What this chapter is about
This chapter takes your understanding of ratios and proportions further. In earlier classes, you learnt what a ratio is and how to check if two ratios are equal (proportion). Now you will explore deeper ideas: direct proportion, inverse proportion, and how to solve real-life problems using these concepts.
Direct proportion means when one quantity increases, the other also increases at the same rate. Inverse proportion means when one quantity increases, the other decreases at the same rate. These ideas appear everywhere — in cooking recipes, map scales, speed-time-distance problems, and sharing costs among friends.
After studying this chapter, you should be able to identify whether two quantities are in direct or inverse proportion, set up the correct equation, and solve for an unknown value. You will also learn to apply these skills to practical situations involving money, work, time, and measurements.
Key ideas
- Two quantities are in direct proportion if their ratio stays constant. When one doubles, the other also doubles. Example: cost and number of items bought at a fixed price per item.
- Two quantities are in inverse proportion if their product stays constant. When one doubles, the other becomes half. Example: more workers finish a job in fewer days.
- To check direct proportion: divide corresponding values — if all answers are equal, it is direct proportion.
- To check inverse proportion: multiply corresponding values — if all products are equal, it is inverse proportion.
- The unitary method finds the value for one unit first, then scales up or down. It works for both types of proportion.
- In direct proportion, if x₁/y₁ = x₂/y₂, you can cross-multiply to find an unknown.
- In inverse proportion, x₁ × y₁ = x₂ × y₂ helps you find an unknown.
- Real-life clues: phrases like "at the same rate" or "per item" suggest direct proportion; phrases like "working together" or "sharing equally" often suggest inverse proportion.
Formulas and facts to remember
- Direct proportion: x₁/y₁ = x₂/y₂ (ratio is constant)
- Inverse proportion: x₁ × y₁ = x₂ × y₂ (product is constant)
- Unitary method for direct proportion: Value for 1 unit = Total value ÷ Number of units
- Unitary method for inverse proportion: If 1 person takes T days, then n persons take T/n days (when work stays the same)
- Scale on maps: Map distance and actual distance are in direct proportion when using the same scale.
Worked examples
Example 1: Direct proportion (cost of mangoes)
If 5 kg of mangoes cost ₹400, what will 8 kg cost at the same rate?
Step 1: Check the type — more mangoes means more cost, so this is direct proportion.
Step 2: Find cost of 1 kg. Cost of 1 kg = 400 ÷ 5 = ₹80
Step 3: Find cost of 8 kg. Cost of 8 kg = 80 × 8 = ₹640
Answer: 8 kg of mangoes cost ₹640.
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Example 2: Inverse proportion (workers and days)
12 workers can build a wall in 10 days. How many days will 15 workers take to build the same wall?
Step 1: Check the type — more workers means fewer days, so this is inverse proportion.
Step 2: Use the product rule. Workers × Days = constant 12 × 10 = 15 × d
Step 3: Solve for d. 120 = 15 × d d = 120 ÷ 15 = 8
Answer: 15 workers will take 8 days.
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Example 3: Using proportion to find map distance
On a map, 2 cm represents 50 km. What distance does 7 cm represent?
Step 1: Map distance and actual distance are in direct proportion.
Step 2: Set up the proportion. 2/50 = 7/x
Step 3: Cross-multiply. 2 × x = 50 × 7 2x = 350 x = 175
Answer: 7 cm on the map represents 175 km.
Common mistakes
- Assuming every problem is direct proportion → first check whether increasing one quantity increases or decreases the other.
- Forgetting to keep units the same before comparing → always convert to the same unit (all in metres, all in hours, etc.).
- In inverse proportion, dividing instead of multiplying → remember, for inverse proportion you multiply corresponding values.
- Setting up the proportion upside down → write both ratios in the same order (first quantity on top, second on bottom, or vice versa, but be consistent).
- Using cross-multiplication wrongly in inverse proportion → cross-multiplication works for equal ratios (direct), not for inverse; use the product formula instead.
Quick revision
- Direct proportion: ratio stays the same; when one goes up, the other goes up.
- Inverse proportion: product stays the same; when one goes up, the other goes down.
- Unitary method: find the value for 1 unit, then multiply or divide as needed.
- Direct → x₁/y₁ = x₂/y₂; Inverse → x₁ × y₁ = x₂ × y₂.
- Always identify the type of proportion before solving.