Step by step
The question
Question: A shop sold 40 shirts.
Large shirts cost $12 and small shirts cost $8.
The shop made $420.
How many of each shirt were sold?
1 Find what we know
What do we know?
40 shirts in all.
Two kinds: large shirts cost $12 each, small shirts cost $8 each.
All the money: $420.
What do we want?
How many large shirts, and how many small shirts.
2 Name the unknown
We don't know how many large shirts.
So we call it x.
The rest are small.
40 in all, take away the x large ones: 40 − x.
For example: if 4 were large, then 40 − 4 = 36 would be small.
So we need only one letter.
Let's put it in a table: how many, the price, and the money.
Large: x shirts at $12 each.
Small: 40 − x shirts at $8 each.
Money = how many times the price.
12 times x is 12x.
8 times all of 40 − x is 8(40 − x).
We keep 40 − x together in brackets.
The total row: 40 shirts in all, and $420 in all.
3 Write the equation
In words: the large money plus the small money makes all the money.
So the equation is 12x + 8(40 − x) = 420.
Now we solve it.
4 Open the brackets
The 8 in front of the bracket multiplies every part inside.
So we write 8 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 − 8x = 4x.
Now we have 4x + 320 = 420.
We want x by itself.
Two things are stuck to x: times 4 and + 320.
The + 320 is on the outside, so it comes off first.
Then the times 4.
6 Remove the + 320
To remove + 320, we take away 320.
We do it on both sides, so they stay equal.
+ 320 and − 320 make zero.
Cross them out.
420 − 320 = 100.
Now we have 4x = 100.
7 Remove the 4
4x means 4 times x.
We want just x.
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4 divided by 4 is 1.
So just x is left.
100 divided by 4 is 25.
So x = 25!
8 Answer the question
Small: 40 − x.
40 − 25 = 15.
So 25 large shirts and 15 small shirts.
9 Check
Let's check in the story.
25 + 15 = 40 shirts, like the story says.
The money: $300 plus $120 makes $420, like the story says.
Does it make sense?
Both are whole numbers from 0 to 40.
That fits 40 shirts.
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.