Step by step
The question
Question: Eva has 5-cent coins and 20-cent coins.
She has 8 more 5-cent coins than 20-cent coins.
The coins are worth €2.65.
How many of each coin does she have?
1 Find what we know
What do we know?
Two kinds of coins: 5 cents and 20 cents each.
There are 8 more 5-cent coins than 20-cent coins.
All the coins are worth €2.65.
What do we want?
How many 20-cent coins, and how many 5-cent coins.
2 Name the unknown
We know the 5-cent coins from the 20-cent coins.
So we start with the 20-cent coins.
We call how many there are x.
8 more: x + 8.
We work in cents, so every number is whole.
$1 is 100 cents.
So €2.65 is 265 cents.
Careful!
There are 8 more 5-cent coins than 20-cent coins.
So the + 8 goes with the 5-cent coins, not the 20-cent coins.
Let's put it in a table: how many coins, the value of one, and the money in cents.
20-cent coins: x of them, 20 cents each.
5-cent coins: x + 8 of them, 5 cents each.
Money = how many times the value of one coin.
20 times x is 20x.
5 times all of x + 8 is 5(x + 8).
We keep x + 8 together in brackets.
The total: 265 cents in all.
3 Write the equation
In words: the 20-cent money plus the 5-cent money makes all the money.
So the equation is 20x + 5(x + 8) = 265.
Now we solve it.
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.
20x + 5x = 25x.
Now we have 25x + 40 = 265.
We want x by itself.
Two things are stuck to x: times 25 and + 40.
The + 40 is on the outside, so it comes off first.
Then the times 25.
6 Remove the + 40
To remove + 40, we take away 40.
We do it on both sides, so they stay equal.
+ 40 and − 40 make zero.
Cross them out.
265 − 40 = 225.
Now we have 25x = 225.
7 Remove the 25
25x means 25 times x.
We want just x.
To remove times 25, we divide by 25.
We do it on both sides, so they stay equal.
25 divided by 25 is 1.
So just x is left.
225 divided by 25 is 9.
So x = 9!
8 Answer the question
5-cent coins: x + 8.
9 + 8 = 17.
So 9 coins of 20 cents, and 17 coins of 5 cents.
9 Check
Let's check in the story.
17 is 8 more than 9, like the story says.
The money: 180 cents plus 85 cents makes 265 cents.
265 cents is €2.65, 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.