How much was in the envelope?

Question

Anna gave an envelope to Ben and told him how much it contained. Ben fished $0.5 from the envelope and gave it to his sister. Later, when he counts the money in the envelope, he finds twice as much as Anna said it contained. Mentally, he adds the $0.5 and realizes that Anna muddled up the sum she gave him. The number of cents was actually the number of dollars, and the number of dollars was the number of cents. How much was in the envelope?

Answer 1

Let d=dollars and c=cents (what was actually in the envelope).

Anna says the envelope contains c dollars and d cents.

When 50c is removed from the envelope, the value remaining in cents is 100d+c-50. What’s left is twice as much as Anna said: 2(100c+d) cents.

So 100d+c-50=200c+2d.

Therefore 98d-199c=50, and d=(199c+50)/98.

d=(196c+3c+50)/98=2c+(3c+50)/98.

If 3c+50=98, 3c=48, c=16.

Therefore d=32+1=33.

So the envelope contained $33.16.