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# What is Joe's age (in years)?

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CEO
Joined: 12 Sep 2015
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What is Joe's age (in years)?  [#permalink]

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10 Jun 2017, 13:58
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4
00:00

Difficulty:

75% (hard)

Question Stats:

49% (01:49) correct 51% (01:17) wrong based on 120 sessions

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What is Joe's age (in years)?

(1) The sum of Joe's age (in year's) and Jan's age (in years) is 39
(2) The product of Joe's age (in year's) and Jan's age (in years) is 380

*kudos for all correct solutions

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Re: What is Joe's age (in years)?  [#permalink]

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10 Jun 2017, 14:31
1
Let Joe's age be x and Jan's age be y

(1) The sum of Joe's age (in year's) and Jan's age (in years) is 39
x+y = 39(we can have many ways in which we can fill x and y)
eg 19+20,15+24
Hence, we cannot arrive at an unique age of Joe. Insufficient

(2) The product of Joe's age (in year's) and Jan's age (in years) is 380
x*y = 380(again we have several ways of giving a value of x and y)
eg 19*20,38*10
Hence, we cannot arrive at an unique age of Joe. Insufficient

On combining the two equations,
We prime factorize $$380 = 19*2^2*5$$
Now we have an option 19,20 which will give a sum of 39 and a product of 380
But we cannot be sure which of the ages is Joe's. It could be either of the ages.
Hence since we cannot arrive at an unique solution for the age of Joe, insufficient(Option E)
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What is Joe's age (in years)?  [#permalink]

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11 Jun 2017, 02:56
1
What is Joe's age (in years)?

(1) The sum of Joe's age (in year's) and Jan's age (in years) is 39

Clearly Insufficient

(2) The product of Joe's age (in year's) and Jan's age (in years) is 380

Clearly Insufficient

Combining 1 & 2:

Without rushing to solve, I know that I can setup equation and solve but there is one problem remains: who is older than the other? There is no statement, for example, that Joe is 3 years more than Jan. Hence, we can't know for sure.

Clearly Insufficient

CEO
Joined: 12 Sep 2015
Posts: 3847
What is Joe's age (in years)?  [#permalink]

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11 Jun 2017, 06:13
Top Contributor
GMATPrepNow wrote:
What is Joe's age (in years)?

(1) The sum of Joe's age (in years) and Jan's age (in years) is 39
(2) The product of Joe's age (in years) and Jan's age (in years) is 380

ASIDE: I created this question to illustrate a common myth that suggests that 2 equations with 2 variables are always solvable.

Target question: What is Joe's age?

Statement 1: The sum of Joe's age (in years) and Jan's age (in years) is 39
There are several scenarios that satisfy statement 1. Here are two:
Case a: Joe is 38 and Jan is 1. In this case, Joe is 38
Case b: Joe is 37 and Jan is 2. In this case, Joe is 37
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The product of Joe's age (in years) and Jan's age (in years) is 380
There are several scenarios that satisfy statement 2. Here are two:
Case a: Joe is 38 and Jan is 10. In this case, Joe is 38
Case b: Joe is 380 and Jan is 1. In this case, Joe is 380
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Let x = Joe's age
Let y = Jan's age

Statement 1 tells us that x + y = 39, which we can rewrite as y = 39 - x
Statement 2 tells us that xy = 380

Replace y in second equation with 39 - x to get: x(39 - x) = 380
Expand: 39x - x² = 380
Rearrange: x² - 39x + 380 = 0
Factor: (x - 20)(x - 19) = 0
So, x = 20 or x = 19
So, Joe is EITHER 20 years old OR 19 years old
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

So, what happened here?
We had two equations with 2 variables (y = 39 - x and xy = 380) yet we were unable to determine the value of x.
The reason is that one of our equations (xy = 380) is a QUADRATIC equation.
The rule that says "we can solve a system of 2 equations with 2 variables" applies only to situations in which the 2 equations are both LINEAR equations. That is, the variables are not raised to any powers greater than 1. Now one might say "But wait, x and y are not raised to powers greater than 1 in the equation xy = 380."
This is true, however, we have a product of 2 variables in xy, so we can think of it as variable², which makes is a QUADRATIC.

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Re: What is Joe's age (in years)?  [#permalink]

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10 Aug 2018, 04:47
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Re: What is Joe's age (in years)?   [#permalink] 10 Aug 2018, 04:47
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