplication (a × d = b × c). For 21 : 6 :: 35 : 10: 21 × 10 = 210 and 6 × 35 = 210 — equal, so it is true. The others fail: 7 × 7 = 49 ≠ 12 × 12 = 144; 8 × 6 = 48 ≠ 3 × 24 = 72; 12 × 12 = 144 ≠ 18 × 28 = 504.
₹4,500 is shared between two people in the ratio 2 : 3. How much does the person with the larger share get?
Total groups = 2 + 3 = 5. Size of one group = 4,500 ÷ 5 = ₹900. The larger share has 3 groups: 3 × 900 = ₹2,700. (The smaller share is 2 × 900 = ₹1,800, and 2,700 + 1,800 = ₹4,500.)
Practice Problems
Try each one yourself first. Then tap to check your full solution.
Easy
Write the ratio 600 : 900 in its simplest form.
Find the HCF of 600 and 900. It is 300 (since 600 = 300 × 2 and 900 = 300 × 3).
Divide both terms by 300: 600 ÷ 300 : 900 ÷ 300 = 2 : 3.
So 600 : 900 in its simplest form is 2 : 3.
A rectangle 30 cm by 18 cm is enlarged with a scale factor of 2. What are its new width and height?
Multiply every length by the scale factor 2.
New width = 30 × 2 = 60 cm.
New height = 18 × 2 = 36 cm.
Both sides were multiplied by the same factor, so the new rectangle (60 by 36) is similar to the old one.
Divide 6 cups of an idli mixture into rice and urad dal in the ratio 2 : 1.
Total groups = 2 + 1 = 3.
Size of one group = 6 ÷ 3 = 2 cups.
Rice = 2 × 2 = 4 cups. Urad dal = 1 × 2 = 2 cups.
(Check: 4 + 2 = 6 cups, and 4 : 2 = 2 : 1 as required.)
Medium
Are the width-to-height ratios of these two photos proportional: 64 : 48 and 96 : 72? Use both the simplest-form test and cross multiplication.
Simplest-form test. HCF of 64 and 48 is 16, so 64 : 48 = 4 : 3. HCF of 96 and 72 is 24, so 96 : 72 = 4 : 3. Both reduce to 4 : 3, so they are proportional.
Cross multiplication. For 64 : 48 :: 96 : 72, check 64 × 72 against 48 × 96. 64 × 72 = 4,608 and 48 × 96 = 4,608. They are equal, so the ratios are proportional.
So 64 : 48 :: 96 : 72 — the two photos are the same shape.
A small farmer sells a 200 g packet of tea for ₹200. A large estate sells a 1 kg packet for ₹800. Are the weight-to-price ratios proportional? Which tea is more expensive per kilogram?
First use the same unit for weight. 1 kg = 1000 g.
Small farmer’s weight : price = 200 : 200, simplest form 1 : 1.
Large estate’s weight : price = 1000 : 800, simplest form 5 : 4 (HCF is 200).
These simplest forms differ, so the ratios are not proportional.
To compare prices, find the cost of 1 kg in each place. The estate’s 1 kg costs ₹800. For the small farmer, 200 g costs ₹200, so 1 kg (five times as much) costs 5 × 200 = ₹1,000.
So the small farmer’s tea is more expensive — ₹1,000 per kg versus ₹800 per kg.
A car travels 90 km in 150 minutes at a steady speed. How far will it travel in 4 hours? (Be careful with units.)
The time and distance are proportional at a steady speed, so set up time : distance.
First fix the units: 4 hours = 4 × 60 = 240 minutes. So the proportion is 150 : 90 :: 240 : x.
Use cross multiplication: 150 × x = 240 × 90, so 150x = 21,600.
Divide: x = 21,600 ÷ 150 = 144.
So the car travels 144 km in 4 hours.
Challenge
A 40 kg mixture has sand and cement in the ratio 3 : 1. How much cement must you ADD so the new ratio of sand to cement becomes 5 : 2? (The sand does not change.)
First find the starting amounts. Total groups = 3 + 1 = 4, so one group = 40 ÷ 4 = 10 kg.
Sand = 3 × 10 = 30 kg. Cement = 1 × 10 = 10 kg.
The sand stays 30 kg. We want the new ratio sand : cement = 5 : 2. So set up 5 : 2 :: 30 : ?, where ? is the new total cement.
In 5 : 2, the cement (second term) is 2/5 of the sand (first term). So the new cement = (2/5) × 30 = 12 kg.
We already have 10 kg of cement, so we must add 12 − 10 = 2 kg of cement.
(Check: new mixture is 30 kg sand and 12 kg cement, and 30 : 12 = 5 : 2 as required.)
A large pizza has the same shape as a small pizza but is scaled up by a factor of 2. The small pizza costs ₹120 and the large costs ₹360. Which gives more pizza for your money?
The large pizza is scaled by a factor of 2 in length. But pizza is an area, so its amount grows by k² = 2² = 4. The large pizza has 4 times as much food as the small one.
Now compare value. Two ways:
Cost per amount. Call the small pizza “1 unit” of food. The large is 4 units. Small: ₹120 for 1 unit = ₹120 per unit. Large: ₹360 for 4 units = 360 ÷ 4 = ₹90 per unit.
Same money. For ₹360 you could buy 3 small pizzas (3 × ₹120), giving 3 units of food. Or 1 large pizza, giving 4 units. The large gives more.
Either way, the large pizza is better value — ₹90 per unit versus ₹120 per unit. This is the area-scales-by-k² idea saving you money in real life.
Summary
- Two figures are similar (same shape) when every matching length changes by the same factor — the same multiplication, not the same subtraction. For a rectangle, the width-to-height ratio must stay the same.
- The scale factor is the single number you multiply every length by. A factor above 1 enlarges; a factor between 0 and 1 shrinks. The scale factor changes the size but never the ratio.
- Length scales by k, but area scales by k². Doubling the side of a shape makes its area four times bigger, because area is two lengths multiplied together.
- A ratio a : b means “for every a of the first, there are b of the second.” The numbers a and b are its terms.
- The simplest form of a ratio comes from dividing both terms by their HCF. Two ratios are proportional (written a : b :: c : d) when their simplest forms match — or, equivalently, when a × d = b × c (cross multiplication).
- To share x in the ratio m : n, split it into m + n equal groups of size x ÷ (m + n), then give each share its number of groups.
What’s Next
You can now spot similar shapes, scale them, simplify and compare ratios, and split amounts fairly. This is the foundation of proportional thinking — and you will use it in science, maps, recipes, money and much more.
That brings Part I of Ganita Prakash to a close. The journey continues in Part II. Next up is Part II, Chapter 1 — Fractions in Disguise, where you will meet numbers that look one way but are secretly fractions in hiding — and the ratio thinking you built here will make them feel familiar. Onward!
Frequently Asked Questions
What makes two images or figures similar?
Two figures are similar when they have the same shape but can be different sizes. This happens when every matching length changes by the same factor. For a rectangle, the width and height must change by the same factor, so the ratio of width to height stays the same. If only one side changes, the shape looks stretched and is not similar.
What is a scale factor?
A scale factor is the number you multiply every length of a figure by to enlarge or shrink it. If a 30 by 20 rectangle is scaled by a factor of 2, each side is multiplied by 2 to give 60 by 40. Because both sides change by the same factor, the new figure is similar to the old one.
Why does the area grow by the square of the scale factor?
Area depends on two lengths multiplied together. If each length is multiplied by k, then the area is multiplied by k times k, which is k squared. So a scale factor of 2 makes lengths twice as long but the area four times as big, because a 2 by 2 square is made of four 1 by 1 squares.
How do you check if two ratios are proportional?
Reduce both ratios to their simplest form by dividing each one by the HCF of its terms. If the simplest forms are exactly the same, the ratios are proportional. A quicker test is cross multiplication: for a : b and c : d, the ratios are proportional when ad equals bc.
How do you share an amount in a given ratio?
Add the two terms of the ratio to find the total number of equal groups. Divide the amount by that total to find the size of one group. Then give each share its number of groups. To share 42 in the ratio 4 : 3, there are 7 groups of 42 divided by 7 equals 6, so the shares are 4 times 6 equals 24 and 3 times 6 equals 18.