The sum of two numbers is 1 and their product is -1. What is

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Wed May 08, 2019 1:39 am

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[GMAT math practice question]

The sum of two numbers is 1 and their product is -1. What is the sum of their cubes?

A. 1
B. 2
C. 3
D. 4
E. 5

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Source: Beat The GMAT — Problem Solving | Wed May 08, 2019 1:39 am

regor60: Master | Next Rank: 500 Posts; Posts: 416; Joined: Thu Oct 15, 2009 11:52 am; Thanked: 27 times

by regor60 » Wed May 08, 2019 7:31 am

Call the numbers X and Y.

So X+Y=1 and XY=-1 given the problem statement.

Let's square X+Y = X^2+2XY+Y^2 = 1.

Since XY=-1, we can substitute: X^2+Y^2-2 = 1. So,

X^2+Y^2 = 3.

Multiplying X^2+Y^2 by X+Y = (X+Y)(X^2+Y^2) = X^3 + Y^3 +XY^2 + YX^2 = (3)(1)

Factor an XY from the last two terms: X^3+Y^3 + XY(Y+X) = 3

Substitute XY=-1 and X+Y=1 into the above yields

X^3+Y^3 -(1)(1) = 3

therefore X^3+Y^3 = 4,D

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Wed May 08, 2019 8:39 am

Max@Math Revolution wrote:[GMAT math practice question]

The sum of two numbers is 1 and their product is -1. What is the sum of their cubes?

A. 1
B. 2
C. 3
D. 4
E. 5

x³ + y³ = (x+y)(x²+y²-xy)

Since x+y=1, we get:
(x+y)² = 1²
x² + y² + 2xy = 1

Substituting xy=-1 into x² + y² + 2xy = 1, we get:
x² + y² + 2(-1) = 1
x² + y² = 3

Substituting x+y=1, x²+y²=3 and xy=-1 into x³ + y³ = (x+y)(x²+y²-xy), we get:
x³ + y³ = (1)(3-(-1)) = (1)(4) = 4

The correct answer is D.

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Fri May 10, 2019 12:03 am

=>

Let the numbers be x and y. Then x + y = 1 and xy = -1.
Since (x+y)^2 = x^2 + 2xy + y^2 = x^2 + y^2 – 2, we have x^2 + y^2 = 3,
and x^3 + y^3 = (x+y)(x^2-xy+y^2) = 1*(x^2+ 1 + y^2) = x^2+y^2+1 = 4.

Therefore, the answer is D.
Answer: D

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8246; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Fri May 10, 2019 4:35 pm

Max@Math Revolution wrote:[GMAT math practice question]

The sum of two numbers is 1 and their product is -1. What is the sum of their cubes?

A. 1
B. 2
C. 3
D. 4
E. 5

We can let a and b be the two numbers. So we have a + b = 1 and ab = -1 and we need to determine the value of a^3 + b^3.

Notice that (a + b)^3 = a^3 + 3a^2*b + 3a*b^2 + b^3. So we have:

1^3 = a^3 + b^3 + 3a^2*b + 3a*b^2

1 = a^3 + b^3 + 3ab(a + b)

1 = a^3 + b^3 + 3(-1)(1)

1 = a^3 + b^3 - 3

4 = a^3 + b^3

Alternate Solution:

Letting a and b denote the numbers, we can use the identity a^3 + b^3 = (a + b)(a^2 - ab + b^2). We already know a + b = 1 and ab = -1, we need to find a^2 + b^2.

Notice that (a + b)^2 = a^2 + 2ab + b^2. Since a + b = 1 and ab = -1, we have

1^2 = a^2 + 2(-1) + b^2

a^2 + b^2 = 1 + 2 = 3.

Now, a^3 + b^3 = (a + b)(a^2 - ab + b^2) = (1)(a^2 + b^2 - ab) = 3 -(-1) = 3 + 1 = 4.

Answer: D

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