Step by step
The question
Question: Gold stamps cost $2 and silver stamps cost $1.
Sam bought 12 fewer silver stamps than gold stamps.
He paid $45.
How many of each stamp did he buy?
1 Find what we know
What do we know?
Two kinds: gold stamps cost $2 each, silver stamps cost $1 each.
There are 12 fewer silver stamps than gold stamps.
All the money: $45.
What do we want?
How many gold stamps, and how many silver stamps.
2 Name the unknown
We know the silver stamps from the gold stamps.
So we start with the gold stamps.
We call how many there are x.
12 fewer: x − 12.
Careful!
There are 12 fewer silver stamps than gold stamps.
So the − 12 goes with the silver stamps, not the gold stamps.
Let's put it in a table: how many, the price, and the money.
Gold stamps: x of them, $2 each.
Silver stamps: x − 12 of them, $1 each.
Money = how many times the price.
2 times x is 2x.
1 times all of x − 12 is 1(x − 12).
We keep x − 12 together in brackets.
The total: $45 in all.
3 Write the equation
In words: the gold money plus the silver money makes all the money.
So the equation is 2x + (x − 12) = 45.
Now we solve it.
4 Open the brackets
A bracket with a plus or nothing in front can just be removed.
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.
2x + x = 3x.
Now we have 3x − 12 = 45.
We want x by itself.
Two things are stuck to x: times 3 and − 12.
The − 12 is on the outside, so it comes off first.
Then the times 3.
6 Remove the − 12
To remove − 12, we add 12.
We do it on both sides, so they stay equal.
− 12 and + 12 make zero.
Cross them out.
45 + 12 = 57.
Now we have 3x = 57.
7 Remove the 3
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just x is left.
57 divided by 3 is 19.
So x = 19!
8 Answer the question
Silver stamps: x − 12.
19 − 12 = 7.
So 19 gold stamps and 7 silver stamps.
9 Check
Let's check in the story.
7 is 12 fewer than 19, like the story says.
The money: $38 plus $7 makes $45, like the story says.
Does it make sense?
Both are whole numbers, not below 0.
That fits stamps.
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.