Data Sufficiency

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

Answer: E
Source: www.gmatprepnow.com
Difficulty level: 550

Brent@GMATPrepNow wrote: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

Answer: E
Source: www.gmatprepnow.com
Difficulty level: 550

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

Answer: E

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.

Nicely articulated Brent. Yes, this question may lead to the incorrect answer C, if someone takes the approach of two variables and assumes that only 'x' can be the age of Joe. Nice question.

-Jay
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