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The sum of two numbers is 13. . .

This topic has 5 expert replies and 1 member reply

The sum of two numbers is 13. . .

Post Sun Sep 24, 2017 7:06 am
The sum of two numbers is 13, and their product is 30. What is the sum of the squares of the two numbers?

(A) -229

(B) -109

(C) 139

(D) 109

(E) 229

The OA is D.

Clearly options A and B are not correct. Can I solve this PS question without finding the values of the two numbers?

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Post Wed Oct 11, 2017 7:04 pm
Hello EconomistGMATTutor, Matt, Rich and Brent. I really appreciate your answer. Thank you all of you.

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Post Tue Dec 12, 2017 5:25 pm
Vincen wrote:
The sum of two numbers is 13, and their product is 30. What is the sum of the squares of the two numbers?

(A) -229

(B) -109

(C) 139

(D) 109

(E) 229
We can let the numbers be a and b. Thus:

a + b = 13 and ab = 30

Since a = 13 - b, we can substitute 13 - b in the equation ab = 30 and we have:

(13 - b)b = 30

13b - b^2 = 30

b^2 - 13b + 30 = 0

(b - 10)(b - 3) = 0

b = 10 or b = 3

Notice that when b = 10, a = 3, and when b = 3, a = 10. Therefore, the two numbers that have a sum of 13 and a product of 39 are 3 and 10, and the sum of their squares is is 9 + 100 = 109.

Alternate Solution:

Let’s square each side of a + b = 13:

(a + b)^2 = 169

a^2 + 2ab + b^2 = 169

Since ab = 30, we can substitute 2ab = 60 in the last equality:

a^2 + b^2 + 60 = 169

a^2 + b^2 = 109

Answer: D

_________________
Jeffrey Miller Head of GMAT Instruction

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Post Wed Oct 11, 2017 7:04 pm
Hello EconomistGMATTutor, Matt, Rich and Brent. I really appreciate your answer. Thank you all of you.

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Post Tue Dec 12, 2017 5:25 pm
Vincen wrote:
The sum of two numbers is 13, and their product is 30. What is the sum of the squares of the two numbers?

(A) -229

(B) -109

(C) 139

(D) 109

(E) 229
We can let the numbers be a and b. Thus:

a + b = 13 and ab = 30

Since a = 13 - b, we can substitute 13 - b in the equation ab = 30 and we have:

(13 - b)b = 30

13b - b^2 = 30

b^2 - 13b + 30 = 0

(b - 10)(b - 3) = 0

b = 10 or b = 3

Notice that when b = 10, a = 3, and when b = 3, a = 10. Therefore, the two numbers that have a sum of 13 and a product of 39 are 3 and 10, and the sum of their squares is is 9 + 100 = 109.

Alternate Solution:

Let’s square each side of a + b = 13:

(a + b)^2 = 169

a^2 + 2ab + b^2 = 169

Since ab = 30, we can substitute 2ab = 60 in the last equality:

a^2 + b^2 + 60 = 169

a^2 + b^2 = 109

Answer: D

_________________
Jeffrey Miller Head of GMAT Instruction

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Post Tue Oct 10, 2017 2:24 pm
Vincen wrote:
The sum of two numbers is 13, and their product is 30. What is the sum of the squares of the two numbers?

(A) -229

(B) -109

(C) 139

(D) 109

(E) 229

The OA is D.

Clearly options A and B are not correct. Can I solve this PS question without finding the values of the two numbers?
Hi Vincen,
Lets have a look at your question.
It can easily be solved using a polynomial identity.

The question states,
"The sum of two numbers is 13, and their product is 30. "
Let the two numbers be 'x' and 'y', then
x + y = 13 --- (i)
xy = 30 ---(ii)

What is the sum of the squares of the two numbers?
x^2 + y^2 = ?

We will use the the identity
(x + y)^2 = x^2 + y^2 + 2xy
Plugin the values of (x + y) and xy from (i) and (ii) in the above identity.

(13)^2 = x^2 + y^2 + 2(30)
169 = x^2 + y^2 + 60
x^2 + y^2 = 169 - 60
x^2 + y^2 = 109
Sum of squares of the two numbers is 109.

Therefore, Option D is correct.

Hope this helps.
I am available if you'd like any follow up.

_________________
GMAT Prep From The Economist
We offer 70+ point score improvement money back guarantee.
Our average student improves 98 points.

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Post Tue Sep 26, 2017 5:19 pm
a + b = 13
ab = 30

a² + b² = (a + b)² - 2ab

a² + b² = 13² - 2*30

a² + b² = 109

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