Mean, Third, and Fourth Proportionals

Mean, third, and fourth proportionals

A mean proportional between two numbers a and b is a number m such that \frac am = \frac mb. This means m^2 = ab, or m = \pm\sqrt{ab}. Since \frac{2}{\pm4} = \frac{\pm4}{8}, both +4 and -4 are mean proportionals between 2 and 8. Type both, like 4, -4 (or +-4).

A third proportional to a and b is the number t for which \dfrac ab = \dfrac bt. In \frac36 = \frac6{12}, 12 is a third proportional to 3 and 6. Solving, t = \dfrac{b^2}{a}.

A fourth proportional to a, b, and c is the number f for which \dfrac ab = \dfrac cf. Since \frac5{12} = \frac{10}{24}, 24 is a fourth proportional to 5, 12, and 10. Solving, f = \dfrac{bc}{a}.

Example. The mean proportionals between \frac{a}{2} and 8a: m^2 = \frac a2\cdot 8a = 4a^2, so m = \pm 2a.

Try one:

✎

Quick check

Find the mean proportionals between 3 and 27.

Exercises

0 of 18 solved

How to type math

Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.

Find the mean proportionals between the numbers. There are two, one positive and one negative; type both, like 6, -6 (or +-6).

The mean proportionals between 1 and 4.

The mean proportionals between 4 and 9.

The mean proportionals between 16 and 4.

The mean proportionals between 3 and 12.

The mean proportionals between (a - b)^2 and 4.

The mean proportionals between \tfrac12 and \tfrac18.

The mean proportionals between \tfrac13 and \tfrac1{27}.

The mean proportionals between \dfrac{4}{a^3} and \dfrac{9}{ax^2}.

Find a third proportional to the numbers in Exercises 1–7 which precede (the t with a : b = b : t).

Exercise 1: 1 and 4

Exercise 2: 4 and 9

Exercise 3: 16 and 4

Exercise 4: 3 and 12

Exercise 5: (a - b)^2 and 4

Exercise 6: \tfrac12 and \tfrac18

Exercise 7: \tfrac13 and \tfrac1{27}

Find a fourth proportional (f with a : b = c : f) to:

A fourth proportional to 1, 2, and 3.

A fourth proportional to 4, 5, and 6.

A fourth proportional to 7, 14, and 5.

A fourth proportional to 5, 12, and a.

A fourth proportional to 7, 21x, and 6x.

A fourth proportional to a, a^2, and a^3.

A fourth proportional to a^3, a^5, and a^4.

A fourth proportional to a + b, a - b, and a^2 - b^2.

A fourth proportional to \dfrac{a - b}{2}, \dfrac{1}{a + b}, and a^2 - b^2.