Six numbers a, b, c, d, e, f are such that ab = 1, bc = ½, cd = 6, de = 2, and…

Concept

When adjacent products of variables are known, choose one variable as a non-zero parameter and express every other variable in terms of it. Products made from alternate variables then become constants because the parameter cancels; finally, multiply every ratio term by a common factor to clear fractions.

Application

1. Let a = x, where x ≠ 0. From ab = 1, b = 1/x.

2. From bc = ½, c = (½)/b = x/2.

3. From cd = 6, d = 6/c = 12/x; therefore ad = x × (12/x) = 12.

4. From de = 2, e = 2/d = x/6; therefore be = (1/x) × (x/6) = 1/6.

5. From ef = ½, f = (½)/e = 3/x; therefore cf = (x/2) × (3/x) = 3/2.

Cross-check

1. Take x = 6. Then (a, b, c, d, e, f) = (6, 1/6, 3, 2, 1, 1/2), which satisfies all five given products.

2. This gives ad:be:cf = 12:1/6:3/2; multiplying every term by 6 gives 72:1:9.

Therefore, ad:be:cf = 72:1:9.