Step by step
The question
Question: Adult tickets cost $12 and student tickets cost $7.
A school sold 5 more student tickets than adult tickets.
It made $415.
How many of each ticket did it sell?
1 Find what we know
What do we know?
Two kinds: adult tickets cost $12 each, student tickets cost $7 each.
There are 5 more student tickets than adult tickets.
All the money: $415.
What do we want?
How many adult tickets, and how many student tickets.
2 Name the unknown
We know the student tickets from the adult tickets.
So we start with the adult tickets.
We call how many there are x.
5 more: x + 5.
Careful!
There are 5 more student tickets than adult tickets.
So the + 5 goes with the student tickets, not the adult tickets.
Let's put it in a table: how many, the price, and the money.
Adult tickets: x of them, $12 each.
Student tickets: x + 5 of them, $7 each.
Money = how many times the price.
12 times x is 12x.
7 times all of x + 5 is 7(x + 5).
We keep x + 5 together in brackets.
The total: $415 in all.
3 Write the equation
In words: the adult money plus the student money makes all the money.
So the equation is 12x + 7(x + 5) = 415.
Now we solve it.
4 Open the brackets
The 7 in front of the bracket multiplies every part inside.
So we write 7 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.
12x + 7x = 19x.
Now we have 19x + 35 = 415.
We want x by itself.
Two things are stuck to x: times 19 and + 35.
The + 35 is on the outside, so it comes off first.
Then the times 19.
6 Remove the + 35
To remove + 35, we take away 35.
We do it on both sides, so they stay equal.
+ 35 and − 35 make zero.
Cross them out.
415 − 35 = 380.
Now we have 19x = 380.
7 Remove the 19
19x means 19 times x.
We want just x.
To remove times 19, we divide by 19.
We do it on both sides, so they stay equal.
19 divided by 19 is 1.
So just x is left.
380 divided by 19 is 20.
So x = 20!
8 Answer the question
Student tickets: x + 5.
20 + 5 = 25.
So 20 adult tickets and 25 student tickets.
9 Check
Let's check in the story.
25 is 5 more than 20, like the story says.
The money: $240 plus $175 makes $415, 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.