The play at the theater charges $7 for adults, $5 for school-age children, and $3 for babies?

The play at the theater charges $7 for adults, $5 for school-age children, and $3 for babies. A group of people went to see the play. There were the same number of school-age children as babies and the number of adults was the same as school-age children and babies combined. If the group paid $1562, how many babies were there?

2 Answers
Dec 19, 2017

Answer:

There were 71 babies.

Explanation:

First define some variables:
Let #x# be the number of adults.
Let #y# be the number of children.
Let #z# be the number of babies.

Now write some equations.
(1) Money: #7x+5y+3z=1562#
(2) Children vs. babies: #y=z#
(3) Adults vs. Children&Babies: #x = y+z#

Subbing (2) into (3) we get that #x=y+y\rightarrow x=2y#, which we'll call (4) for reference.

Subbing (2) and (4) into (1) we get:
#7(2y) +5y + 3(y) = 1562#
#14y+5y+3y=1562#
#22y=1562#
#y=71#

If #y=71#, then by (2), #z=71# also. By (4), #x=142#.

Subbing these values back into (1) confirms that this solution works.

Since #z# is the number of babies, there were 71 babies.

Dec 19, 2017

Answer:

#There  were  71  babies.#

Explanation:

Problems like this can be confusing for two reasons:

  1. It's hard to find a good way to express all those amounts
  2. There are two kinds of data to consider:
    . . . ~ The NUMBER of tickets of each type that were sold
    . . . ~ The monetary VALUE of each type of ticket

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

First find a way to express the NUMBER sold of each type of ticket

Let #x# represent the number of babies' tickets that were sold

Babies' tickets . . . . . . . . . . . .#x# #larr# number of babies' tickets
Same number . . . . . . . . . . . ..#x# #larr# number of children's tickets
The combined number . . . #2x# #larr# number of adults' tickets
~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Now find a way to express the VALUE of the tickets sold

#x  tickets#  @ #$3   ea# . . . . . . #3x# #larr# value of babies' tickets

#x  tickets#  @ #$5  ea# . . . . . . #5x# #larr# value of children's tickets

#2x tickets# @ #$7  ea# . . . .#7(2x)# #larr# value of adults' tickets
~ ~ ~ ~ ~ ~ ~ ~ ~

Now you can set up the equation

[babies' value] plus [children's value] plus [adults' value] = $1562
[ . . . . . #3x# . . . .] .#+# .[ . . . . . #5x# . . . . . ] .#+# . [ . . . #7(2x)# . .]  =   1562

Solve for #x#, already defined as "the number of babies' tickets"

1) Clear the parentheses by distributing the 7
#3x + 5x + 14x = 1562#

2) Combine like terms
#22x = 1562#

3) Divide both sides by 22 to isolate #x#, already defined as "the number of tickets for babies"
#x = 71# #larr# the number of tickets for babies

Answer:
There were 71 babies
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Check:

#71  babies#     @ #$3  ea# . . . . . . . #$213#

#71# #ch##ildren# @ #$5  ea# . . . . . . . #$355#

#142  ad##u##l##ts#   @ #$7  ea# . . . . . . . #$994#
#color(white)(........................................)#..............
#color(white)(......................)##Total#. . . . . . #$1562#

#Check# !