Step by step
The question
Question: Adult tickets cost $12 and child tickets cost $8.
A show sold 25 fewer adult tickets than child tickets.
It made $1000.
How many of each ticket did it sell?
1 Find what we know
What do we know?
Two kinds: adult tickets cost $12 each, child tickets cost $8 each.
There are 25 fewer adult tickets than child tickets.
All the money: $1000.
What do we want?
How many child tickets, and how many adult tickets.
2 Name the unknown
We know the adult tickets from the child tickets.
So we start with the child tickets.
We call how many there are x.
25 fewer: x − 25.
Careful!
There are 25 fewer adult tickets than child tickets.
So the − 25 goes with the adult tickets, not the child tickets.
Let's put it in a table: how many, the price, and the money.
Child tickets: x of them, $8 each.
Adult tickets: x − 25 of them, $12 each.
Money = how many times the price.
8 times x is 8x.
12 times all of x − 25 is 12(x − 25).
We keep x − 25 together in brackets.
The total: $1000 in all.
3 Write the equation
In words: the child money plus the adult money makes all the money.
So the equation is 8x + 12(x − 25) = 1000.
Now we solve it.
4 Open the brackets
The 12 in front of the bracket multiplies every part inside.
So we write 12 times each part.
Work out each multiplication.
5 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x terms go together.
8x + 12x = 20x.
Now we have 20x − 300 = 1000.
We want x by itself.
Two things are stuck to x: times 20 and − 300.
The − 300 is on the outside, so it comes off first.
Then the times 20.
6 Remove the − 300
To remove − 300, we add 300.
We do it on both sides, so they stay equal.
− 300 and + 300 make zero.
Cross them out.
1000 + 300 = 1300.
Now we have 20x = 1300.
7 Remove the 20
20x means 20 times x.
We want just x.
To remove times 20, we divide by 20.
We do it on both sides, so they stay equal.
20 divided by 20 is 1.
So just x is left.
1300 divided by 20 is 65.
So x = 65!
8 Answer the question
Adult tickets: x − 25.
65 − 25 = 40.
So 65 child tickets and 40 adult tickets.
9 Check
Let's check in the story.
40 is 25 fewer than 65, like the story says.
The money: $520 plus $480 makes $1000, like the story says.
Does it make sense?
Both are whole numbers, not below 0.
That fits tickets.
So the answer is right!
Let's look back.
First: find what we know.
Then: name the unknown.
Then: write the equation.
Then: solve it.
Then: answer the question.
Last: check.