Question

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?

Answer 1

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.

Answer 2

`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` !