What this proportion calculator does
A proportion is a statement that two ratios are equal, written as a : b = c : d. This calculator solves the missing value in that equation. Enter any three of the four values — a, b, c, or x — leave the fourth blank, and click Calculate. The tool cross-multiplies to find the unknown, shows every step of the working, and verifies the answer by checking that both ratios reduce to the same value. If you fill in all four values instead, it checks whether the proportion holds and tells you so.
Formula and how to solve a proportion
The core technique is cross-multiplication. Starting from the proportion a : b = c : x, multiply the outer pair and the inner pair and set them equal.
Rearranging for each possible unknown gives four variants:
a = (b · c) / x
b = (a · x) / c
c = (a · x) / b
Variable definitions
- a, b — the two terms of the first ratio.
- c, x — the two terms of the second ratio.
- The colon ":" means "is to" — a : b is read "a is to b".
- Proportions are also written as fractions: a/b = c/x.
Worked examples
Example 1: Solve 3 : 4 = 9 : x
- Set up the cross-multiplication: 3 × x = 4 × 9.
- Simplify the right side: 3x = 36.
- Divide both sides by 3: x = 12.
- Check: 9 : 12 simplifies to 3 : 4. ✓
Example 2: Solve 5 : x = 15 : 21
- Cross-multiply: 5 × 21 = 15 × x.
- Simplify: 105 = 15x.
- Divide both sides by 15: x = 7.
- Check: 5 : 7 and 15 : 21 both equal 5/7. ✓
Example 3: Recipe scaling — 4 servings need 200 g flour; how much for 10 servings?
- Set up the proportion: 4 : 200 = 10 : x.
- Cross-multiply: 4 × x = 200 × 10 = 2000.
- Divide by 4: x = 500.
- Result: 500 g of flour for 10 servings.
Example 4: Map scale — 1 cm on the map represents 5 km in reality. A road measures 7.5 cm on the map. How long is it in reality?
- Proportion: 1 : 5 = 7.5 : x.
- Cross-multiply: 1 × x = 5 × 7.5 = 37.5.
- Since the coefficient of x is 1, x = 37.5 directly.
- Result: 37.5 km in reality.
Example 5: Verifying a proportion — does 6 : 9 = 10 : 15 hold?
- Cross-multiply: 6 × 15 = 90 and 9 × 10 = 90.
- Both products are equal, so the proportion holds.
- Both ratios simplify to 2 : 3.
- Result: Yes, 6 : 9 = 10 : 15 is a valid proportion.
Where proportions are used in Class 9 & 10
Proportions appear across the Punjab board syllabus in problems that involve scaling or constant ratios. Common contexts include:
- Similar triangles — corresponding sides are in constant proportion. If one triangle is twice as large, every side doubles.
- Direct variation — the cost of items, distance travelled at constant speed, and other linear relationships.
- Chemistry stoichiometry — mole ratios in a balanced equation are proportions.
- Physics numericals — Ohm's law, gas laws, and speed problems are all proportionality statements.
- Map and scale problems — distances on a drawing or map versus the real distance.
Common mistakes to avoid
- Mixing up which term goes on top. Once you write a : b = c : x, keep the order consistent. Swapping a and b changes the answer entirely.
- Cross-multiplying the wrong pairs. The rule is a × x = b × c. Multiply the outer pair first, then the inner pair. Multiplying a × b or c × x is a common slip.
- Using the wrong formula for inverse proportions. This calculator handles direct proportions only. Inverse proportions are a × b = c × x, not a : b = c : x. If the problem says "as one increases the other decreases," it is probably inverse.
- Forgetting to check the answer. Substitute x back and verify both ratios simplify to the same value. This catches most algebraic mistakes.
- Zero denominators. A ratio with denominator zero is undefined, so proportions cannot have a zero in the b or x position where it would cause division by zero.
- Assuming units do not matter. The two ratios must have matching units. Kilometres-to-hours versus metres-to-hours is a mismatch — convert units before setting up the proportion.
FAQ
What is a proportion?
A proportion is a statement that two ratios are equal. Written as a : b = c : d, it means the relationship between a and b is the same as the relationship between c and d. For example, 2 : 3 = 6 : 9 because both ratios simplify to 2 : 3.
How does cross-multiplication work?
In a proportion a : b = c : d, multiply the outer pair (a and d) and the inner pair (b and c) and set them equal: a times d = b times c. This follows from multiplying both sides of a/b = c/d by bd. To solve for an unknown, rearrange the equation and divide.
What is the difference between a direct and an inverse proportion?
In a direct proportion, as one quantity increases the other increases at the same rate. In an inverse proportion, as one increases the other decreases so their product stays constant. This calculator handles direct proportions of the form a : b = c : x. Inverse proportions are of the form a times b = c times x.
Can a proportion have a zero term?
A numerator can be zero: 0 : 5 = 0 : 3 is a valid proportion. A denominator cannot be zero, because division by zero is undefined. This calculator will show an error if you try to use zero where it would cause division by zero.
How do I check that my answer is correct?
Substitute your value of x back into the proportion and simplify both ratios. They should reduce to the same lowest-terms form. Alternatively, compute each ratio as a decimal — the two decimals should match. This calculator shows the check automatically.
When is a proportion used in real problems?
Proportions solve scaling problems: a recipe for 4 people scaled to 10, a map distance converted to real distance, the cost of N items given the cost of M, and similar triangles in geometry. Any time two quantities scale together at a constant rate, you have a direct proportion.
Related tools
Related Hira Academy resources
Proportions underpin similar triangles, scale drawings, and direct variation problems in the BISE Punjab syllabus. Revise the theory with our free Class 9 notes and Class 10 notes.