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Joanna bought only $0.15 stamps and $0.29 stamps. How many $0.15 stamp

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jabhatta2

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27 Mar 2021, 15:41

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GMATGuruNY

Consider the following equation:
2x + 3y = 30.

If x and y are nonnegative integers, the following solutions are possible:
x=15, y=0
x=12, y=2
x=9, y=4
x=6, y=6
x=3, y=8
x=0, y=10

Notice the following:
The value of x changes in increments of 3 (the coefficient for y).
The value of y changes in increments of 2 (the coefficient for x).
This pattern will be exhibited by any fully reduced equation that has two variables constrained to nonnegative integers.

Hi GMATGuruNY - i really liked this explanation
Could you explain WHY the values of x increase by 3 (coefficient of y). I know 3 is the co-efficient of y but I am struggling to understand why that plays a role specifically.

GMATGuruNY

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28 Mar 2021, 09:36

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jabhatta2

GMATGuruNY

Given 2x+3y = 30, one integer option for 2x+3y is as follows:
2*15 + 3*0

If the value x decreases by 3 -- the coefficient for y -- we get:
2x = 2(15-3) = 2*15 - 2*3
If the value y increases by 2 -- the coefficient for x -- we get:
3y = 3(0+2) = 3*0 + 3*2

Resulting sum for 2x+3y:
(2*15 - 2*3) + (3*0 + 3*2)
2*15 + 3*0
Because the blue terms cancel out, the value of 2x+3y remains the same.

Bambi2021

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21 Jun 2021, 15:42

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For statement 1:

440 is a multiple of 5, just like 15, so 29 must become a multiple of 5.

29*5 = 145
29*10 = 290
29*15 = 435

Only 145 and 290 is possible.

440-145 = 295, which is not divisible by 15, so only 290 works.

Crytiocanalyst

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01 Aug 2021, 05:42

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Statement 1:
15x+29y = 440

15x + 29x = 440
44x = 440
x=10, y=10

solution for 15x+29y=440 is as follows:
x=10, y=10
other solutions for the same was not available
SUFFICIENT.

Statement 2:
x=y
Here, x and y can take any values
Therefore INSUFFICIENT.

Hence IMO A

jasminelin

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Magophon

Here is the method to never fail to answer correctly Diophantine-equations-related Data Sufficiency problems.

1. First, be sure that the 2 variables must be non-negative integers or positive integers and that each statement provides a linear equation relating the 2 variables. Furthermore, be sure that the 2 equations are not equivalent (2x+3y=20 and 6x+9y=60 are equivalent) and are reduced to the form: ax + by = c whith integral coefficients and constant term such that GCF (a,b)=1.

2. Find an initial Solution: "Take advantage" of the fact that statements never contradict each other and thus system of equations constructed with both statements have always at least one solution. So resolve the system of equations.

3. Unicity: Once you arrive to a solution, say (x0, y0), go back to the first statement alone, for example, and check the unicity of the solution using only that statement by applying the test below. In case the solution is unique, statement 2 is superfluous and statement 1 is sufficient. The answer is A or D. In case the solution is not unique the answer is B, C or E.

Apply the test on statement (2). And update your answer.

If there is more than one solution using each statement alone then the answer is C.

-------------------------------------------------------------------------------------------------------------------------------------------------
Now here is the rule that indicates whether or not a non-negative integer solution is unique to an equation:
Suppose the equation be: ax+by=c (reduced with a, b, c positive integers. i.e. GCF(a,b)=1)
If (x0-b)<0 AND (y0-a)<0 then there is no other non-negative integer solution than (x0, y0) and the corresponding statement is sufficent.
If (x0-b)>=0 OR (y0-a)>=0 then other non-negative integers solutions exist and the statement is not sufficient.

If the variables must be positive the test is:
If (x0 - b)<=0 AND (y0 - a)<=0 then there is no other positive integer solution than (x0, y0) and the corresponding statement is sufficent.
If (x0 - b)>0 OR (y0 - a)>0 then other positive integers solutions exist and the statement is not sufficient.

Note: The test is to subtract each coefficient from the solution found for the opposite variable.
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Let's apply this to a real GMAT problem:
A man buys some juice boxes. The boxes are from two different brands, A and B. How many boxes of brand A did the man buy if he bought $5.29 worth of boxes?
(1) The price of brand A box is $0.81 and the price of brand B box is $0.31
(2) The total amount of boxes is 9

Variables here must be positive integers -number of juice boxes- since it is suggested that some juice boxes are from brand A and the rest from brand B.
1. The equations provided are:
(1) 0.81A + 0.31B = 5.29.
(2) A + B = 9
Which are reduced to:
(1) 81A + 31B = 529
(2) A + B = 9,
which is a system of reduced, linear, non-equivalent equations.

2. Find an initial Solution:
(1) 81A + 31B = 529. GCF(31, 81)=1.
(2) A + B = 9

Mutliplying (2) by 31 and subtracting it from (1) we get:
50A=250 so A=5 and B=4.
An initial solution is (5, 4)

3. Unicity:
Unicity for statement (1):
81(5) + 31(4) = 529
Since
(5 - 31) <=0 AND (4 - 81)<=0 then there no other positive solution than (5, 4) so statement (1) is sufficient.

Unicity for statement (2):
It is obvious that statement (2) alone is not sufficient but the test is still applicable.
1(5) + 1(4) = 9
Since
(5 - 1) >0 OR (4 - 1) > 0 then there are other positive solutions than (5, 4) so statement (2) is not sufficient.
Answer A.

Hope this helps.

Would this still be the case if the equation were written 0.81x + 0.31y = 5.29
(x,y) remains (5,4)
but (5 - 0.31) AND (4-0.81) are both greater than 0
but there is no other solution

jaky_nguyen

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udaymathapati

Joanna bought only $0.15 stamps and $0.29 stamps. How many $0.15 stamps did she buy?

(1) She bought $4.40 worth of stamps.

(2) She bought an equal number of $0.15 stamps and $0.29 stamps.

Another solution to find statement 1 only has one valid pair of values:

Let C = number of $0.15 stamps purchased
Let E = number of $0.29 stamps purchased

Statement 1:
0.15C + 0.29E = 4.40 or 15C + 29E =440
-> C = (440-29E)/15
OR C = (450-30E+E-10)/15 = 30-2E +(E-10)/15;
C is integer, then E must be 10,25,... but E can not be 25 because 29*25 > 440
-> E must be only 10
-> Statement 1: Sufficient

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