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The sum of two numbers is 13. . .

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by Vincen » 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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by Brent@GMATPrepNow » Sun Sep 24, 2017 7:15 am
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
I suggest that you start looking for a number pair that ADD to 13 and have a PRODUCT of 30
It doesn't take long to discover that 3 and 10 work

What is the sum of the squares of the two numbers?
3² + 10² = 9 + 100 = 109

Answer: D

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by [email protected] » Mon Sep 25, 2017 3:46 pm
Hi Vincen,

Certain Quant questions on Test Day can be solved rather easily with a bit of 'brute force' math - simply list out the possibilities until you find the one that 'fits' the given information.

Here, we're told that the SUM of two numbers is 13 and the PRODUCT of those two numbers is 30. By definition, BOTH of those numbers must be positive. Since the sum, product and all 5 answer choices are all INTEGERS, it's highly likely that the two numbers will be integers as well.

2 Positive integers that sum to 13:
1 and 12; product = 12
2 and 11; product = 22
3 and 10; product = 30

The question asks us for the sum of the squares of the two values....
3^2 + 10^2 = 109

Final Answer: D

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by Matt@VeritasPrep » 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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by EconomistGMATTutor » 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.
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by Vincen » 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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by Jeff@TargetTestPrep » 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

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