Question

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?

Answer 1

21 of the expensive kind

Explanation:

Answer 2

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