Shortcut Trick
The fourth proportional to A, B, and C is calculated as (B × C) ÷ A.
Here, A = a3 + a2b + ab2 = a(a2 + ab + b2), B = a + b, and C = a2 + ab + b2.
Fourth Proportional = [(a + b) × (a2 + ab + b2)] ÷ [a(a2 + ab + b2)] = (a + b) ÷ a = 1 + (b/a).
∴ The correct answer is 1+(b)/(a).
Alternate Method
Given:
First term (A) = a3 + a2b + ab2
Second term (B) = a + b
Third term (C) = a2 + ab + b2
Let the fourth proportional be x.
Formula Used:
For four quantities A, B, C, and x to be in proportion (A : B :: C : x):
Product of extremes = Product of means ⇒ A × x = B × C
Calculations:
⇒ x = (B × C) ÷ A
⇒ x = [(a + b) × (a2 + ab + b2)] ÷ (a3 + a2b + ab2)
⇒ Factorizing the denominator: a3 + a2b + ab2 = a(a2 + ab + b2)
⇒ x = [(a + b) × (a2 + ab + b2)] ÷ [a × (a2 + ab + b2)]
⇒ Canceling the common term (a2 + ab + b2):
⇒ x = (a + b) ÷ a
⇒ x = (a/a) + (b/a)
⇒ x = 1 + (b/a)
∴ The correct answer is 1+(b)/(a).
Additional Information
Fourth Proportional
If four quantities a, b, c, and d are in proportion (a:b = c:d), then d is the fourth proportional to a, b, and c. It is calculated as d = (bc)/a.
Third Proportional
If three quantities a, b, and c are in continued proportion (a:b = b:c), then c is the third proportional to a and b. Formula: c = b2/a.
Mean Proportional
The mean proportional between two numbers a and b is the square root of their product, given by √(ab).
Basic Proportion Property
In any proportion, the Product of Extremes (first and fourth terms) is always equal to the Product of Means (second and third terms).