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 assume the amount of money Ben's mom said she put in the envelope was x dollars and y cents.
According to the given information, we know that Ben spent $0.50 from the envelope, which means that the amount left in the envelope is (x*100 + y) - 0.50.
We also know that the amount left in the envelope is twice the amount his mom said she put in, so we can set up the equation: (x*100 + y) - 0.50 = 2(x + y*100).
Expanding and rearranging the equation, we get: x*100 + y - 0.50 = 2x + 200y.
Subtracting x from both sides gives us: 100y - 0.50 = x + 200y.
Now let's simplify the equation further: 100y - x = 0.50 + 200y.
Subtracting 200y from both sides, we have: -100y - x = 0.50.
Since we know that x and y are both positive integers, the term on the left side of the equation (-100y - x) cannot be greater than 0.50. In fact, the only solution that satisfies this condition is y = 0 and x = 1.
Therefore, Ben's mom actually put $1 in the envelope.
Let's start by assigning variables to the given information to make it easier to work with.
Let:
- x be the number of dollars Ben's mom named.
- y be the number of cents Ben's mom named.
Based on the given information, we know that Ben spent $0.50, which is equal to 50 cents.
We are also given the information that when Ben opens the envelope, he finds twice as much money left in the envelope as his mom said she had placed there.
So, the amount left in the envelope is 2 * y dollars and 2 * x cents.
According to Ben, the amount left in the envelope (in dollars and cents) plus the amount he spent at the yard sale ($0.50) equals the amount his mom named (which is x dollars and y cents).
Based on this information, we can create the following equation:
(2 * y) dollars + (2 * x) cents + $0.50 = x dollars + y cents
Now let's convert all the values from cents to dollars:
(2 * y) dollars + (2 * x / 100) dollars + $0.50 = x dollars + y / 100 dollars
Combining like terms:
(2 * y + 2 * x / 100) dollars + $0.50 = (x + y / 100) dollars
As we know that the number of dollars Ben's mom named was actually the number of cents, and the number of cents she named was actually the number of dollars, we can rewrite the equation as:
(2 * y + 2 * x / 100) dollars + $0.50 = (y + x / 100) dollars
Since the equation holds true, the coefficient of dollars on both sides of the equation should be the same. Let's equate them:
2 * y + 2 * x / 100 = y + x / 100
Simplifying the equation further:
2 * y + 2 * x / 100 = y + x / 100
2 * y = y + x / 100 - 2 * x / 100
2 * y = y + x / 100 - 2 * x / 100
2 * y = y - x / 100
2 * y - y = - x / 100
y = - x / 100
Since the value of y cannot be negative, we can disregard this solution.
Therefore, we cannot determine the exact amount of money Ben's mom put in the envelope with the given information.