Word Problems on Proportion
You have learned that a proportion is a statement that two ratios are equal. For example, 2:3 = 4:6 is a proportion. Now it is time to use proportions to solve real-life word problems.
Proportions are used everywhere — in cooking (scaling recipes), in maps (reading distances), in shopping (unit prices), and in measurement (shadows and heights). The key technique is cross multiplication: if a/b = c/d, then a × d = b × c.
Once you set up the proportion correctly, finding the unknown value is just simple multiplication and division.
What is Word Problems on Proportion?
Definition: A proportion states that two ratios are equal.
If a:b = c:d, then a/b = c/d.
If a/b = c/d, then a × d = b × c
Steps to solve proportion word problems:
- Read the problem and identify the two quantities being compared.
- Write the known ratio.
- Set up the proportion with the unknown (call it x).
- Cross multiply and solve for x.
- Write the answer with the correct unit.
Types and Properties
1. Scaling Recipes
If 2 cups of flour makes 10 cookies, how much flour for 25 cookies? Set up: 2/10 = x/25.
2. Map and Model Problems
If 1 cm on a map = 5 km in real life, what does 3.5 cm represent? Set up: 1/5 = 3.5/x.
3. Price and Quantity Problems
If 5 notebooks cost Rs. 150, how much do 8 notebooks cost? Set up: 5/150 = 8/x.
4. Shadow and Height Problems
If a 6 m pole casts a 4 m shadow, how tall is a tree with a 10 m shadow? Set up: 6/4 = x/10.
Solved Examples
Example 1: Example 1: Recipe scaling
Problem: A recipe uses 3 cups of sugar for 12 cupcakes. How much sugar for 20 cupcakes?
Solution:
- 3/12 = x/20
- Cross multiply: 3 × 20 = 12 × x → 60 = 12x → x = 5.
Answer: 5 cups of sugar.
Example 2: Example 2: Map scale
Problem: On a map, 2 cm represents 50 km. What distance does 7 cm represent?
Solution:
- 2/50 = 7/x
- 2x = 50 × 7 = 350 → x = 175.
Answer: 175 km.
Example 3: Example 3: Cost problem
Problem: 4 kg of rice costs Rs. 200. How much does 7 kg cost?
Solution:
- 4/200 = 7/x
- 4x = 200 × 7 = 1400 → x = 350.
Answer: Rs. 350.
Example 4: Example 4: Shadow problem
Problem: A 5 m pole casts a 3 m shadow. A building casts a 12 m shadow. Find the building's height.
Solution:
- 5/3 = x/12
- 3x = 5 × 12 = 60 → x = 20.
Answer: Building height = 20 m.
Example 5: Example 5: Distance and time
Problem: A car travels 120 km in 2 hours. At the same speed, how far in 5 hours?
Solution:
- 120/2 = x/5
- 2x = 120 × 5 = 600 → x = 300.
Answer: 300 km.
Example 6: Example 6: Workers and time
Problem: 6 workers paint a wall in 8 hours. If all workers work at the same rate, how many hours for 3 workers?
Solution:
- More workers = less time. This is inverse proportion.
- 6 × 8 = 3 × x → 48 = 3x → x = 16.
Answer: 16 hours.
Example 7: Example 7: Model building
Problem: A model car is built at 1:20 scale. The model is 25 cm long. Find the real car's length.
Solution:
- 1/20 = 25/x → x = 25 × 20 = 500 cm = 5 m.
Answer: Real car = 5 m.
Example 8: Example 8: Mixing paint
Problem: To get orange, mix red and yellow in ratio 3:2. If you use 9 litres of red, how much yellow?
Solution:
- 3/2 = 9/x → 3x = 18 → x = 6.
Answer: 6 litres of yellow.
Example 9: Example 9: Earning money
Problem: Ravi earns Rs. 450 in 3 days. How much in 10 days at the same rate?
Solution:
- 450/3 = x/10 → 3x = 4500 → x = 1500.
Answer: Rs. 1,500.
Example 10: Example 10: Fuel consumption
Problem: A car uses 5 litres of petrol for 60 km. How much petrol for 150 km?
Solution:
- 5/60 = x/150 → 60x = 750 → x = 12.5.
Answer: 12.5 litres.
Real-World Applications
Cooking: Scaling recipes up or down uses proportion. If a recipe for 4 people needs 2 cups of flour, then for 10 people you need 5 cups.
Maps: Map scales are proportions. If 1 cm = 50 km, then 3 cm = 150 km.
Shopping: Comparing prices per unit (Rs. per kg) and finding the cost of different quantities uses proportion.
Construction: Scale models of buildings use proportions. A 1:100 model means 1 cm on the model = 100 cm (1 m) in reality.
Science: Shadow measurement to find heights uses the proportion between object height and shadow length.
Key Points to Remember
- A proportion says two ratios are equal: a/b = c/d.
- Cross multiply to solve: a × d = b × c.
- Set up the proportion with the unknown as x.
- Make sure both ratios compare the same quantities in the same order.
- Direct proportion: as one increases, the other increases at the same rate.
- Check your answer by substituting back into the proportion.
- Always write the answer with the correct unit.
- Draw a table if needed to organise the known and unknown values.
Practice Problems
- If 3 pencils cost Rs. 15, how much do 10 pencils cost?
- On a map, 4 cm = 100 km. What distance does 6.5 cm represent?
- A recipe needs 2 eggs for 8 pancakes. How many eggs for 20 pancakes?
- A 4 m pole casts a 6 m shadow. A tree casts a 15 m shadow. Find the tree's height.
- A car travels 180 km on 12 litres. How far on 20 litres?
- If 5 workers can do a job in 12 days, how many days for 10 workers?
- A model is 1:50 scale. The real building is 25 m tall. How tall is the model?
- To make green paint, mix blue and yellow 2:5. If you have 8 litres of blue, how much yellow do you need?
Frequently Asked Questions
Q1. What is a proportion?
A proportion is a statement that two ratios are equal. For example, 2/3 = 4/6 is a proportion.
Q2. What is cross multiplication?
If a/b = c/d, then a × d = b × c. This is the main technique for solving proportion problems.
Q3. How do I set up a proportion for a word problem?
Identify the two quantities. Write the known ratio on one side and the ratio with the unknown on the other. Make sure both ratios have the same quantities in the same positions.
Q4. What is the difference between ratio and proportion?
A ratio compares two quantities (3:4). A proportion says two ratios are equal (3:4 = 6:8). Proportion is an equation between two ratios.
Q5. Are all proportion problems direct proportion?
No. In direct proportion, both quantities increase together. In inverse proportion, one increases while the other decreases (like workers and time).
Q6. How do I check my answer?
Substitute the answer back into the proportion. Both sides should be equal. For example, if 3/12 = 5/20, check: 3 × 20 = 60 and 12 × 5 = 60. Equal, so correct.
Q7. Can proportion involve decimals?
Yes. For example, 2.5/5 = x/10. Cross multiply: 2.5 × 10 = 5x, so x = 5.
Q8. Why is proportion useful?
Proportion helps solve problems where two quantities change at the same rate. It is used in cooking, maps, shopping, construction, and science.
