Step by step
The question
Question: A purse has 10-cent coins and 50-cent coins.
There are 3 more 10-cent coins than 50-cent coins.
The coins are worth $3.30.
How many of each coin are in the purse?
1 Find what we know
What do we know?
Two kinds of coins: 10 cents and 50 cents each.
There are 3 more 10-cent coins than 50-cent coins.
All the coins are worth $3.30.
What do we want?
How many 50-cent coins, and how many 10-cent coins.
2 Name the unknown
We know the 10-cent coins from the 50-cent coins.
So we start with the 50-cent coins.
We call how many there are x.
3 more: x + 3.
We work in cents, so every number is whole.
$1 is 100 cents.
So $3.30 is 330 cents.
Careful!
There are 3 more 10-cent coins than 50-cent coins.
So the + 3 goes with the 10-cent coins, not the 50-cent coins.
Let's put it in a table: how many coins, the value of one, and the money in cents.
50-cent coins: x of them, 50 cents each.
10-cent coins: x + 3 of them, 10 cents each.
Money = how many times the value of one coin.
50 times x is 50x.
10 times all of x + 3 is 10(x + 3).
We keep x + 3 together in brackets.
The total: 330 cents in all.
3 Write the equation
In words: the 50-cent money plus the 10-cent money makes all the money.
So the equation is 50x + 10(x + 3) = 330.
Now we solve it.
4 Open the brackets
The 10 in front of the bracket multiplies every part inside.
So we write 10 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.
50x + 10x = 60x.
Now we have 60x + 30 = 330.
We want x by itself.
Two things are stuck to x: times 60 and + 30.
The + 30 is on the outside, so it comes off first.
Then the times 60.
6 Remove the + 30
To remove + 30, we take away 30.
We do it on both sides, so they stay equal.
+ 30 and − 30 make zero.
Cross them out.
330 − 30 = 300.
Now we have 60x = 300.
7 Remove the 60
60x means 60 times x.
We want just x.
To remove times 60, we divide by 60.
We do it on both sides, so they stay equal.
60 divided by 60 is 1.
So just x is left.
300 divided by 60 is 5.
So x = 5!
8 Answer the question
10-cent coins: x + 3.
5 + 3 = 8.
So 5 coins of 50 cents, and 8 coins of 10 cents.
9 Check
Let's check in the story.
8 is 3 more than 5, like the story says.
The money: 250 cents plus 80 cents makes 330 cents.
330 cents is $3.30, 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.