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.
a. How much money did Ben's Mom actually put in the envelope?
Let's say Ben's Mom said she put x dollars in the envelope and y cents in the envelope.
According to the problem, the number of dollars she named was actually the number of cents, so the actual amount of dollars in the envelope is y dollars.
The number of cents she named was actually the number of dollars she put in, so the actual amount of cents in the envelope is x cents.
We know that after spending $0.50 at the yard sale, Ben was left with twice the amount of money his Mom said she placed in the envelope.
So after spending $0.50, Ben is left with (2 * y dollars) + x cents.
We can set up the equation:
(2 * y dollars) + x cents = (y dollars) * 100 cents
Simplifying the equation, we get:
2y + x = 100y
Rearranging the equation, we get:
2y - 100y = -x
Combining like terms, we get:
-98y = -x
Since the left side of the equation is in terms of y and the right side is in terms of x, we can conclude that -98y must be equal to -x.
Therefore, the number of cents x must be equal to the number of dollars y multiplied by -98.
So, the amount of money Ben's Mom actually put in the envelope is y dollars and x cents, which is y * 100 cents and y * -98 dollars.
Let's solve this step-by-step:
Let's assume the amount of money Ben's mom said she put in the envelope is 'x' dollars and 'y' cents. According to the given information, we know that the number of dollars she named was actually the number of cents, and the number of cents she named was actually the number of dollars she put in.
So, in cents, the amount she named is 'x' dollars, which is equal to 100x cents.
Similarly, in dollars, the amount she named is 'y' cents.
After spending $0.50 at the yard sale, Ben realizes that there is twice as much money left in the envelope as his mom said she put in. This means:
2 * (100x + y) = (100x + y) + 0.50
Let's solve this equation step-by-step:
Expanding the left side of the equation:
200x + 2y = 100x + y + 0.50
Combining like terms:
200x - 100x + 2y - y = 0.50
Simplifying:
100x + y = 0.50
Now, we know that the amount she named is 'x' dollars, which is equal to 100x cents.
So, the equation becomes:
x + y = 0.50
Since we have two equations:
100x + y = 0.50 (equation 1)
x + y = 0.50 (equation 2)
We can solve these equations simultaneously.
Subtracting equation 2 from equation 1:
(100x + y) - (x + y) = 0.50 - 0.50
Simplifying:
99x = 0
Dividing both sides by 99:
x = 0
Plugging this value back into equation 2:
0 + y = 0.50
Simplifying:
y = 0.50
Therefore, Ben's mom actually put $0.50 in the envelope.