At The Bakery, Lew Spent A Total Of $6.00 For One Kind Of GMAT Data Sufficiency

Question: At the bakery, Lew spent a total of $6.00 for one kind of cupcake and one kind of doughnut. How many doughnuts did he buy?

(1) The price of 2 doughnuts was $0.10 less than the price of 3 cupcakes.
(2) The average (arithmetic mean) price of 1 doughnut and 1 cupcake was $0.35.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are not sufficient.

Solutions and Explanation
Approach Solution : 1

Let the cost of one donut be D
Let the cost of one cup cake be C
Consider the number of donuts bought = x
Consider the number of cupcakes bought = y
Therefore, xD + yC = $6.00
Statement - 1 : The price of 2 doughnuts was $0.10 less than the price of 3 cupcakes
This implies that, 2D = 3C - 0.10
The value of x or y cannot be determined
Therefore this statement is not sufficient.
Statement - 2 : The average (arithmetic mean) price of 1 doughnut and 1 cupcake was $0.35
This implies that, (D + C) / 2 = 0.35
This has the same problem as the previous statement.
Therefore this statement is not sufficient.
By combining both the statements,
While we can find solutions for D and C, we are unable to find solutions for how much of each has been purchased.

Correct Answer: (E)

Approach Solution : 2
Consider two linear equations of the following form: Ax + By = C and Px + Qy = R, where A, B, P, and Q are the coefficients.
Scenario 1: Infinite solutions if A/P = B/Q = C/R
Scenario 2: There is no solution if A/P = B/Q but not if C/R.
Scenario 3: A/P is not equal to B/Q and the solution is unique
In actuality, the following also results from the common form of a straight line: y = mx + c
For this issue:
Let the cost of a doughnut be $d and that of a cupcake be $c.
Let's say there are x cupcakes and y doughnuts,
which equals xc + yd = 6—-----(i)
Statement - 1 : The price of 2 doughnuts was $0.10 less than the price of 3 cupcakes
Therefore, 2d = 3c - 0.1—-----(ii)
Since there are too many variables, (i) and (ii) would not produce a unique solution in this case.
Therefore this is not sufficient.
Statement - 2 : The average (arithmetic mean) price of 1 doughnut and 1 cupcake was $0.35
Therefore, c = $0.50 and d = $0.20—---(iii)
The equations (ii), and (iii) can be solved to determine the price of each.
However, we are unable to determine x or y
Therefore this is not sufficient.
Let us combine both statements,
Using the c and d values from (i),
we get 5x + 2y = 60
There should be an infinite number of solutions given the two unknowns in this case.
However, we are aware that the numbers x and y are positive (additional constraint).
The following values should be checked because there may not be an infinite number of solutions:
First-step resolution is, x = 12 and y = 0
To obtain the following solutions, decrease x by 2 (the y-coefficient) and increase y by 5 (the x-coefficient):
x=10, y=5, x=8, y=10, x=6, y=15, x=4, y=20, x=2. y = 25 x = 0, y = 30 (none of the equations need to be solved, the process is just being outlined here).
Therefore, we lack a unique value and so this is insufficient.

Correct Answer: (E)

Approach Solution : 3
The target answer requires the number of doughnuts that Lew bought.
Let D be the NUMBER of donuts that were bought.
Let C be the NUMBER of cupcakes that were bought.
let X be (in CENTS) be the Per-donut price,
Let Y represent the cost per cupcake (in CENTS)
Aside: Since there are 4 different variables, the target question probably requires 4 equations.
Given: Lew spent $6.00 on a single variety of cupcake and a single variety of doughnut.
That is to say, Lew spent 600 Cents.
Therefore we could say, DX + CY = 600. Let this be an equation.
Statement - 1 : The price of 2 doughnuts was $0.10 less than the price of 3 cupcakes
This implies that, 2X = 3Y - 10.
This is not sufficient.
Statement - 2 : The average (arithmetic mean) price of 1 doughnut and 1 cupcake was $0.35
This implies that, 1X + 1Y = 70
This is not sufficient.
From the equations from the statements, let us say that there are three equations in total. This suggests that the combined statements are probably insufficient.
Let us combine both the statements. This system can be solved to give X = 40 and Y = 30.
We obtain the following when we can enter these values into our first equation, DX + CY = 600: D(40) + C(30) = 600
Change this to: 40D + 30C = 600
Both sides are divided by 10 to get, 4D + 3C = 60
This equation has a number of solutions.
Let us write two from them,
D = 3 and C = 16 in case a. In this instance, Lew purchased three donuts, which is the response to the main query.
D = 6 and C = 12 in case b. In this instance, Lew purchased 6 donuts, which is the response to the main query.
The combined statements are insufficient because we lack certainty to respond to the target question.

Correct Answer: (E)

“At the bakery, Lew spent a total of $6.00 for one kind of” - is a topic of the GMAT Quantitative reasoning section of GMAT. GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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At The Bakery, Lew Spent A Total Of $6.00 For One Kind Of GMAT Data Sufficiency

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