Direct Proportion
Direct proportion describes a relationship between two quantities where both increase or decrease at the same rate. When one quantity doubles, the other also doubles. When one becomes half, the other also becomes half.
For example, if 1 kg of apples costs Rs 120, then 2 kg costs Rs 240, 3 kg costs Rs 360, and so on. The cost increases in direct proportion to the weight.
Direct proportion is one of the most practical mathematical concepts, used in everyday calculations like shopping, cooking, map reading, and unit conversions.
What is Direct Proportion?
Definition: Two quantities x and y are said to be in direct proportion if:
x/y = k (constant)
Or equivalently:
x₁/y₁ = x₂/y₂
Where:
- k is the constant of proportionality
- As x increases, y increases by the same factor
- As x decreases, y decreases by the same factor
- The ratio x/y remains the same throughout
Symbol: x ∝ y (read as "x is directly proportional to y")
This means: y = kx where k is a fixed constant.
Methods
Method 1: Unitary Method
- Find the value of one unit first.
- Then multiply to find the value of the required number of units.
Method 2: Proportion Method
- Set up the proportion: x₁/y₁ = x₂/y₂
- Cross multiply to find the unknown.
Method 3: Constant of Proportionality
- Find k = x/y from the given pair.
- Use y = x/k (or x = ky) to find the unknown.
How to identify direct proportion:
- Ask: "If I increase one quantity, does the other increase too?"
- If yes → likely direct proportion.
- If no (one increases while the other decreases) → likely inverse proportion.
Solved Examples
Example 1: Example 1: Cost of items
Problem: If 5 notebooks cost Rs 150, find the cost of 12 notebooks.
Solution:
Given:
- 5 notebooks = Rs 150
- 12 notebooks = Rs ?
Using proportion:
- 5/150 = 12/x
- Cross multiply: 5x = 150 × 12
- 5x = 1800
- x = 360
Answer: 12 notebooks cost Rs 360.
Example 2: Example 2: Using unitary method
Problem: A car travels 240 km in 4 hours at constant speed. How far will it travel in 7 hours?
Solution:
Given:
- 4 hours → 240 km
- 7 hours → ? km
Unitary method:
- 1 hour → 240/4 = 60 km
- 7 hours → 60 × 7 = 420 km
Answer: The car will travel 420 km in 7 hours.
Example 3: Example 3: Map scale
Problem: On a map, 2 cm represents 50 km. What distance does 7.5 cm represent?
Solution:
Given:
- 2 cm = 50 km
- 7.5 cm = ? km
Using proportion:
- 2/50 = 7.5/x
- 2x = 50 × 7.5
- 2x = 375
- x = 187.5
Answer: 7.5 cm represents 187.5 km.
Example 4: Example 4: Recipe scaling
Problem: A recipe for 4 people needs 3 cups of rice. How many cups are needed for 10 people?
Solution:
Given:
- 4 people → 3 cups
- 10 people → ? cups
Using proportion:
- 4/3 = 10/x
- 4x = 30
- x = 7.5
Answer: 7.5 cups of rice are needed for 10 people.
Example 5: Example 5: Verify direct proportion
Problem: Check if the following data is in direct proportion:
x: 2, 4, 6, 8
y: 5, 10, 15, 20
Solution:
Check the ratio x/y for each pair:
- 2/5 = 0.4
- 4/10 = 0.4
- 6/15 = 0.4
- 8/20 = 0.4
All ratios are equal (0.4). The constant of proportionality k = 0.4.
Answer: Yes, x and y are in direct proportion.
Example 6: Example 6: Currency conversion
Problem: If 1 USD = Rs 83, how many rupees will you get for 25 USD?
Solution:
Given:
- 1 USD = Rs 83
- 25 USD = Rs ?
Direct proportion:
- 1/83 = 25/x
- x = 83 × 25 = 2075
Answer: 25 USD = Rs 2,075.
Example 7: Example 7: Find the constant of proportionality
Problem: y is directly proportional to x. When x = 6, y = 18. Find y when x = 10.
Solution:
Given:
- y ∝ x, so y = kx
- When x = 6, y = 18: 18 = k × 6, so k = 3
Finding y when x = 10:
- y = kx = 3 × 10 = 30
Answer: When x = 10, y = 30.
Example 8: Example 8: Wages problem
Problem: A worker earns Rs 4,500 in 6 days. How much will the worker earn in 15 days?
Solution:
Given:
- 6 days → Rs 4,500
- 15 days → Rs ?
Using proportion:
- 6/4500 = 15/x
- 6x = 4500 × 15
- 6x = 67500
- x = 11250
Answer: The worker earns Rs 11,250 in 15 days.
Example 9: Example 9: Not in direct proportion
Problem: Is the following data in direct proportion?
x: 1, 2, 3, 4
y: 3, 5, 7, 9
Solution:
Check ratios:
- 1/3 = 0.333
- 2/5 = 0.400
- 3/7 = 0.429
- 4/9 = 0.444
The ratios are NOT equal.
Answer: No, x and y are NOT in direct proportion. (They follow a linear relation y = 2x + 1, but not a proportional one.)
Example 10: Example 10: Fuel consumption
Problem: A car uses 8 litres of petrol to travel 120 km. How much petrol is needed for 450 km?
Solution:
Given:
- 120 km → 8 litres
- 450 km → ? litres
Using proportion:
- 120/8 = 450/x
- 120x = 8 × 450
- 120x = 3600
- x = 30
Answer: 30 litres of petrol are needed for 450 km.
Real-World Applications
Real-world applications of direct proportion:
- Shopping: If 1 kg of rice costs Rs 60, then 5 kg costs Rs 300. Cost is directly proportional to quantity.
- Cooking: Scaling recipes up or down — ingredients are in direct proportion to the number of servings.
- Maps and scale drawings: Distance on map is directly proportional to actual distance.
- Travel: At constant speed, distance is directly proportional to time.
- Currency exchange: Amount in one currency is directly proportional to the amount in another (at a fixed exchange rate).
- Shadows: The length of a shadow is directly proportional to the height of the object (at the same time of day).
- Wages: Daily/hourly wages are directly proportional to the number of days/hours worked.
Key Points to Remember
- Two quantities are in direct proportion if their ratio remains constant: x/y = k.
- If x doubles, y doubles. If x halves, y halves.
- The equation is y = kx, where k is the constant of proportionality.
- To solve: use x₁/y₁ = x₂/y₂ and cross multiply.
- The unitary method finds the value of one unit first, then scales up.
- The graph of direct proportion is a straight line through the origin.
- To check if data is in direct proportion, verify that x/y is the same for all pairs.
- Direct proportion means both quantities change in the same direction (both increase or both decrease).
- Do not confuse with inverse proportion, where one increases as the other decreases.
- A linear relation y = mx + c is direct proportion ONLY if c = 0.
Practice Problems
- If 8 pens cost Rs 120, find the cost of 15 pens.
- A car travels 180 km in 3 hours. How far will it travel in 5 hours at the same speed?
- Check if the data is in direct proportion: x = {2, 5, 8, 10}, y = {6, 15, 24, 30}.
- On a map, 3 cm represents 75 km. What distance does 8 cm represent?
- y is directly proportional to x. When x = 4, y = 20. Find y when x = 9.
- A machine produces 120 parts in 8 hours. How many parts will it produce in 14 hours?
- If 2.5 kg of flour costs Rs 87.50, find the cost of 6 kg.
- Is the following data in direct proportion? x = {1, 2, 3, 4}, y = {4, 7, 10, 13}. Justify your answer.
Frequently Asked Questions
Q1. What is direct proportion?
Two quantities are in direct proportion when their ratio is constant. If one quantity increases, the other increases by the same factor. The relationship is x/y = k (constant).
Q2. What is the symbol for 'directly proportional'?
The symbol is ∝ (alpha). x ∝ y means 'x is directly proportional to y.' This means x = ky for some constant k.
Q3. What is the constant of proportionality?
The constant of proportionality (k) is the fixed ratio between the two quantities. If y = kx, then k = y/x. For example, if 1 pen costs Rs 10, then k = 10.
Q4. What is the difference between direct and inverse proportion?
In direct proportion, both quantities increase or decrease together (x/y = k). In inverse proportion, one increases as the other decreases (x × y = k).
Q5. What does the graph of direct proportion look like?
It is a straight line passing through the origin (0, 0). The slope of the line equals the constant of proportionality k.
Q6. Is y = 2x + 3 a direct proportion?
No. In direct proportion, the equation must be y = kx (passing through the origin). y = 2x + 3 has a y-intercept of 3, so it is a linear relation but NOT a direct proportion.
Q7. What is the unitary method?
The unitary method finds the value of one unit first, then multiplies to find the required value. Example: If 5 kg costs Rs 200, then 1 kg costs 200/5 = Rs 40, and 8 kg costs 40 × 8 = Rs 320.
Q8. How do you verify if data is in direct proportion?
Calculate the ratio x/y for each data pair. If all ratios are equal, the data is in direct proportion. If any ratio is different, it is not.
Q9. Can time and distance be in direct proportion?
Yes, but only when speed is constant. At constant speed, distance = speed × time, so distance is directly proportional to time.
Q10. Give 3 examples of direct proportion from daily life.
(1) Number of litres of petrol and total cost. (2) Number of hours worked and total wages. (3) Weight of fruits bought and total price. In all cases, doubling one quantity doubles the other.
