Find the fourth proportional to 21, 18 and 7.

Concept: For three quantities a, b and c that form a proportion a : b :: c : d, the product of the extremes equals the product of the means, i.e. a × d = b × c. So the fourth proportional d is given by d = (b × c) / a.

Application: Substitute the given values step by step.

1. Take a = 21, b = 18 and c = 7, since the proportion is 21 : 18 :: 7 : d.

2. Apply the extremes-means rule: 21 × d = 18 × 7.

3. Compute the product of the means: 18 × 7 = 126.

4. Solve for d: d = 126 / 21 = 6.

Cross-check: Simplify the first ratio 21 : 18 to its lowest terms by dividing both terms by 3, giving 7 : 6. The second ratio must equal this same simplified ratio, so d = 6 is confirmed independently.