What are the fourth proportional, third proportional and mean proportional, with an example of each?
For a : b = c : d, d is the fourth proportional, d = bc/a (4, 9, 12 gives 27). For a : b = b : c, c is the third proportional, c = b^2/a (4, 6 gives 9). The mean proportional between a and b is the square root of ab (9 and 16 give 12).
Explain it simply
A proportion says two ratios are equal, like the same recipe scaled up. If 4 cups of rice need 9 cups of water, how much water for 12 cups of rice? That missing number is the fourth proportional. When the middle term repeats (a : b = b : c), the numbers form a geometric progression, and b is the "mean" sitting between a and c.
Example
1. Fourth proportional to 4, 9, 12. 4 : 9 = 12 : x, so 4x = 9 x 12 = 108, x = 27.
2. Third proportional to 4 and 6. 4 : 6 = 6 : x, so 4x = 36, x = 9.
3. Mean proportional between 9 and 16. 9 : x = x : 16, so x^2 = 144, x = 12.
Check by ratios: 4/9 = 12/27 (both 0.444...), 4/6 = 6/9 (both 0.666...), 9/12 = 12/16 (both 0.75).
console.log((9 * 12) / 4); // Output: 27
console.log((6 * 6) / 4); // Output: 9
console.log(Math.sqrt(9 * 16)); // Output: 12The core rule behind all three: in a : b = c : d, the product of the extremes equals the product of the means, ad = bc.
What to say in the interview
"All three come from product of extremes equals product of means. Fourth proportional is bc over a, third proportional is b squared over a, and the mean proportional is the square root of ab."
Common mistakes
- Taking the mean proportional as the average (12.5) instead of the square root of the product (12).
- Writing the terms in the wrong order; order matters in a proportion.