Step by step
The question
Question: A jar has 5-cent coins and 10-cent coins.
There are 9 more 5-cent coins than 10-cent coins.
The coins are worth $2.25.
How many of each coin are in the jar?
1 Find what we know
What do we know?
Two kinds of coins: 5 cents and 10 cents each.
There are 9 more 5-cent coins than 10-cent coins.
All the coins are worth $2.25.
What do we want?
How many 10-cent coins, and how many 5-cent coins.
2 Name the unknown
We know the 5-cent coins from the 10-cent coins.
So we start with the 10-cent coins.
We call how many there are x.
9 more: x + 9.
We work in cents, so every number is whole.
$1 is 100 cents.
So $2.25 is 225 cents.
Careful!
There are 9 more 5-cent coins than 10-cent coins.
So the + 9 goes with the 5-cent coins, not the 10-cent coins.
Let's put it in a table: how many coins, the value of one, and the money in cents.
10-cent coins: x of them, 10 cents each.
5-cent coins: x + 9 of them, 5 cents each.
Money = how many times the value of one coin.
10 times x is 10x.
5 times all of x + 9 is 5(x + 9).
We keep x + 9 together in brackets.
The total: 225 cents in all.
3 Write the equation
In words: the 10-cent money plus the 5-cent money makes all the money.
So the equation is 10x + 5(x + 9) = 225.
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.
10x + 5x = 15x.
Now we have 15x + 45 = 225.
We want x by itself.
Two things are stuck to x: times 15 and + 45.
The + 45 is on the outside, so it comes off first.
Then the times 15.
6 Remove the + 45
To remove + 45, we take away 45.
We do it on both sides, so they stay equal.
+ 45 and − 45 make zero.
Cross them out.
225 − 45 = 180.
Now we have 15x = 180.
7 Remove the 15
15x means 15 times x.
We want just x.
To remove times 15, we divide by 15.
We do it on both sides, so they stay equal.
15 divided by 15 is 1.
So just x is left.
180 divided by 15 is 12.
So x = 12!
8 Answer the question
5-cent coins: x + 9.
12 + 9 = 21.
So 12 coins of 10 cents, and 21 coins of 5 cents.
9 Check
Let's check in the story.
21 is 9 more than 12, like the story says.
The money: 120 cents plus 105 cents makes 225 cents.
225 cents is $2.25, 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.