A proportional relationship is a relationship between two quantities where one is always a constant multiple of the other: y = kx. The graph is a straight line through the origin, the ratio y/x is always the same number k, and doubling x doubles y. Use the calculator to find the constant of proportionality and to solve any proportional relationship for x or y.
Proportional Relationship Calculator
y = kx — find k, then find the missing value.
What Is a Proportional Relationship?
Two quantities have a proportional relationship when their ratio is always the same. If y is proportional to x, then y = kx for some constant k, called the constant of proportionality. For example, if 3 notebooks cost $12, the constant is $4 per notebook, and the cost of any number of notebooks is y = 4x.
The Equation y = kx and the Constant k
In y = kx, k = y ÷ x. Find k from any known pair, then use it for every other pair. If x = 3 gives y = 12, then k = 12 ÷ 3 = 4, and for x = 7, y = 4 × 7 = 28. The constant k is also the unit rate and the slope of the graph.
How to Tell If a Relationship Is Proportional
Check three things: (1) the ratio y/x is the same for every pair, (2) the graph is a straight line, and (3) the line passes through the origin (0, 0). If any of these fails, the relationship is not proportional. A taxi fare with a fixed starting charge is not proportional, because at x = 0 the fare is not $0. A table where y/x changes from pair to pair is not proportional either.
Proportional Relationships in Tables and Graphs
In a table, divide each y by its x: if every answer is the same k, the table shows a proportional relationship. On a graph, plot the pairs: a proportional relationship makes a straight line through the origin, and the steeper the line, the larger k is. Real examples include distance at a steady speed (distance = speed × time), cost at a fixed unit price, and converting currencies at a fixed rate.
Worked Examples
Example 1: A car travels 150 miles in 3 hours at a steady speed. How far in 7 hours? k = 150 ÷ 3 = 50 mph, so y = 50 × 7 = 350 miles. Example 2: 5 kg of apples cost $15. What is the cost of 8 kg? k = 15 ÷ 5 = $3 per kg, so y = 3 × 8 = $24. Example 3: Is the table (2, 6), (4, 12), (5, 14) proportional? 6 ÷ 2 = 3, 12 ÷ 4 = 3, but 14 ÷ 5 = 2.8 — the ratios are not all equal, so it is not proportional.
Proportional Relationship FAQs
What is the constant of proportionality?
It is the number k in y = kx, found by k = y ÷ x. It is the unit rate — for example, the price per item or the speed in miles per hour.
Does a proportional graph always go through the origin?
Yes. Because y = kx gives y = 0 when x = 0, the graph of a proportional relationship is a straight line through (0, 0).
How is a proportional relationship different from a proportion?
A proportion is one equation, a/b = c/d. A proportional relationship is the whole pattern: every pair in it satisfies the same proportion, y = kx.
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