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CAT 2019 Slot 1 — QA Question 13

Number TheoryEasy

The product of two positive numbers is 616. If the ratio of the difference of their cubes to the cube of their difference is 157 : 3, then the sum of the two numbers is

Answer & solution

Correct answer: 50

  • A

    58

  • B

    85

  • 50

  • D

    95

Solution

Easy

Let the numbers be x,yx,y with xy=616xy=616. Expand the cube of the difference using (x−y)3=x3−y3−3xy(x−y)(x-y)^3=x^3-y^3-3xy(x-y), plug the given ratio in, and the common factor (x−y)(x-y) cancels. A small "add xyxy" trick then turns x2+y2+xyx^2+y^2+xy into the perfect square (x+y)2(x+y)^2.

1

Set up the given relations. Product and ratio of (difference of cubes) to (cube of difference).

xy=616⇒ x3−y3(x−y)3=1573(given ratio)\begin{aligned} &xy=616\\ &\Rightarrow\ \frac{x^3-y^3}{(x-y)^3}=\frac{157}{3}\quad\text{(given ratio)} \end{aligned}
2

Expand the denominator. Use the identity for (x−y)3(x-y)^3 and substitute.

(x−y)3=x3−y3−3xy(x−y)⇒ 3(x3−y3)=157[(x3−y3)−3xy(x−y)](cross-multiply)⇒ 154(x3−y3)=3⋅157⋅xy (x−y)(collect x3−y3)\begin{aligned} &(x-y)^3=x^3-y^3-3xy(x-y)\\ &\Rightarrow\ 3(x^3-y^3)=157\big[(x^3-y^3)-3xy(x-y)\big]\quad\text{(cross-multiply)}\\ &\Rightarrow\ 154(x^3-y^3)=3\cdot157\cdot xy\,(x-y)\quad\text{(collect }x^3-y^3\text{)} \end{aligned}
3

Use xy=616xy=616 and cancel (x−y)(x-y). Factor x3−y3=(x−y)(x2+y2+xy)x^3-y^3=(x-y)(x^2+y^2+xy).

154(x−y)(x2+y2+xy)=3⋅157⋅616 (x−y)(from step 1)⇒ 154(x2+y2+xy)=290136(x≠y, cancel (x−y))⇒ x2+y2+xy=1884\begin{aligned} &154(x-y)(x^2+y^2+xy)=3\cdot157\cdot616\,(x-y)\quad\text{(from step 1)}\\ &\Rightarrow\ 154(x^2+y^2+xy)=290136\quad\text{(}x\ne y\text{, cancel }(x-y)\text{)}\\ &\Rightarrow\ x^2+y^2+xy=1884 \end{aligned}
4

Complete the square. Add one more xyxy to both sides to build (x+y)2(x+y)^2.

x2+y2+2xy=1884+xy=1884+616(add xy=616)⇒ (x+y)2=2500⇒ x+y=50\begin{aligned} &x^2+y^2+2xy=1884+xy=1884+616\quad\text{(add }xy=616\text{)}\\ &\Rightarrow\ (x+y)^2=2500\\ &\Rightarrow\ x+y=50 \end{aligned}
x+y=50x+y=50

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CAT 2019 Slot 1 QA Q13 — Number Theory solution | TheCATExam