Quote:
Alice bought a certain number of 30 cent stamps, 35 cent stamps and 40 cent stamps. She spent a total of $4.20 in buying these stamps. Did she buy more than 5 stamps of any of the three values?
(1) The number of 35 cent stamps and 40 cent stamps that Alice bought are equal.
(2) The number of 30 cent stamps and 40 cent stamps that Alice bought are equal. The number of 35 cent stamps that she bought was not more than the number of 40 cent stamps that she bought.
Fun one!
Let's say that x is the number of 30 cent stamps, y is the number of 35 cent stamps, and z is the number of 40 cent stamps. So, in the question stem, we have an equation:
30x + 35y + 40z = 420 (it's 420 and not 4.20 because we're using cents rather than dollars!)
Divide everything by 10 to simplify the math:
3x + 3.5y + 4z = 42
The question is whether she bought more than 5 stamps of any of the three values. That is, is x, y, or z greater than 5?
Also note that we know x, y, and z are non-negative integers, since you can't buy a fractional or negative number of stamps.
Statement 1: The number of 35 cent stamps and 40 cent stamps that Alice bought are equal.
In other words, y = z. So, the equation simplifies as follows:
3x + 3.5y + 4z = 42
3x + 7.5y = 42
Okay, one of the values
could be bigger than 5, for instance, if x = 14 and y = 0. So, it's possible to get a "yes" answer. Can we also get a "no," where both of the values are smaller than 5?
Let's try the biggest values possible that are still smaller than 5. So, if y = 4, then we have this:
3x + 7.5(4) = 42
3x + 30 = 42
3x = 12
x = 4
Therefore, x = 4, y = 4, z = 4 is a valid solution that fits this statement, and gives us a "no" answer (none of the values are greater than 5.)
Therefore, this statement is
insufficient.
Statement 2 The number of 30 cent stamps and 40 cent stamps that Alice bought are equal. The number of 35 cent stamps that she bought was not more than the number of 40 cent stamps that she bought.
This says that x = z, and y<=z.
The second case we tested above fits the bill: x = y = z = 4. In this case, we get an answer of "no." Can we also get a "yes" to show that this is insufficient?
To do that, we should simplify the equation:
3x + 3.5y + 4z = 42
3x + 3.5y + 4x = 42
7x + 3.5y = 42
I multiplied by 2 to get rid of the decimal:
14x + 7y = 84
Then, divide by 7:
2x + y = 12
The constraint from this statement is that y is no bigger than z. Since x and z are equal, we know that y can't be any bigger than x. Is there a solution that fits, where one of the values is greater than 5?
x = 6 and y = 0 works (and that implies that z = 6 as well.)
Therefore, this statement is also
insufficient.
Statements 1 + 2 togetherInterestingly, we already found a case that works with both statements and gives a "yes" answer: x = y = z = 4.
Can we find a case that works with both statements and gives a "no" answer?
Take the info from both statements: x = y, x = z. That is, all three values must be equal.
Plug that into the equation:
3x + 3.5x + 4x = 42
10.5x = 42
21x = 84
x = 4
In other words, x = y = z = 4 is the only solution that still works. So,
both statements together are sufficient and the correct answer is C.