Step by step
The question
Question: Leo has 40 coins.
Some are 5-cent coins and the rest are 25-cent coins.
The coins are worth $5.20.
How many of each coin does he have?
1 Find what we know
What do we know?
40 coins in all.
Two kinds of coins: 5 cents and 25 cents each.
All the coins are worth $5.20.
What do we want?
How many 25-cent coins, and how many 5-cent coins.
2 Name the unknown
We don't know how many 25-cent coins.
So we call it x.
The rest are 5-cent coins.
40 in all, take away the x 25-cent coins: 40 − x.
For example: if 10 are 25-cent coins, then 40 − 10 = 30 are 5-cent coins.
So we need only one letter.
We work in cents, so every number is whole.
$1 is 100 cents.
So $5.20 is 520 cents.
Let's put it in a table: how many coins, the value of one, and the money in cents.
25-cent coins: x of them, 25 cents each.
5-cent coins: 40 − x of them, 5 cents each.
Money = how many times the value of one coin.
25 times x is 25x.
5 times all of 40 − x is 5(40 − x).
We keep 40 − x together in brackets.
The total row: 40 coins in all, and 520 cents in all.
3 Write the equation
In words: the 25-cent money plus the 5-cent money makes all the money.
So the equation is 25x + 5(40 − x) = 520.
Now we solve it.
Careful!
Don't mix dollars and cents.
The coins are counted in cents.
So all the money is 520 cents, not 5.20.
4 Open the brackets
The 5 in front of the bracket multiplies every part inside.
So we write 5 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.
25x − 5x = 20x.
Now we have 20x + 200 = 520.
We want x by itself.
Two things are stuck to x: times 20 and + 200.
The + 200 is on the outside, so it comes off first.
Then the times 20.
6 Remove the + 200
To remove + 200, we take away 200.
We do it on both sides, so they stay equal.
+ 200 and − 200 make zero.
Cross them out.
520 − 200 = 320.
Now we have 20x = 320.
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.
320 divided by 20 is 16.
So x = 16!
8 Answer the question
5-cent coins: 40 − x.
40 − 16 = 24.
So 16 coins of 25 cents, and 24 coins of 5 cents.
9 Check
Let's check in the story.
16 + 24 = 40 coins, like the story says.
The money: 400 cents plus 120 cents makes 520 cents.
520 cents is $5.20, like the story says.
Does it make sense?
Both are whole numbers, not below 0.
That fits coins.
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.