A woman bought 100 Christmas cards. For the ones that sing a song when you open them, she paid 30 cents each. For the rest she paid 5 cents each. If the cards cost $10.25 in all, how many of the expensive kind did she buy?
Let the number of sing-a-songs be x Let the number of ordinarys be y
Value Value Type Number of of of of EACH ALL card cards card cards ------------------------------------------- sing-a-songs x $0.30 $0.30x ordinarys y $0.05 $0.05y ------------------------------------------- TOTALS 100 ----- $10.25
The first equation comes from the second column.
x + y = 100
The second equation comes from the last column.
0.3x + 0.05y = 10.25
Get rid of decimals by multiplying every term by 100:
30x + 5y = 1025
So we have the system of equations: .
We solve by substitution. Solve the first equation for y:
x + y = 100 y = 100 - x
Substitute (100 - x) for y in 30x + 5y = 1025
30x + 5(100 - x) = 1025 30x + 500 - 5x = 1025 25x + 500 = 1025 25x = 525 x = 21 = the number of sing-a-song cards.
Substitute in y = 100 - x y = 100 - (21) y = 79 ordinarys.
Checking: 21 sing-a-songs is $6.30 and 79 ordinarys is $3.95 That's 100 cards. And indeed $6.30 + $3.95 = $10.25
