Direct proportion and cross multiplication
Quick answerA direct proportion is the equality of two ratios: A/B = C/D. To find a missing value, use cross multiplication: A × D = B × C, then isolate the unknown — for example, if D is unknown, D = (B × C) / A. Direct proportions describe quantities that rise and fall together at the same rate.
Cross multiplication works because if two fractions are equal, multiplying both sides of the equation by both denominators clears the fractions and leaves a simple product equation. This is the same math behind scaling a recipe, converting a currency at a fixed rate, or figuring out how far you'll travel at a constant speed.
Worked example (illustrative): say 2 lb of apples cost $10 at a given store. How much would 5 lb cost, at the same price per pound? Set up 2/10 = 5/x, cross multiply to get 2x = 50, and divide to get x = $25. This is a hypothetical price used only to demonstrate the method — actual prices vary by store and season.
The same setup applies to the aspect-ratio problem of finding a missing screen dimension. For a 16:9 screen with a width of 1920 px, height = 1920 × (9/16) = 1080 px — the ratio width/height stays constant across every valid resolution in that aspect ratio.
Inverse proportion — the workers-and-days problem
Quick answerIn an inverse proportion, as one quantity increases, the other decreases at the same rate, so their product stays constant: A × B = C × D. For 6 workers finishing a job in 12 days, 6 × 12 = 8 × d gives d = 72 / 8 = 9 days for 8 workers. Cross multiplication does not apply here — the products are set equal directly.
The reason this is inverse rather than direct is that the total amount of work is fixed. If you assume every worker contributes the same output per day, then adding workers doesn't change the job size — it just splits the same total work across more people, so each person needs fewer days to finish their share. This is a simplifying assumption; in a real job, output per worker, coordination overhead, and other real-world factors can change the actual number of days.
Other classic inverse-proportion setups include taps filling a pool (more taps, less time) and speed vs. travel time at a fixed distance (faster speed, less time). Whenever "more of one thing means less of the other, at the same rate," check whether the product — not the ratio — is what stays constant.
Direct vs. inverse proportion — example problems (illustrative numbers)
| Problem | Type | Result |
|---|
| 2 lb of apples cost $10 — 5 lb? | Direct | $25 |
| 80 mph for 3 hours — distance? | Direct | 240 mi |
| 6 workers finish a job in 12 days — 8 workers? | Inverse | 9 days |
| 4 taps fill a pool in 3 hours — 6 taps? | Inverse | 2 hours |
Compound proportion — when more than one factor changes
Quick answerA compound proportion applies when a result depends on more than one quantity at once, some directly and some inversely. Multiply the known result by a new/old ratio for each directly proportional factor, and an old/new ratio for each inversely proportional factor. Example: 4 workers produce 120 units in 6 hours — how many units do 6 workers produce in 8 hours? 120 × (6/4) × (8/6) = 240 units.
Both factors here — the number of workers and the number of hours worked — are directly proportional to the output, so both ratios are new/old. If a factor were inversely related instead (say, fewer defects per unit as more inspection time is added), you'd flip that factor's ratio to old/new before multiplying. Most real-world staffing, output, and cost estimates that involve two or more changing variables are solved this way.
Simplifying a ratio to lowest terms
Quick answerTo simplify a ratio, divide every term by their greatest common divisor (GCD). The GCD of 12:18 is 6, so dividing each term by 6 gives 2:3. For a ratio with decimals, first multiply every term by a power of 10 to make the terms whole numbers, then find the GCD as usual.
Simplifying makes a ratio easier to read and compare — a screen resolution of 1920:1080 simplifies to 16:9, which is the aspect ratio most people actually recognize. Beyond the simplified form, it can help to see the ratio as a percent, a decimal, or a fraction, since different contexts call for different representations of the same relationship. Two ratios are equal exactly when their cross products match (A × D = B × C); if they aren't equal, the one with the larger decimal value is the larger ratio.
Frequently asked questions
How do you solve a proportion?
A proportion is the equality of two ratios: A/B = C/D. To find the missing value from three known terms, use cross multiplication: A × D = B × C. Then isolate the unknown — for example, if D is unknown, D = (B × C) / A. In an inverse proportion, the products are equal instead: A × B = C × D.
What is the difference between a direct and an inverse proportion?
In a direct proportion, two quantities rise and fall together at the same rate, so their ratio stays constant (A/B = C/D). In an inverse proportion, one quantity rises while the other falls at the same rate, so their product stays constant (A × B = C × D). Cross multiplication only applies to a direct proportion.
How many workers-and-days does it take to finish a job faster?
Workers and days are an inverse proportion: more workers means fewer days, assuming the same total amount of work and a constant rate per worker. If 6 workers finish a job in 12 days, then 6 × 12 = 8 × d gives d = 72 / 8 = 9 days for 8 workers. This is a simplified example that assumes equal productivity per worker.
How do you simplify a ratio to lowest terms?
Divide every term of the ratio by their greatest common divisor (GCD). For example, the GCD of 12:18 is 6, so dividing each term by 6 gives 2:3. For decimal ratios, first multiply every term by a power of 10 to make them whole numbers before finding the GCD.
What is a compound proportion, and when do you need one?
A compound proportion is used when a result depends on more than one quantity at once, some directly and some inversely. For example, if 4 workers produce 120 units in 6 hours, then 6 workers working 8 hours produce 120 × (6/4) × (8/6) = 240 units — multiplying the known result by a new/old ratio for each direct factor and an old/new ratio for each inverse factor.