Word Problems on Proportions
Word problems on direct and inverse proportion appear in everyday situations — speed and time, workers and days, cost and quantity, maps and scales. Solving them requires first identifying whether the quantities are in direct or inverse proportion.
Direct proportion: When one quantity increases, the other also increases at the same rate. (More items → more cost.)
Inverse proportion: When one quantity increases, the other decreases. (More workers → fewer days.)
What is Word Problems on Proportions?
Direct Proportion: x and y are in direct proportion if x/y = k (constant), i.e., x₁/y₁ = x₂/y₂.
Inverse Proportion: x and y are in inverse proportion if xy = k (constant), i.e., x₁y₁ = x₂y₂.
Direct: x₁/y₁ = x₂/y₂
Inverse: x₁ × y₁ = x₂ × y₂
Methods
Steps to solve proportion word problems:
- Identify the two quantities involved.
- Determine if they are in direct or inverse proportion.
- Set up the equation using the appropriate formula.
- Solve for the unknown.
Quick test:
- Ask: "If one increases, does the other increase or decrease?"
- Both increase → direct
- One increases, other decreases → inverse
Solved Examples
Example 1: Example 1: Cost and quantity (direct)
Problem: If 5 kg of rice costs Rs 400, how much does 8 kg cost?
Solution:
- More rice → more cost. Direct proportion.
- 5/400 = 8/x
- x = (400 × 8)/5 = Rs 640
Answer: 8 kg costs Rs 640.
Example 2: Example 2: Workers and days (inverse)
Problem: 12 workers can build a wall in 10 days. How many days will 15 workers take?
Solution:
- More workers → fewer days. Inverse proportion.
- 12 × 10 = 15 × x
- x = 120/15 = 8 days
Answer: 15 workers take 8 days.
Example 3: Example 3: Speed and time (inverse)
Problem: A car travelling at 60 km/h takes 4 hours to reach a destination. How long at 80 km/h?
Solution:
- Higher speed → less time. Inverse proportion.
- 60 × 4 = 80 × x
- x = 240/80 = 3 hours
Answer: At 80 km/h, it takes 3 hours.
Example 4: Example 4: Map scale (direct)
Problem: On a map, 2 cm represents 5 km. What distance does 7 cm represent?
Solution:
- More cm on map → more actual distance. Direct proportion.
- 2/5 = 7/x
- x = (5 × 7)/2 = 17.5 km
Answer: 7 cm represents 17.5 km.
Example 5: Example 5: Pipes filling a tank (inverse)
Problem: 6 pipes can fill a tank in 90 minutes. How long will 9 pipes take?
Solution:
- More pipes → less time. Inverse proportion.
- 6 × 90 = 9 × x
- x = 540/9 = 60 minutes
Answer: 9 pipes take 60 minutes.
Example 6: Example 6: Food supply (inverse)
Problem: A camp has food for 120 people for 15 days. If 30 more people join, how long will the food last?
Solution:
- More people → fewer days. Inverse proportion.
- Total people = 120 + 30 = 150
- 120 × 15 = 150 × x
- x = 1800/150 = 12 days
Answer: Food lasts 12 days.
Example 7: Example 7: Petrol and distance (direct)
Problem: A car uses 12 litres of petrol to travel 180 km. How much petrol for 300 km?
Solution:
- More distance → more petrol. Direct proportion.
- 12/180 = x/300
- x = (12 × 300)/180 = 20 litres
Answer: 20 litres needed for 300 km.
Example 8: Example 8: Machines and hours (inverse)
Problem: 8 machines produce 480 articles in 6 hours. How many hours for 5 machines to produce the same?
Solution:
- Fewer machines → more hours. Inverse proportion.
- 8 × 6 = 5 × x
- x = 48/5 = 9.6 hours = 9 hours 36 minutes
Answer: 5 machines take 9 hours 36 minutes.
Example 9: Example 9: Number of pages and days (direct)
Problem: A student reads 40 pages in 2 hours. How many pages in 5 hours at the same rate?
Solution:
- More hours → more pages. Direct proportion.
- 40/2 = x/5
- x = (40 × 5)/2 = 100 pages
Answer: 100 pages in 5 hours.
Example 10: Example 10: Mixed problem
Problem: 18 men can dig a trench in 12 days working 8 hours per day. How many days will 24 men take working 6 hours per day?
Solution:
- Total work = 18 × 12 × 8 = 1728 man-hours
- New: 24 men × 6 hours × d days = 1728
- 144d = 1728
- d = 12 days
Answer: 12 days.
Real-World Applications
Where proportion problems appear:
- Travel: Speed, distance, time calculations.
- Cooking: Scaling recipes for more or fewer servings.
- Construction: Workers, days, and project planning.
- Maps: Scale conversions between map distance and real distance.
- Business: Cost, production, and resource allocation.
Key Points to Remember
- Direct proportion: Both quantities increase or decrease together. x₁/y₁ = x₂/y₂.
- Inverse proportion: One increases while the other decreases. x₁y₁ = x₂y₂.
- Always identify the type of proportion first.
- For compound proportion (3 quantities), use: total work = men × days × hours.
- More workers = fewer days (inverse). More items = more cost (direct).
- Higher speed = less time for same distance (inverse).
- More people = food lasts fewer days (inverse).
- Map scales are always direct proportion.
Practice Problems
- If 15 books cost Rs 1,200, what do 25 books cost?
- 10 workers finish a job in 18 days. How many days for 15 workers?
- A car travels 240 km in 4 hours. How far in 7 hours at the same speed?
- A factory produces 500 items with 10 machines in 5 hours. How many hours with 8 machines?
- Food for 200 soldiers lasts 30 days. How long for 250 soldiers?
- On a map, 3 cm = 12 km. Find the actual distance for 8 cm.
Frequently Asked Questions
Q1. How do I know if it's direct or inverse proportion?
Ask: if one quantity increases, does the other increase (direct) or decrease (inverse)? More workers → fewer days = inverse. More items → more cost = direct.
Q2. Can I use the unitary method instead?
Yes. Find the value for 1 unit first, then multiply. For direct: cost of 1 item = total cost ÷ items. For inverse: work done by 1 worker = total work ÷ workers.
Q3. What if three quantities are involved?
Use compound proportion. Calculate total work = men × days × hours, then solve for the unknown using the same total work.
Q4. Is speed and distance direct or inverse?
Speed and distance (for fixed time) are in direct proportion. Speed and time (for fixed distance) are in inverse proportion.
Q5. What is the unitary method?
A method where you first find the value for 1 unit, then use it to find the value for the required number of units.
Q6. Can two quantities be neither direct nor inverse?
Yes. Not all relationships are proportional. For example, the area of a circle and its radius are not in direct proportion (area = πr², which is quadratic, not linear).
