Direct Proportion

Direct proportion happens when two quantities change together, keeping their ratio the same constant value. This means if one goes up, the other does by the same factor, as speed and distance travelled. We use the symbol "∝" to relate the proportions. Let's explore direct proportion.

Table of Contents

• Direct Proportion Definition
• Direct Proportion Formula
• Direct Proportion Graph
• Direct Proportion Solved Examples
• Direct Proportion Practice Questions

Direct Proportion Definition

Two quantities are in direct proportion when an increase in one quantity causes a proportional increase in the other, and a decrease causes a proportional decrease. The ratio between the two quantities always stays the same.

You are buying oranges at a fixed price per orange. The more oranges you buy, the more you pay, and the relationship between quantity and cost never changes. That fixed, equal link is what makes it a direct proportion. We write it as y ∝ x, which means "y is proportional to x."

Direct Proportion Formula

The formula for direct proportion is written as y = k × x. Here, k is a special number called the constant of proportionality. It never changes as long as the two quantities stay proportional. You can always find k by dividing y by x: k = y ÷ x. If you get the same answer every time you divide, the quantities are in direct proportion.

For example: if oranges cost ₹6 each, then k = 6. So y = 6 × x, where x is the number of oranges and y is the total cost. The ratio y ÷ x will always equal 6, no matter how many oranges you buy.

Direct Proportion Graph

When you draw a direct proportion on a graph, it always gives you a perfectly straight line that begins at the origin, the point (0, 0). This is the most important visual rule to remember. If the line is straight but starts from any other point, the relationship is not a direct proportion. The slope of the line equals the value of k. A steeper slope means a larger k, which means y grows faster for every one unit increase in x.

Direct Proportion Solved Examples

Example 1: A car travels 150 km using 10 litres of fuel. How far will it travel on 16 litres?

Solution: First, find k. Since distance and fuel are directly proportional, k = 150 ÷ 10 = 15 km per litre. Now use the formula: distance = 15 × 16 = 240 km. The diagram below shows the three steps clearly.

Example 2: A worker earns ₹420 for 7 hours of work. How much does she earn for 11 hours?

Solution: k = 420 ÷ 7 = ₹60 per hour. Earnings for 11 hours = 60 × 11 = ₹660.

Example 3: Check if these pairs are in direct proportion (4, 20), (7, 35), (9, 45).

Solution: Divide y by x in each case: 20 ÷ 4 = 5, 35 ÷ 7 = 5, 45 ÷ 9 = 5. The ratio is the same every time. So yes, they are in direct proportion with k = 5.

Direct Proportion Practice Questions

1. If y is directly proportional to x, and y = 18 when x = 3, find y when x = 7?

2. A recipe uses 250 g of flour to make 20 biscuits. How many grams of flour are needed to make 52 biscuits?

3. If 5 metres of ribbon cost ₹45, what is the cost of 13 metres?

4. A machine fills 240 bottles in 4 hours. How many bottles does it fill in 9 hours?

5. A tap drips 90 ml of water in 6 minutes. How much water drips in 25 minutes?