Proportion and ratio calculator

Direct and inverse proportion

In a direct proportion, when one quantity doubles the other doubles too (price and quantity). In an inverse proportion, when one doubles the other is cut in half (workers and time to finish a job).

\[ \text{direct} \ :\ x = \dfrac{B \times C}{A} \]

\[ \text{inverse} \ :\ x = \dfrac{A \times B}{C} \]

Cross multiplication

Two ratios are equal when their cross products are equal: \( \dfrac{A}{B} = \dfrac{C}{x} \) means \( A \times x = B \times C \). Divide both sides by A to isolate x. This is the same idea as a ratio table, where you move from one column to the other by multiplying by B ÷ A.

When it is not obvious that the situation is proportional, check first: the quotient B ÷ A should be the same for every pair of values.

Proportion calculator: a step-by-step table method

To set up a proportion, use a two-row table. First, write the two quantities in columns. Then place the known value under its matching value. Next, mark the unknown x. Finally, solve with cross multiplication.

For example, 3 pounds cost $4.50. What do 10 pounds cost? The math is 4.50 × 10 ÷ 3, or $15. So you also find the unit price of $1.50 per pound. Likewise, 2 cups of flour for 4 servings give 3 cups for 6 servings: 2 × 6 ÷ 4.

Proportion calculator: inverse proportion examples

Inverse proportion confuses many people. Yet the idea is simple: more people means less time. If 8 workers finish a job in 6 days, 12 workers finish it in 8 × 6 ÷ 12, or 4 days. In the same way, a car at 45 mph takes 2 hours, and at 60 mph it takes 45 × 2 ÷ 60, or 1.5 hours.

However, always check the direction. In fact, if one quantity rises, the other must fall. Therefore, a larger result than expected should raise a flag.

Proportion calculator and percentages

A proportion also solves percent problems. To find 25% of 80, set 100% equal to 80 and 25% equal to x. The math 80 × 25 ÷ 100 gives 20. To find a percent, flip the question: 18 out of 60 is 18 × 100 ÷ 60, or 30%.

For speed, use the percentage calculator. For a sale, open the discount calculator. Finally, to change units, try the unit converter or the recipe converter.

Proportion calculator: common mistakes

Three mistakes come up often. First, people mix units, such as ounces with pounds. Second, they apply a direct rule to an inverse case. Third, they forget that some situations are not proportional, like a child's height by age.

To practice, invent simple cases. For example, 12 eggs for 3 cakes give 20 eggs for 5 cakes. That way, you quickly check whether your instinct is right.

Try one new problem each week, and the method soon becomes automatic. Also, write the units next to each number, since that habit catches most slips before they cost you a point or a dollar.

Proportion calculator: everyday examples

This method helps around the house. For example, a recipe for 4 people calls for 2 cups of rice. For 10 people, compute 2 × 10 ÷ 4, or 5 cups. Likewise, if your car uses 3 gallons for 90 miles, it uses 7 gallons for 210 miles. Then a full 12 gallon tank gives you 12 × 90 ÷ 3, or 360 miles.

So you settle quantities, distances and prices fast. However, if the price changes in steps, the method no longer applies. Also, always double-check the units before you trust the answer.