Direct Proportion – Meaning, Formula & Word Problems

The concept of direct proportion plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you are working on speed-distance problems, budgeting, or even basic science experiments, understanding direct proportion will make calculations easier and concepts clearer.

What Is Direct Proportion?

A direct proportion is defined as a relationship between two quantities where an increase in one causes a proportional increase in the other (or decrease leads to proportional decrease). You’ll find this concept applied in ratio and proportion, science measurements, and practical problems like shopping and recipes.

Key Formula for Direct Proportion

Here’s the standard formula: \( y = kx \)
Where:

The formula means that as x increases or decreases, y does the same in direct relation, keeping the ratio \( \frac{y}{x} = k \) always constant.

Direct Proportion Symbol & Variants

In maths, we write direct proportion as y ∝ x, which reads as "y is directly proportional to x". Replacing the symbol with an equals sign and k gives the equation \( y = kx \). Other ways you may see this are "proportional relationship", or "linear relation", especially in graphs or science topics.

Real-Life Examples of Direct Proportion

• If you buy more candies, you pay more money (price and quantity are directly proportional).
• The duration of a road trip increases if you travel more kilometers at the same speed.
• Increasing the number of workers increases the amount of work done.
• The weight of apples you buy increases with the price (if cost per kg is fixed).

Step-by-Step Illustration

Let’s see how to solve a direct proportion question:

1. Read the question and spot the two directly related quantities.

2. Write down their values as \( x_1, y_1 \) (first pair) and \( x_2, y_2 \) (second pair, one unknown).

3. Use \( \frac{x_1}{y_1} = \frac{x_2}{y_2} \) or set up the equation using \( y = kx \).

4. Substitute the values and solve for the unknown.

Example: A bus travels 120 km in 4 hours. How much time will it take to travel 240 km at the same speed?

- \( x_1 = 120, y_1 = 4, x_2 = 240, y_2 = ? \)
- \( \frac{120}{4} = \frac{240}{y_2} \)
- \( y_2 = \frac{240 \times 4}{120} = 8 \) hours.
So, the bus takes 8 hours.

Speed Trick or Exam Shortcut

To solve direct proportion questions faster, remember: set up the ratios straight away and cross-multiply. For example, in Vedantu classes, teachers often guide students to quickly write \( \frac{x_1}{y_1} = \frac{x_2}{y_2} \) and solve in just two steps, which saves time in competitive exams and quizzes.

Shortcut Example: If 5 pens cost Rs. 60, how much do 8 pens cost?

1. Ratio: \( \frac{5}{60} = \frac{8}{y} \)

2. \( y = \frac{8 \times 60}{5} = Rs. 96 \).

Direct Proportion vs Inverse Proportion

Feature	Direct Proportion	Inverse Proportion
Relationship	One increases, the other increases	One increases, the other decreases
Formula	y = kx	xy = k
Example	Distance & Cost: buy more, pay more	Speed & Time: more speed, less time

To explore more, see Direct and Inverse Proportion on Vedantu.

How Direct Proportion Looks on a Graph

The graph of direct proportion is a straight line passing through the origin (0,0) with a positive slope. This means as x increases, y increases at a constant rate. For example, plotting number of books read (x) against total pages (y) gives a straight line.

Common Mistakes in Direct Proportion

• Confusing direct with inverse proportion (always check if both values rise/fall together).
• Forgetting to set the correct ratio (using product formula instead of ratio for direct).
• Mixing up the known and unknown pairs during cross multiplication.

Practice Problems – Try These Yourself

• If 7 apples cost Rs. 140, how much will 12 apples cost?
• A car covers 60 km in 1 hour. How far will it go in 5 hours at the same speed?
• 10 notebooks weigh 2 kg. What is the weight of 25 notebooks?
• Marks scored are directly proportional to the number of hours studied. If Riya gets 70 marks for 14 hours of study, how many hours for 50 marks?

Connections with Other Maths Concepts

Mastering direct proportion helps with ratio and proportion, percentage problems (percentage of a number), and even solving equations (simple equations). This means understanding direct proportion makes other chapters easier too.

Classroom Tip

An easy way to remember direct proportion: When both values move in the "SAME DIRECTION" (up/up or down/down), use the direct proportion formula. Vedantu’s teachers use real-life stories and ratio tables to help students visualize these connections during live classes.

We explored direct proportion—including the key formula, examples, mistakes, and comparison with related ideas. Keep practicing with Vedantu to get better at identifying and solving direct proportion problems in exams and real-life situations.

For more learning, check out: Ratio and Proportion, Direct and Inverse Proportion, Proportion Problems, and Solving Equations with Variables.