Step by step
The question
Question: A school sold 100 tickets.
Adult tickets cost $5 and child tickets cost $3.
The school made $401.
How many of each ticket were sold?
1 Find what we know
What do we know?
100 tickets in all.
Two kinds: adult tickets cost $5 each, child tickets cost $3 each.
All the money: $401.
What do we want?
How many adult tickets, and how many child tickets.
2 Name the unknown
We don't know how many adult tickets.
So we call it x.
The rest are child.
100 in all, take away the x adult ones: 100 − x.
For example: if 10 were adult, then 100 − 10 = 90 would be child.
So we need only one letter.
Let's put it in a table: how many, the price, and the money.
Adult: x tickets at $5 each.
Child: 100 − x tickets at $3 each.
Money = how many times the price.
5 times x is 5x.
3 times all of 100 − x is 3(100 − x).
We keep 100 − x together in brackets.
The total row: 100 tickets in all, and $401 in all.
3 Write the equation
In words: the adult money plus the child money makes all the money.
So the equation is 5x + 3(100 − x) = 401.
Now we solve it.
4 Open the brackets
The 3 in front of the bracket multiplies every part inside.
So we write 3 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.
5x − 3x = 2x.
Now we have 2x + 300 = 401.
We want x by itself.
Two things are stuck to x: times 2 and + 300.
The + 300 is on the outside, so it comes off first.
Then the times 2.
6 Remove the + 300
To remove + 300, we take away 300.
We do it on both sides, so they stay equal.
+ 300 and − 300 make zero.
Cross them out.
401 − 300 = 101.
Now we have 2x = 101.
7 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
So x = 101/2!
8 Answer the question
x = 101/2 is not a whole number.
There are no half tickets.
So there is no answer.
Let's look back.
First: find what we know.
Then: name the unknown.
Then: write the equation.
Then: solve it.
Last: answer the question.