Similarity and change by the same factor
A 60 × 40 photo shrunk to 30 × 20 looks exactly the same, only smaller. But a photo cut to 40 × 20 by removing 20 from each side looks squeezed.
Two shapes look similar when every side is MULTIPLIED by the same factor. Adding or subtracting the same amount from each side does not keep the shape.
In the figure: The first two have the same shape (factor 1/2). The third has a different shape.
Rule 1: Find the factor — Factor = new size ÷ old size.
- Easy: 15 → 45: factor 45 ÷ 15 = 3.
- Medium: 8 → 6: factor 6 ÷ 8 = 3/4.
- Exam-type: Width 60 → 75: factor 5/4, so height 40 becomes 40 × 5/4 = 50.
Rule 2: Use the same factor on the other side — Once you know the factor from one side, multiply the other side by it.
- Easy: 80 × 50 halved gives 40 × 25.
- Medium: 40 × 30 with width 60: factor 3/2, height 45.
- Exam-type: A map 20 × 16 cut to width 5: factor 1/4, height 4.
Rule 3: Same amount is not same factor — Compare the ratios of the sides, not the differences.
- Easy: 20 × 10 → 30 × 20: 2 : 1 and 3 : 2 differ, not similar.
- Medium: 60 × 40 → 40 × 20: factors 2/3 and 1/2 differ.
- Exam-type: 90 × 60 and 60 × 45: 3 : 2 and 4 : 3, not similar.
If the sides become k times, the area becomes k × k times: a 5 × 3 photo enlarged 4 times has area 20 × 12 = 240, which is 16 times 15.
Adding the same number to both sides and expecting the same look. Test: 20 × 10 plus 10 each gives 30 × 20, which is a different shape.
Multiply, do not add. Memory clue: "Times keeps the shape, plus bends the shape".
Spot the difference
| Point | Same factor (multiply) | Same amount (add/subtract) |
|---|---|---|
| 6 × 4 becomes | 12 × 8 | 10 × 8 |
| Ratio of sides | 3 : 2 → 3 : 2 | 3 : 2 → 5 : 4 |
| Looks similar? | yes | no |
Practice questions
- 1A 60 × 40 picture is halved in width. Find the new height.
- 2Find the factor of change from 15 to 45.
- 3Which looks similar to 4 × 3: 12 × 9 or 8 × 5?
- 4A 40 × 30 photo is enlarged to width 60. Find the height.
- 5A map 20 × 16 is reduced to width 5. Find the height.
- 6Does 30 × 20 look similar to 20 × 10?
- 7A 5 × 3 photo is enlarged 4 times. Find the new area.
- 8Pictures 90 × 60 and 60 × 45: are they similar?
Show answers (try them yourself first)
- 120
- 23
- 312 × 9
- 445
- 54
- 6No: 3 : 2 and 2 : 1 differ
- 7240
- 8No: 3 : 2 and 4 : 3 differ
Ratios and their terms
In a class of 40 students, 15 are boys and 25 are girls. We say the ratio of boys to girls is 15 : 25, which tells us for every 15 boys there are 25 girls.
A ratio a : b compares two quantities of the same kind. The numbers a and b are the terms. The order matters: a : b and b : a are different.
Ratios and their terms
In the app you can move this figure with your finger.
Open in the appIn the figure: Change the sliders to see how the two quantities compare.
Rule 1: Write the ratio — Write the quantity named first on the left of the colon.
- Easy: 12 boys and 18 girls: boys : girls = 12 : 18.
- Medium: 8 black and 6 white balls: white : all = 6 : 14.
- Exam-type: MATHEMATICS: vowels : consonants = 4 : 7.
Rule 2: Same units first — Convert both quantities to the same unit before writing the ratio.
- Easy: 2 m and 50 cm: 200 : 50.
- Medium: 3 m and 150 cm: 300 : 150.
- Exam-type: 1 hour and 20 minutes: 60 : 20.
Rule 3: Use the ratio to find a missing amount — Find the number of equal groups, then multiply the other term by it.
- Easy: Apples : oranges = 3 : 5, 12 apples: 12 = 3 × 4, so oranges = 5 × 4 = 20.
- Medium: a : b = 5 : 2 and b = 14: 14 = 2 × 7, a = 5 × 7 = 35.
- Exam-type: Boys : girls = 1 : 2 with 18 girls: 9 boys.
A ratio has no unit. 200 : 50 simplified is 4 : 1, and it is the same whether you use centimetres or metres.
Writing the terms in the wrong order. "Boys to girls" is boys first. Also mixing units, such as 2 m : 50 cm written as 2 : 50.
First named goes first. Memory clue: "A to B means A before the colon".
Spot the difference
| Point | 3 : 5 | 5 : 3 |
|---|---|---|
| Meaning | for every 3 of the first, 5 of the second | for every 5 of the first, 3 of the second |
| Same ratio? | no | no |
| First term | 3 | 5 |
Practice questions
- 1A class has 12 boys and 18 girls. Write boys : girls.
- 2What are the terms of the ratio 7 : 4?
- 3Write the ratio of 2 m to 50 cm using the same unit.
- 4Apples : oranges = 3 : 5 and there are 12 apples. How many oranges?
- 5Out of 40 students, 15 are boys. Write boys : girls.
- 6a : b = 5 : 2 and b = 14. Find a.
- 7A bag has 8 black and 6 white balls. Write white : all balls.
- 8Is the ratio 3 : 5 the same as 5 : 3? Give a reason.
Show answers (try them yourself first)
- 112 : 18
- 27 and 4
- 3200 : 50
- 420
- 515 : 25
- 635
- 76 : 14
- 8No, the order matters: for every 3 of the first, 5 of the second
Simplest form and proportion
Two lemonade recipes: 4 glasses with 6 spoons of sugar, and 6 glasses with 9 spoons. Reduce both to 2 : 3 and you know they taste equally sweet.
To write a ratio in its simplest form, divide both terms by their HCF. Two ratios are in proportion when their simplest forms are equal. We write a : b :: c : d.
In the figure: All three reduce to 3 : 2, so their shapes are in proportion.
Rule 1: Simplify — Divide both terms by the HCF.
- Easy: 60 : 40, HCF 20, gives 3 : 2.
- Medium: 48 : 36, HCF 12, gives 4 : 3.
- Exam-type: 18 : 27 : 36, HCF 9, gives 2 : 3 : 4.
Rule 2: Decimals, fractions and units — Remove the decimals or fractions first, or change to the same unit, then simplify.
- Easy: 1.5 : 2.5 → 15 : 25 → 3 : 5.
- Medium: 2/3 : 5/6 → 4/6 : 5/6 → 4 : 5.
- Exam-type: 1 m : 40 cm → 100 : 40 → 5 : 2.
Rule 3: Test for proportion — Simplify both ratios. Same answer means in proportion.
- Easy: 6 : 9 and 10 : 15 are both 2 : 3: proportional.
- Medium: 5 : 8 and 35 : 56 are both 5 : 8: proportional.
- Exam-type: 4 : 9 and 20 : 40: the second is 1 : 2, so not proportional.
A proportion a : b :: c : d is true when both terms of a : b are multiplied by the same factor to get c : d.
Checking only one term. 4 : 9 and 20 : 40: 4 × 5 = 20 but 9 × 5 = 45, not 40, so they are not proportional.
Divide by the HCF until nothing common is left. Memory clue: "Shrink both until they cannot shrink".
Spot the difference
| Point | Ratio | Proportion |
|---|---|---|
| What is it? | comparison of two quantities | equality of two ratios |
| Example | 6 : 9 | 6 : 9 :: 10 : 15 |
| Terms | 2 | 4 |
Practice questions
- 1Simplify 60 : 40.
- 2Simplify 24 : 36.
- 3Are 6 : 9 and 10 : 15 proportional?
- 4Simplify 1.5 : 2.5.
- 5Simplify 100 : 250.
- 6Are 5 : 8 and 35 : 56 proportional?
- 7Simplify 18 : 27 : 36.
- 8Are 4 : 9 and 20 : 40 proportional? Why?
Show answers (try them yourself first)
- 13 : 2
- 22 : 3
- 3Yes, both are 2 : 3
- 43 : 5
- 52 : 5
- 6Yes, both are 5 : 8
- 72 : 3 : 4
- 8No: 4 : 9 stays 4 : 9 but 20 : 40 is 1 : 2
Solving problems with proportional reasoning
A painter knows that 3 litres of paint cover 12 m² of wall. For a 20 m² wall, she does not guess: 20 is 12 × 5/3, so she needs 3 × 5/3 = 5 litres.
In a proportional problem, all quantities change by the same factor. Method 1: find the factor of change. Method 2: find the amount for ONE unit, then multiply.
In the figure: Both rows are multiplied by the same factor from one column to the next.
Rule 1: Factor method — Find how many times the known amount is repeated, then repeat the other quantity the same number of times.
- Easy: 3 : 7 :: 12 : x: 3 × 4 = 12, so x = 7 × 4 = 28.
- Medium: 5 : 9 :: 35 : x: 5 × 7 = 35, so x = 9 × 7 = 63.
- Exam-type: Rice : water = 2 : 5; 6 cups rice: 6 = 2 × 3, water = 5 × 3 = 15 cups.
Rule 2: Unit method — Find the amount for 1, then multiply for the amount you want.
- Easy: 6 notebooks cost ₹90: one costs ₹15, so 10 cost ₹150.
- Medium: A tap fills 40 litres in 8 minutes: 5 litres per minute, so 100 litres in 20 minutes.
- Exam-type: A train covers 240 km in 4 hours: 60 km per hour, so 420 km needs 7 hours.
Rule 3: Compare two situations — Reduce both ratios. Equal simplest forms mean the same strength, taste or speed.
- Easy: Juice : water 4 : 6 and 6 : 9 are both 2 : 3: same taste.
- Medium: 60 bricks in 3 m and 40 bricks in 2 m: both 20 per metre.
- Exam-type: Cement : sand = 1 : 4; 12 bags of sand need 12 ÷ 4 = 3 bags of cement.
Direct proportion means: if one quantity becomes 3 times, the other also becomes 3 times (more paint, more area).
Adding instead of multiplying: "3 litres for 12 m², so 5 litres for 14 m²" is wrong. It should be 20 m².
Go to ONE first, then go to what you want. Memory clue: "One first, then many".
Spot the difference
| Point | Factor method | Unit method |
|---|---|---|
| Idea | how many times repeated | value of one |
| Best when | numbers divide nicely | the factor is awkward |
| Example | 3 : 7 :: 12 : 28 | ₹90 for 6 → ₹15 for 1 |
Practice questions
- 13 litres of paint cover 12 m². How many litres for 20 m²?
- 2If 3 : 7 :: 12 : x, find x.
- 36 notebooks cost ₹90. Find the cost of 10.
- 4A car goes 150 km in 3 hours. How far in 9 hours at the same speed?
- 5A map scale is 1 cm : 5 km. A road is 7.5 cm on the map. Find its real length.
- 6A recipe for 4 people uses 300 g of flour. How much for 10 people?
- 7Mix A: 4 cups of juice and 6 of water. Mix B: 6 of juice and 9 of water. Same taste?
- 8A photo 36 cm wide and 24 cm tall is copied to a height of 15 cm. Find the new width.
Show answers (try them yourself first)
- 15 litres
- 228
- 3₹150
- 4450 km
- 537.5 km
- 6750 g
- 7Yes, both are 2 : 3
- 822.5 cm
Sharing in a ratio and unit conversion
Two friends earn ₹1200 together from a stall and agree to share it in the ratio 3 : 2 because one worked more hours. The money is cut into 5 equal parts of ₹240; they get 3 parts and 2 parts.
To share a total in the ratio a : b: (1) add the terms to get the number of parts, (2) divide the total by it to get the value of 1 part, (3) multiply each term by that value.
In the figure: The bar is cut into 5 equal parts; one person gets 3 parts and the other gets 2 parts.
Rule 1: Share a total — Parts = sum of terms; one part = total ÷ parts.
- Easy: ₹1200 in 3 : 2: 5 parts, ₹240 each, shares ₹720 and ₹480.
- Medium: 64 in 5 : 3: 8 parts of 8; the parts are 40 and 24.
- Exam-type: Angles of a triangle in 2 : 3 : 4: 9 parts, 20° each, angles 40°, 60°, 80°.
Rule 2: Use a given part or difference — Match the known amount to its number of parts to find one part.
- Easy: Shares 5 : 4 and the smaller is 36: 1 part = 9, total 81.
- Medium: Numbers 7 : 3 differ by 20: 4 parts = 20, 1 part = 5, numbers 35 and 15.
- Exam-type: Ages 7 : 2 and the daughter is 8: 1 part = 4, mother = 28.
Rule 3: Unit conversions — A conversion is a proportion: multiply or divide by the conversion factor.
- Easy: 3.5 km = 3.5 × 1000 = 3500 m.
- Medium: 10 inches = 10 × 2.54 = 25.4 cm.
- Exam-type: 36 km/h = 36 × 1000 ÷ 3600 = 10 m/s; 3 hectares = 30,000 m².
Check shares: they must add up to the total (720 + 480 = 1200) and must be in the given ratio.
Dividing the total by one term instead of the sum of the terms: ₹1200 ÷ 3 is wrong; the right divisor is 3 + 2 = 5.
Add, divide, multiply. Memory clue: "ADM: Add the parts, Divide the total, Multiply each term".
Spot the difference
| Point | Wrong | Right |
|---|---|---|
| ₹1200 in 3 : 2 | ₹1200 ÷ 3 = ₹400 | ₹1200 ÷ 5 = ₹240 |
| Shares | ₹1200 and ₹800 | ₹720 and ₹480 |
| Total of shares | ₹2000 | ₹1200 |
Practice questions
- 1Share ₹1200 in the ratio 3 : 2. Find the larger share.
- 2Share 35 sweets in the ratio 4 : 3. Find the larger share.
- 3Convert 3.5 km to metres.
- 4Divide 64 in the ratio 5 : 3. Find the smaller part.
- 5Two shares are in the ratio 5 : 4 and the smaller is 36. Find the total.
- 6Convert 5 kg 250 g to grams.
- 7Angles of a triangle are in the ratio 2 : 3 : 4. Find the largest angle.
- 8Two numbers are in the ratio 7 : 3 and differ by 20. Find the larger.
Show answers (try them yourself first)
- 1₹720
- 220
- 33500 m
- 424
- 581
- 65250 g
- 780°
- 835