Ben's Mom gives him an envelope of cash to bring to his Dad's store. She tells him how much she put in the envelope.
On his way to his Dad's store, Ben sees a vinyl Abbey Road album at a yard sale for $0.50. Since Ben has no cash of his own, he dips into the envelope to buy the album.
When he gets to his Dad's store, Ben opens the envelope only to find twice as much money left in the envelope as his Mom said she had placed there! Ben mentally adds the $0.50 he spent at the yard sale to the amount left and realizes that his Mom muddled up the dollars and cents when she told him how much the envelope held. The number of dollars she named was actually the number of cents; the number of cents she named was actually the number of dollars she put in.
How much money did Ben's Mom actually put in the envelope?
Let's say Ben's mom originally said she put x dollars and y cents in the envelope.
According to the problem, Ben's mom mixed up the dollars and cents. So the actual amount of money she put in the envelope is y dollars and x cents.
When Ben opens the envelope, he finds twice as much money as was left in the envelope. If we express this mathematically, it can be written as: 2 * (y * 100 + x) = y * 100 + x - 50.
Simplifying this equation, we have: 200y + 2x = 100y + x - 50.
Bringing like terms to one side of the equation: 100y - x = -50.
Now, we can solve for y (the number of dollars) in terms of x (the number of cents):
100y = x - 50.
Dividing both sides by 100:
y = (x - 50)/100.
Since y represents the number of dollars, it must be an integer. Therefore, x must be greater than or equal to 50 for y to be an integer.
The problem stated that the amount left in the envelope was twice the original amount, so y * 100 + x = 2 * (y * 100 + x - 50).
Simplifying, we have: y * 100 + x = 200y + 2x - 100.
Bringing like terms to one side: 100y - x = -100.
We already had the equation 100y - x = -50. Setting these two equations equal to each other: -50 = -100.
This implies that x = 0.
Substituting this value of x into y = (x - 50)/100, we find y = (0 - 50)/100 = -50/100 = -0.5.
Since the amount of money cannot be negative, this solution does not make sense. Therefore, x cannot be 0.
Hence, there is no possible solution for this problem. It seems that there is a mistake in the given information or the problem itself.
Let's assume the number of dollars Ben's Mom named is D and the number of cents she named is C.
According to the problem, the number of dollars she named (D) is actually the number of cents she put in the envelope, and the number of cents she named (C) is actually the number of dollars she put in.
Now, given that Ben spent $0.50 on the album, we can equate the equation as follows:
(C - 50 cents) + (D * 100 cents) = 2 * (C + D * 100)
Let's simplify this equation step-by-step:
C - 50 + D * 100 = 2 * C + 2 * D * 100
Combining like terms:
D * 100 - 2 * D * 100 = -C + 2 * C - 50
Simplifying both sides:
-100 * D = C - 50
Finally, since we know that C is the number of dollars Ben's Mom put in the envelope, and D is the number of cents she put in, we need to find values for C and D that satisfy the above equation.
Let's try some values:
If C = 100 and D = 50 cents, then:
-100 * 50 = 100 - 50
-5000 = 50
This does not satisfy the equation.
If C = 500 and D = 250 cents, then:
-100 * 250 = 500 - 50
-25000 = 450
This also does not satisfy the equation.
If C = 1000 and D = 500 cents, then:
-100 * 500 = 1000 - 50
-50000 = 950
This also does not satisfy the equation.
Therefore, there must be an error in the problem statement, and we cannot determine the actual amount of money Ben's Mom put in the envelope with the given information.