Direct variation (one goes up, the other goes up)
1 kg of apples costs ₹120. 2 kg costs ₹240 and 3 kg costs ₹360. When the weight grows, the money grows by the same number of times.
If the ratio of two quantities always stays the same, they vary directly. That fixed ratio is the constant of variation (k).
In the figure: Weight 1, 2, 3 kg → cost ₹120, 240, 360: all points lie on a straight line through (0, 0); at every point cost ÷ weight = 120.
Formula: y ÷ x = k (or y = k × x)
In words: Once you find k, y = k × x for any x.
Direct variation (one goes up, the other goes up)
In the app you can move this figure with your finger.
Open in the app- Easy: 5 kg of apples cost ₹600. Price of 8 kg? k = 600 ÷ 5 = 120 ; 120 × 8 = ₹960.
- Medium: m ∝ n. When n = 9, m = 36. Find m when n = 14. k = 36 ÷ 9 = 4 ; m = 4 × 14 = 56.
- Exam-type: y ∝ √x. When x = 16, y = 12. Find y when x = 64. k = 12 ÷ √16 = 12 ÷ 4 = 3 ; y = 3 × √64 = 3 × 8 = 24.
The sign "∝" means "varies directly": y ∝ x says y and x vary directly.
Always find k as y ÷ x and keep the same order every time.
Once k is found, filling a table is easy: multiply every new x by k. E.g. with k = 120, 4 kg cost 120 × 4 = ₹480. Memory clue: Direct: y ÷ x stays the same; "both grow hand in hand".
Spot the difference
| Point | y ∝ x | y ∝ √x |
|---|---|---|
| The constant part | y ÷ x | y ÷ √x |
| If x is multiplied by 4, y is | multiplied by 4 | multiplied by 2 |
| k = 3, x = 16: y | 48 | 12 |
| Graph | straight line through the origin | curve through the origin |
Practice questions
- 12 kg of rice cost ₹90. The cost of 5 kg?
- 26 notebooks cost ₹72. The cost of 1 notebook?
- 3p ∝ q. When q = 4, p = 20. If q = 9, p?
- 43 m of cloth cost ₹270. How many metres for ₹630?
- 5A tap gives 40 litres in 5 minutes. How many litres in 12 minutes?
- 6y ∝ x. When x = 7, y = 28. If y = 60, x?
- 7y ∝ √x. When x = 9, y = 15. If x = 36, y?
- 8y ∝ x. When x = 10, y = 35. Find y when x = 4, and x when y = 63.
Show answers (try them yourself first)
- 1k = 90 ÷ 2 = 45; 45 × 5 = ₹225.
- 272 ÷ 6 = ₹12.
- 3k = 20 ÷ 4 = 5; p = 5 × 9 = 45.
- 4k = 270 ÷ 3 = 90; 630 ÷ 90 = 7 m.
- 5k = 40 ÷ 5 = 8; 8 × 12 = 96 litres.
- 6k = 28 ÷ 7 = 4; x = 60 ÷ 4 = 15.
- 7k = 15 ÷ 3 = 5; y = 5 × 6 = 30.
- 8k = 35 ÷ 10 = 3.5; y = 3.5 × 4 = 14; x = 63 ÷ 3.5 = 18.
Inverse variation (one goes up, the other comes down)
6 workers build a wall in 10 days. 12 workers need only 5 days. Twice the workers, half the days. 6 × 10 = 60 and 12 × 5 = 60: the product stays the same.
If the product of two quantities always stays the same, they vary inversely.
In the figure: Workers 3, 6, 12 → days 20, 10, 5: the product is always 60; the points lie on a curve that never touches the axes.
Formula: x × y = k (or y = k ÷ x)
In words: If one grows by some number of times, the other shrinks by the same number of times.
Inverse variation (one goes up, the other comes down)
In the app you can move this figure with your finger.
Open in the app- Easy: 6 workers finish in 10 days. How many days for 4 workers? 6 × 10 = 60 ; 60 ÷ 4 = 15 days.
- Medium: At 40 km/h the trip takes 6 hours. At 60 km/h how long? 40 × 6 = 240 ; 240 ÷ 60 = 4 hours.
- Exam-type: x ∝ 1/y. When y = 8, x = 15. Find x when y = 24. k = 15 × 8 = 120 ; x = 120 ÷ 24 = 5.
In inverse variation x × y = k stays the same. If x grows by some number of times, y shrinks by the same number of times.
Workers–days and speed–time are inverse. Price–weight is direct. First ask: when one goes up, does the other go up or down?
In inverse variation first find the whole work = men × days, then divide. Memory clue: Inverse: x × y stays the same; "one up, the other down".
Spot the difference
| Point | Direct | Inverse |
|---|---|---|
| If one goes up, the other | goes up | goes down |
| Stays the same | ratio y ÷ x | product x × y |
| Example | weight and price | workers and days |
| Graph | straight line (through origin) | curve |
Practice questions
- 110 workers finish a job in 12 days. How many days for 6 workers?
- 2x × y = 36. If x = 4, y?
- 3At 50 km/h it takes 3 hours. How many hours at 75 km/h?
- 4y ∝ 1/x. When x = 5, y = 8. If x = 20, y?
- 5Food lasts 20 people for 15 days. For 25 people, how many days?
- 6A job takes 8 people 9 days. How many people are needed to finish it in 6 days?
- 76 people finish a job in 20 days. After 4 days, 2 people leave. In how many more days do the rest finish the job?
- 8x ∝ 1/y. When y = 6, x = 12. If x = 9, y?
Show answers (try them yourself first)
- 110 × 12 = 120; 120 ÷ 6 = 20 days.
- 236 ÷ 4 = 9.
- 350 × 3 = 150; 150 ÷ 75 = 2 hours.
- 4k = 5 × 8 = 40; y = 40 ÷ 20 = 2.
- 520 × 15 = 300; 300 ÷ 25 = 12 days.
- 68 × 9 = 72; 72 ÷ 6 = 12 people.
- 7Total work = 6 × 20 = 120; done in 4 days 6 × 4 = 24; left 96; with 4 people: 96 ÷ 4 = 24 days.
- 8k = 6 × 12 = 72; y = 72 ÷ 9 = 8.
Graphs and how to recognise the type
Travelling at 20 km an hour: 1 h – 20 km, 2 h – 40 km, 3 h – 60 km. The points lie on a straight line. But to cover 60 km, a bigger speed means less time: 10 → 6 h, 20 → 3 h, 30 → 2 h. These points lie on a curve.
In the figure: Left: (1, 20), (2, 40), (3, 60) on a straight line through (0, 0); right: (2, 30), (3, 20), (6, 10) on a curve.
In words: A direct variation graph is a straight line through the origin. An inverse variation graph is a curve that never touches the axes.
Graphs and how to recognise the type
In the app you can move this figure with your finger.
Open in the appGraphs and how to recognise the type
In the app you can move this figure with your finger.
Open in the app- Easy: Points (1, 20), (2, 40), (3, 60): y ÷ x = 20 every time and the line passes through (0, 0) → direct.
- Medium: Points (2, 30), (3, 20), (6, 10): x × y = 60 every time and the graph is a curve → inverse.
- Exam-type: Table x: 1, 2, 4 and y: 5, 10, 15. y ÷ x = 5, 5, 3.75 is not constant; x × y = 5, 20, 60 is not constant either → neither direct nor inverse.
If the graph passes through the origin (0, 0), it is direct variation.
A curved graph is not always inverse variation; check whether x × y stays the same.
First check both in the table: y ÷ x constant → direct; x × y constant → inverse. Memory clue: First the table, then the graph: direct is a straight line, inverse is a curve.
Spot the difference
| Point | Direct graph | Inverse graph |
|---|---|---|
| Shape | straight line | curve |
| Through (0, 0) | yes | no |
| Constant in the table | y ÷ x | x × y |
| Example points | (1, 20), (2, 40) | (2, 30), (3, 20) |
Practice questions
- 14 notebooks cost ₹60. What do 7 notebooks cost?
- 23 pens cost ₹21. What do 10 pens cost?
- 3Speed and time for a fixed distance: direct or inverse?
- 4a ∝ b. When b = 6, a = 18. Find a when b = 11.
- 58 men finish a job in 15 days. How many days will 12 men take?
- 6x ∝ 1/y. When y = 4, x = 9. Find x when y = 12.
- 7y ∝ √x. When x = 25, y = 10. Find y when x = 64.
- 8A car at 45 km/h covers a distance in 4 hours. What speed is needed to cover it in 3 hours?
Show answers (try them yourself first)
- 160 ÷ 4 = 15 ; 15 × 7 = ₹105.
- 221 ÷ 3 = 7 ; 7 × 10 = ₹70.
- 3Inverse (more speed, less time).
- 4k = 18 ÷ 6 = 3 ; a = 3 × 11 = 33.
- 58 × 15 = 120 ; 120 ÷ 12 = 10 days.
- 6k = 9 × 4 = 36 ; x = 36 ÷ 12 = 3.
- 7k = 10 ÷ √25 = 10 ÷ 5 = 2 ; y = 2 × √64 = 2 × 8 = 16.
- 845 × 4 = 180 ; 180 ÷ 3 = 60 km/h.