Shortcut Trick
Fourth proportional of a, b, c = (b × c) / a = (5 × 3) / 7 = 15 / 7
Third proportional of a, b = b2 / a = 132 / 7 = 169 / 7
Required ratio = (15 / 7) : (169 / 7) = 15 : 169
∴ The correct answer is 15 : 169.
Alternate Method
Given:
First set of numbers for fourth proportional: 7, 5, and 3
Second set of numbers for third proportional: 7 and 13
Formula Used:
For fourth proportional x: a : b :: c : x ⇒ x = (b × c) / a
For third proportional y: a : b :: b : y ⇒ y = b2 / a
Fourth Proportional (x)
7 : 5 :: 3 : x
x = 15 / 7
Third Proportional (y)
7 : 13 :: 13 : y
y = 169 / 7
Final Ratio (x : y)
15/7 : 169/7 = 15 : 169
Calculations:
⇒ Let the fourth proportional of 7, 5, and 3 be x.
⇒ 7 : 5 :: 3 : x
⇒ 7 / 5 = 3 / x
⇒ 7 × x = 5 × 3
⇒ x = 15 / 7
⇒ Let the third proportional of 7 and 13 be y.
⇒ 7 : 13 :: 13 : y
⇒ 7 / 13 = 13 / y
⇒ 7 × y = 13 × 13
⇒ y = 169 / 7
⇒ The required ratio = x : y
⇒ x : y = (15 / 7) : (169 / 7)
⇒ x : y = 15 : 169
∴ The correct answer is 15 : 169.
Additional Information
Third Proportional
If two terms a and b are given, their third proportional c satisfies the relation a : b = b : c. Formula: c = b2 / a.
Fourth Proportional
If three terms a, b, and c are given, their fourth proportional d satisfies the relation a : b = c : d. Formula: d = (b × c) / a.
Mean Proportional
If a, b, and c are in continued proportion (a : b = b : c), then b is called the mean proportional between a and c. Formula: b = √(a × c).
Duplicate & Compound Ratio
The duplicate ratio of a : b is given by a2 : b2. The compound ratio of two different ratios a : b and c : d is calculated as ac : bd.