Step by step
The question
Question: Ben has 15 coins.
Some are 10-cent coins and the rest are 50-cent coins.
The coins are worth $3.50.
How many of each coin does he have?
1 Find what we know
What do we know?
15 coins in all.
Two kinds of coins: 10 cents and 50 cents each.
All the coins are worth $3.50.
What do we want?
How many 50-cent coins, and how many 10-cent coins.
2 Name the unknown
We don't know how many 50-cent coins.
So we call it x.
The rest are 10-cent coins.
15 in all, take away the x 50-cent coins: 15 − x.
For example: if 10 are 50-cent coins, then 15 − 10 = 5 are 10-cent coins.
So we need only one letter.
We work in cents, so every number is whole.
$1 is 100 cents.
So $3.50 is 350 cents.
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: 15 − x of them, 10 cents each.
Money = how many times the value of one coin.
50 times x is 50x.
10 times all of 15 − x is 10(15 − x).
We keep 15 − x together in brackets.
The total row: 15 coins in all, and 350 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(15 − x) = 350.
Now we solve it.
Careful!
Don't mix dollars and cents.
The coins are counted in cents.
So all the money is 350 cents, not 3.50.
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 = 40x.
Now we have 40x + 150 = 350.
We want x by itself.
Two things are stuck to x: times 40 and + 150.
The + 150 is on the outside, so it comes off first.
Then the times 40.
6 Remove the + 150
To remove + 150, we take away 150.
We do it on both sides, so they stay equal.
+ 150 and − 150 make zero.
Cross them out.
350 − 150 = 200.
Now we have 40x = 200.
7 Remove the 40
40x means 40 times x.
We want just x.
To remove times 40, we divide by 40.
We do it on both sides, so they stay equal.
40 divided by 40 is 1.
So just x is left.
200 divided by 40 is 5.
So x = 5!
8 Answer the question
10-cent coins: 15 − x.
15 − 5 = 10.
So 5 coins of 50 cents, and 10 coins of 10 cents.
9 Check
Let's check in the story.
5 + 10 = 15 coins, like the story says.
The money: 250 cents plus 100 cents makes 350 cents.
350 cents is $3.50, 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.