1. Chapter at a Glance
A recipe for 4 people needs 2 cups of rice; how much for 10 people? A map shows 1 cm for 5 km; how far apart are two towns 7.4 cm apart on the map? A photo is enlarged — will it look stretched? All these questions need proportional reasoning: understanding how two quantities change together. This chapter builds the ideas of ratio, equivalent ratios and proportion, and the methods to solve proportion problems.
- Ratios and how to simplify and compare them.
- Equivalent ratios and when two ratios are in proportion.
- Solving proportions using the cross-product rule and the unitary method.
- Dividing a quantity in a given ratio.
- Scale drawings, maps and recipes.
2. Ratio
A ratio compares two quantities of the same kind in the same units by division. The ratio of a to b is written a : b (or a/b). It tells how many times one quantity is of the other.
- A class has 18 girls and 12 boys. Girls : boys = 18 : 12 = 3 : 2.
- Convert to the same units first: 50 cm : 2 m = 50 : 200 = 1 : 4.
- A ratio has no units.
Simplest form
Divide both terms by their HCF: 36 : 60 → HCF 12 → 3 : 5.
Comparing ratios
Write them as fractions with a common denominator (or compare cross products). 3 : 4 vs 5 : 7 → 3/4 = 21/28, 5/7 = 20/28 → 3 : 4 is greater.
3. Equivalent Ratios and Proportion
Ratios like 2 : 3, 4 : 6, 10 : 15 are equivalent — multiply or divide both terms by the same number. When two ratios are equal, the four quantities are in proportion: a : b = c : d, also written a : b :: c : d.
a : b = c : d exactly when a × d = b × c (product of extremes = product of means).
Are 15, 45, 40, 120 in proportion?
Solution: 15 × 120 = 1800; 45 × 40 = 1800. Equal, so yes (15 : 45 = 40 : 120 = 1 : 3).
Find x if 6 : x = 9 : 24.
Solution: 6 × 24 = 9x → x = 144 ÷ 9 = 16.
4. The Unitary Method
Find the value of one unit first, then multiply.
12 notebooks cost ₹540. Find the cost of 20 notebooks.
Solution: 1 notebook = 540 ÷ 12 = ₹45. 20 notebooks = 20 × 45 = ₹900. (Or by ratio: 12 : 20 = 540 : x → x = 900.)
5. Dividing in a Ratio
To divide a quantity Q in the ratio a : b, the parts are Q × a/(a + b) and Q × b/(a + b).
Divide ₹1,200 between Anu and Bina in the ratio 3 : 5.
Solution: Total parts = 8. One part = 150. Anu = 3 × 150 = ₹450; Bina = 5 × 150 = ₹750.
For three people in the ratio a : b : c, total parts = a + b + c and the method is the same.
6. Scale Drawings, Maps and Recipes
- A map scale 1 : 50,000 means 1 cm on the map = 50,000 cm = 0.5 km on the ground.
- A model car on the scale 1 : 40 that is 10 cm long represents a car 400 cm = 4 m long.
- When a photo is resized, its length and width must be multiplied by the same factor; otherwise it looks stretched.
- Recipes scale proportionally: if 4 people need 2 cups of rice, 10 people need 10 × (2 ÷ 4) = 5 cups.
On a map with scale 1 cm : 5 km, two towns are 7.4 cm apart. What is the actual distance?
Solution: 7.4 × 5 = 37 km.
7. Solved Problems (NCERT Style)
Green paint is made by mixing blue and yellow in the ratio 2 : 3. How much blue is needed with 7.5 L of yellow?
Solution: 2 : 3 = x : 7.5 → 3x = 15 → x = 5 L.
The ratio of ages of a father and son is 7 : 2. The father is 35. Find the son's age and their ratio after 5 years.
Solution: One part = 5; son = 10. After 5 years: 40 : 15 = 8 : 3.
Pack A: 500 g for ₹60. Pack B: 750 g for ₹84. Which is cheaper per gram?
Solution: A: ₹0.12 per g; B: ₹0.112 per g. Pack B is cheaper.
8. Master Formula Table
| Idea | Rule |
|---|---|
| Ratio | a : b = a/b (same units) |
| Simplest form | divide by HCF |
| Proportion | a : b = c : d ⇔ ad = bc |
| Unitary method | value of 1 → value of many |
| Dividing Q in a : b | Qa/(a+b), Qb/(a+b) |
| Map scale 1 : n | actual = map × n |
| Fourth proportional of a, b, c | d = bc/a |
9. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Ratio | Comparison by division | Simplify, compare |
| Proportion | Equal ratios | Cross products |
| Unitary method | Find one unit | Cost, time, quantity |
| Sharing | Total parts | Divide money |
| Scale | Same factor for all lengths | Maps, models |
10. Mind Map
- Same units
- Simplest form
- Compare
- Equivalent ratios
- ad = bc
- Unitary
- Dividing in ratio
- Maps, models
- Recipes, mixing
- Best buy
11. Tips and Tricks
- Always convert to the same units before forming a ratio.
- In sharing problems, first find "one part".
- Compare prices using cost per unit (per gram, per litre).
- Writing a ratio with units (it has none).
- Adding the same number to both terms (changes the ratio).
- Reversing the order of a ratio (3 : 5 is not 5 : 3).
12. Important One-liners
- A ratio compares quantities of the same kind.
- A ratio has no units.
- Equivalent ratios are formed by multiplying or dividing both terms by the same number.
- In a proportion, product of extremes = product of means.
- The unitary method finds the value of one unit first.
- Scaling a picture needs the same factor for length and width.
13. Exam-pattern Questions with Detailed Answers
The ratio 45 : 60 in simplest form is:
- 9 : 12
- 3 : 4
- 4 : 3
- 15 : 20
Show answer
HCF 15: 3 : 4.
Answer: (B)
Find the ratio of 80 paise to ₹2.
Show answer
80 : 200 = 2 : 5.
Answer: 2 : 5
If 5 pens cost ₹60, find the cost of 8 pens.
Show answer
1 pen = ₹12; 8 pens = ₹96.
Answer: ₹96
Find x if 3 : 8 = x : 56.
Show answer
8x = 168 → x = 21.
Answer: 21
Divide 72 marbles among three children in the ratio 2 : 3 : 4.
Show answer
9 parts; one part = 8. Shares: 16, 24, 32.
Answer: 16, 24, 32
Which ratio is greatest?
- 2 : 3
- 5 : 8
- 3 : 4
- 7 : 12
Show answer
As decimals: 0.667, 0.625, 0.75, 0.583. Greatest: 3 : 4.
Answer: (C)
A model of a building uses the scale 1 : 200. The model is 45 cm tall. How tall is the building in metres?
Show answer
45 × 200 = 9000 cm = 90 m.
Answer: 90 m
A photo 6 cm × 4 cm is enlarged so its length becomes 15 cm. What should its width be so it doesn't look stretched?
Show answer
Scale factor = 15 ÷ 6 = 2.5. Width = 4 × 2.5 = 10 cm.
Answer: 10 cm
Two numbers are in the ratio 5 : 7. If 6 is subtracted from each, the ratio becomes 2 : 3. Find the numbers.
Show answer
Let the numbers be 5k and 7k. (5k − 6) : (7k − 6) = 2 : 3 → 3(5k − 6) = 2(7k − 6) → 15k − 18 = 14k − 12 → k = 6.
Numbers: 30 and 42. Check: 24 : 36 = 2 : 3 ✔.
Answer: 30 and 42
In a school, the ratio of boys to girls is 4 : 5. If there are 360 students, how many more girls should join to make the ratio 2 : 3?
Show answer
Boys = 160, girls = 200. Need 160 : g = 2 : 3 → g = 240. More girls = 40.
Answer: 40
A recipe for 6 people needs 450 g flour and 3 eggs. How much flour and how many eggs for 10 people?
Show answer
Per person: 75 g flour, ½ egg. For 10: 750 g flour, 5 eggs.
Answer: 750 g flour, 5 eggs
A map has scale 1 : 25,000. Two villages are 6.2 km apart. How far apart are they on the map?
Show answer
6.2 km = 6,20,000 cm. Map distance = 6,20,000 ÷ 25,000 = 24.8 cm.
Answer: 24.8 cm
14. Quick Revision Summary
- Ratio compares like quantities in the same units; simplify with the HCF.
- Proportion: a : b = c : d ⇔ ad = bc.
- Unitary method: find one unit, then scale up.
- Divide in a : b using total parts.
- Scales and enlargements use the same factor for every length.