Proportion Calculator
Solve direct proportion (A/B = C/D) and inverse proportion (A × B = C × D) equations with step-by-step cross-multiplication proofs.
Proportion Equation Solver
Select proportion type and target variable to solve.
Proportion Output
Quick Summary
Our proportion calculator solver with steps solves direct proportion equations ($\A / B = \C / D$) and inverse proportion equations ($A \cdot B = C \cdot D$). It automatically isolates any unknown variable ($A, B, C, \text{or } D$), cross-multiplies terms, and displays complete algebraic proof steps.
How It Works: Direct vs Inverse Proportion Rules
A proportion states that two ratios or rates are equal:
1. **Direct Proportion ($\A / B = \C / D$):** As one variable increases, the other increases proportionally. Cross multiply $A \cdot D = B \cdot C$ and isolate $X$ (e.g. $D = \B \cdot C / A$).
2. **Inverse Proportion ($A \cdot B = C \cdot D$):** As one variable increases, the other decreases proportionally. Constant product $k = A \cdot B \implies D = \A \cdot B / C$.
3. **Unit Rate Verification:** In direct proportions, the unit rate $\A / B$ equals $\C / D$.
Formula Explanation
Your proportion calculations use the direct and inverse cross-multiplication equations:
Step-by-Step Worked Example
Here is a detailed 5-step breakdown for solving the direct proportion 4/6 = 10/x:
- Step 1 (Set Up Proportion Equation): $\4 / 6 = \10 / x$.
- Step 2 (Cross Multiply Terms): $4 \times x = 6 \times 10 \implies 4x = \mathbf{60}$.
- Step 3 (Divide Both Sides by 4): $x = \60 / 4$.
- Step 4 (Compute Target Solution): $x = \mathbf{15}$.
- Step 5 (Verify Proportion Equality): $\4 / 6 = 0.6667$ and $\10 / 15 = 0.6667$, delivering **x = 15**!
Calculation Examples: Real-World Scenario Comparison
Compare proportion solver results across direct and inverse real-world problems:
| Proportion Problem | Proportion Type | Formula Applied | Solved Variable Result | Full Proportion Statement |
|---|---|---|---|---|
| 4/6 = 10/D | Direct | D = (6 × 10) / 4 | D = 15 | 4 : 6 = 10 : 15 |
| 3/5 = C/25 | Direct | C = (3 × 25) / 5 | C = 15 | 3 : 5 = 15 : 25 |
| 4 workers in 6 days = 8 workers in D days | Inverse | D = (4 × 6) / 8 | D = 3 days | 4 × 6 = 8 × 3 (24) |
| 60 mph for 2 hrs = 40 mph for D hrs | Inverse | D = (60 × 2) / 40 | D = 3 hours | 60 × 2 = 40 × 3 (120) |
Benefits of Using the Proportion Calculator
Utilizing this calculator provides key mathematical, academic, and real-world problem solving benefits:
- Handles Both Direct & Inverse Proportions: Easily toggle between direct scaling ($A/B = C/D$) and inverse rates ($A \cdot B = C \cdot D$).
- Solves Any Position (A, B, C, or D): Solves for any missing term anywhere in the proportion equation.
- Physics & Speed/Time Calculation: Calculate speed, distance, time, or gas law problems ($P_1 V_1 = P_2 V_2$).
- Construction & Recipe Scaling: Accurately scale mortar mixes or large catering recipes without calculation errors.
Frequently Asked Questions (FAQ)
What is a proportion in mathematics?
A proportion is an equation stating that two ratios or rates are equal (e.g. A/B = C/D).
What is the difference between direct and inverse proportion?
In direct proportion, as one variable increases, the other increases proportionally (A/B = C/D). In inverse proportion, as one increases, the other decreases proportionally (A·B = C·D).
How do you solve a proportion using cross-multiplication?
Multiply opposite terms diagonally (A × D = B × C), then divide by the known coefficient to isolate the unknown variable.
What is 4/6 = 10/x solved for x?
4x = 60 ⇒ x = 15.
What is 3/5 = x/25 solved for x?
5x = 75 ⇒ x = 15.
What is an example of inverse proportion in real life?
Worker-time problems: 4 workers take 6 days to complete a job, so 8 workers take 3 days (4 × 6 = 8 × 3 = 24).
What are extremes and means in a proportion?
In the proportion A:B = C:D, A and D are called the "extremes", while B and C are called the "means" (Product of extremes = Product of means).
How do you know if a word problem is direct or inverse?
If more of X requires more of Y (like buying more apples costs more money), it is direct. If more of X requires less of Y (like driving faster takes less time), it is inverse.
Can proportions contain negative numbers or decimals?
Yes, proportion formulas apply equally to positive integers, negative numbers, and decimals.
Why is cross-multiplication valid?
Cross-multiplication is mathematically valid because multiplying both sides of A/B = C/D by (B × D) eliminates the denominators, yielding A × D = B × C.