Connect a ratio, a fraction and a linear rule

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Connect a ratio, a fraction and a linear rule

A fixed ratio y: x = 3: 1 gives y/x = 3 for nonzero x, or y = 3x. This is a direct-proportion rule whose graph passes through the origin. The value of y triples the value of x.

A part-to-whole fraction also creates a linear relationship. If blue is 3/5 of a varying total T, then B = (3/5)T. The coefficient belongs to the comparison with the total, not to the 3: 2 comparison between blue and gold.

Not every straight line represents a fixed ratio. The rule y = 3x + 2 has the same gradient as y = 3x but a nonzero intercept. Its ratio y/x varies with x, so the fixed-ratio interpretation fails.

Worked example: Blue is 3/5 of a total of 50. Find B.

Use B = (3/5)T.

Substitute T = 50.

Answer: 30.

Check your understanding: Does y = 3x + 2 describe y directly proportional to x?

Answer: No.

Why: The additional constant prevents a constant y/x ratio.