Given that (a^3 + 3ab^2)/(b^3 + 3a^2b) = 63/62, find the ratio a:b

Question

Given that  \frac{a^3 + 3ab^2}{b^3 + 3a^2b} = \frac{63}{62} , find the ratio a : b

(A)  \frac{5}{3}

(B)  \frac{3}{5}

(C)  \frac{2}{3}

(D)  \frac{3}{2}

(E)  \frac{6}{7}

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Arvind123 · Accepted Answer

(Num+Den)/(Num-Den) = 125/1

==> (a+b)^3/(a-b)^3=125/1

Solve for a+b=5, a-b=1

Answer is 3/2.

SaquibHGMATWhiz · Answer

Solution:

This question is based on a concept named  Componendo and Dividendo  which is taught in 10th grade 
 According to this concept if we have 2 ratios such that  \frac{a}{b}=\frac{c}{d} , then we can write  \frac{a+b}{a-b}=\frac{c+d}{c-d}

We have  \frac{a^3 + 3ab^2}{b^3 + 3a^2b} = \frac{63}{62}

⇒\frac{a^3 + 3ab^2+b^3 + 3a^2b}{a^3 + 3ab^2-b^3-3a^2b} = \frac{63+62}{63-62}

⇒\frac{(a+b)^3}{(a-b)^3}=\frac{125}{1}

⇒\frac{a+b}{a-b}=5

⇒a+b=5a-5b

⇒b+5b=5a-a

⇒6b=4a

⇒a:b=6:4=3:2

Hence the right answer is    Option D

Nielgmat · Answer

I did complete the cube in numerator and simplify the expression a little..
But I am struggling to find an easier method other than brute force..
Can someone help??

Posted from my mobile device

FarhinR · Answer

Nielgmat

I think the ratio 63:62 states that the value of a is greater than the value of b, considering the equations are same except for the terms a and b. So, it helps to eliminate the answer choices in which denominator > numerator. So, eliminate (B), (C) and (E).
Therefore, out of the answer choices (A) and (D), (D) has a difference of 1 between the numerator and denominator just like 63 and 62. Hence, the answer is (D).

Used this approach considering the time limitation, there could be other better ways.

Aryan19852 · Answer

Is my method correct?
a / b = 63/62 which can be written as 3(21)/2(21)
21 gets cancelled and ratio remains 3/2

My method is wrong sorry I made a mistake

Oppenheimer1945 · Answer

Let a/b=k:
applying componendo-dividendo:

(k^3+3k)/(1+3k^2)=63/62

(k+1)^3/(k-1)^3=125/1

k+1/k-1=5

k/1=6/4

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Given that (a^3 + 3ab^2)/(b^3 + 3a^2b) = 63/62, find the ratio a:b

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Given that (a^3 + 3ab^2)/(b^3 + 3a^2b) = 63/62, find the ratio a:b

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

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My Rewards