Proportion Calculator

Take the direct case first: 2.5 kg costs 10, so 4 kg costs what? Price per kilogram is what stays fixed here, so this is directly proportional. The constant is 10 ÷ 2.5 = 4 per kilogram. Cross multiplying gives 2.5 × answer = 4 × 10, so the answer is 40 ÷ 2.5 = 16. Check both cross products: 2.5 × 16 = 40 and 4 × 10 = 40. They match, so the arithmetic held. Now the inverse case: four workers take six days, so three workers take what? Nothing about a rate is fixed here — what is fixed is the job. Four workers for six days is twenty-four worker-days of work, and that number does not change because you sent someone home. So 4 × 6 = 3 × answer, and the answer is 24 ÷ 3 = 8 days. Try cross multiplying that instead, as if it were direct. You would write 4 × answer = 3 × 6, giving 18 ÷ 4 = 4.5 days. Three workers finishing in four and a half days when four workers needed six. The arithmetic is flawless and the answer is nonsense, because it answers a question nobody asked — it describes a situation where fewer workers somehow means less time, which is what direct proportion says. The graph tells them apart at a glance. Plot the direct pairs and they lie on a straight line running through the origin: zero kilograms costs zero, and the slope is the price per kilogram. Plot the inverse pairs and they lie on a curve that bends towards both axes and touches neither — because with a fixed amount of work, no number of workers ever finishes in no time, and no length of time ever completes it with no workers. One more worth doing. 7 : 3 = 5 : ? directly gives 15 ÷ 7, which is 15/7, or about 2.143. It is not a whole number, and it should not be forced into one halfway through. Among direct proportions built from whole numbers between 1 and 20, only 29.5% have whole-number answers. Keep the fraction until the moment you write the answer down, then round once.