Ratio, Proportion & Partnership • Topic 2 of 4

Proportion & Variation

In a proportion a:b = c:d, the product of the means equals the product of the extremes (b x c = a x d). Direct variation means y = kx (both rise together); inverse variation means xy = k (one rises as the other falls). The mean proportional between a and b is root(ab); the third proportional to a and b is b^2/a.

✅ Solved examples

1. Find x if 6:9 = x:15.
6 x 15 = 9 x x -> 90 = 9x -> x = 10.
2. Mean proportional between 4 and 16?
root(4 x 16) = root 64 = 8.
3. Third proportional to 8 and 12?
12^2 / 8 = 144/8 = 18.
4. If 12 men finish a job in 8 days, how many days for 16 men (inverse)?
Men x days constant: 12 x 8 = 96 = 16 x d -> d = 6 days.

✏️ Practice — try these, take hints as needed

1. Find x: 4:7 = 12:x.
4x = 7 x 12.
4x = 84.
—
21
2. Mean proportional of 9 and 25?
root(9 x 25).
root 225.
—
15
3. Third proportional to 6 and 18?
18^2/6.
324/6.
—
54
4. 15 workers, 12 days. Days for 20 workers?
15 x 12 = 180.
180/20.
—
9
5. y varies directly as x; y = 12 when x = 4. y at x = 7?
k = 3.
y = 3x.
3 x 7.
21

📝 Topic test — 8 questions

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