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A store sells erasers for 0.23$ per piece and pencil for

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A store sells erasers for 0.23$ per piece and pencil for

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Source: Veritas Prep

A store sells erasers for 0.23$ per piece and pencil for 0.11$ per piece. How many eraser and pencils did Jessica buy?

1) She bought 5 erasers.
2) She spent a total of 1.70$.

The OA is B

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Consider the following equation:
5x + 7y = 70.

If x and y are nonnegative integers, the following solutions are possible:
x=0, y=10
x=7, y=5
x=14, y=0.

Notice the following:
The value of x changes in increments of 7 (the coefficient for y).
The value of y changes in increments of 5 (the coefficient for x).
This pattern will be exhibited by any fully reduced equation that has two variables constrained to nonnegative integers.

BTGmoderatorLU wrote:
Source: Veritas Prep

A store sells erasers for 0.23$ per piece and pencil for 0.11$ per piece. How many eraser and pencils did Jessica buy?

1) She bought 5 erasers.
2) She spent a total of 1.70$.
Statement 1:
Since the number of pencils can be any nonnegative value, INSUFFICIENT.

Statement 2:
23E + 11P = 170.
This equation is constrained to nonnegative integers.

Test a case that also satisfies Statement 1.
If E = 5, we get:
23*5 + 11P = 170
115 + 11P = 170
11P = 55
P = 5.

Thus, one solution for 23E + 11P = 170 is E=5 and P=5.
Since the value of E may change only in increments of 11 -- the coefficient for P -- we get the following alternate options for E:
16, 27, 38...
All of these alternate options for E will yield a sum greater than 170 and thus are not viable.
Thus, the only viable solution for 23E + 11P = 170 is E=5 and P=5.
SUFFICIENT.

The correct answer is B.

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BTGmoderatorLU wrote:
Source: Veritas Prep

A store sells erasers for 0.23$ per piece and pencil for 0.11$ per piece. How many eraser and pencils did Jessica buy?

1) She bought 5 erasers.
2) She spent a total of 1.70$.

The OA is B
Say Jessica bought x numbers of erasers and y x numbers of pencils.

Thus, she spent a total of 0.23x + 0.11y to buy (x + y) items.

We have to get the value of (x + y).

Let's take each statement one by one.

1) She bought 5 erasers.

No information about the numbers of pencils. Insufficient.

2) She spent a total of 1.70$.

=> 0.23x + 0.11y = 1.70

23x + 11y = 170

x = (170 - 11y)/23

x = (161 + 9 - 11y)/23

x = 161/23 + (9 - 11y)/23

x = 7 + (9 - 11y)/23

Since x and y are positive integers, 9 - 11y must be a multiple of 23. With some hit and trail, you'll find one eligible value for y = 5.

At y = 5, we have x = 7 + (9 - 11y)/23 => x = 7 + (9 - 11*5)/23 => x = 7 + (9 - 55)/23 => x = 7 + (-46/23) => x = 7 - 2 = 5.

So, one solution is x = y = 5, thus, x + y = 10. There is no need to try for higher qualified values of y since at higher values x would turn out to be negative, unqualified value.

So, the unique value of x + y = 10. Sufficient.

The correct answer: B

Hope this helps!

-Jay
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This question illustrates a common trap on the GMAT.

As Jay has shown, with statement 2, we're able to write the equation 23x + 11y = 170 , and in high school we learned that, if we're given 1 equation with 2 variables, we cannot find the value of either variable.
However, if we restrict the variables to positive integers within a certain range of values, then there are times when we can, indeed, find the value of a variable if we're given 1 equation with 2 variables.

Here's a very similar Official GMAT question to practice with: https://www.beatthegmat.com/why-does-the-variable-rule-not-apply-here-t111112.html

Cheers,
Brent

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