Question

Joe and Mary each pay the same price for a T-shirt. Before they buy it, Mary has $\$ 2$ more than Joe. To buy the T-shirt, Joe spends $ \$1 $ less than $\frac{2}{5}$ of his money, and Mary spends $\frac{1}{3}$ of her money. What was the total amount of money Joe and Mary originally had altogether?

Answer 1

52

Explanation:

Suppose Joe had $J$ dollars to start with. Then Mary had $J + 2$ dollars. Since they both spent the same amount on their shirts, $\frac{2}{5} J - 1 = \frac{1}{3} (J + 2)$. Distributing, we get $\frac{2}{5} J - 1 = \frac{1}{3}J + \frac{2}{3}$. Isolating $J$ on the left side, $\frac{1}{15}J = \frac{5}{3}$. Therefore, $J = 25$. Thus, their original total amount of money was $J + (J + 2) = \boxed{52}$ dollars.

Answer 2

Let Joe initially have J dollars, and Mary have M dollars. Let the price of the T-shirt be T dollars.

i) "Before they buy it, Mary has 2$ more than Joe":

means M=J+2

ii)

"To buy the T-shirt, Joe spends 1 $ less than \frac{2}{5} of his money"

so
`T= (2)/(5)J-1`

iii)

"Mary spends \frac{1}{3} of her money"

means
`T= (1)/(3)M`

equalize equations ii) and iii):


`(2)/(5)J-1= (1)/(3)M` (they are both equal to T)

substitute M=J+2 from equation i:


`(2)/(5)J-1= (1)/(3)(J+2)`


`(2J-5)/(5)= (J+2)/(3)`


`3(2J-5)=5(J+2)`

6J-15=5J+10
J=25

so M=25+2=27 (dollars)

Together Mary and Joe had 27+25=52 (dollars)


Answer: 52 $