Step by step
The question
Question: Omar has 5-cent coins and 25-cent coins.
He has 6 more 5-cent coins than 25-cent coins.
The coins are worth $1.50.
How many of each coin does he have?
1 Find what we know
What do we know?
Two kinds of coins: 5 cents and 25 cents each.
There are 6 more 5-cent coins than 25-cent coins.
All the coins are worth $1.50.
What do we want?
How many 25-cent coins, and how many 5-cent coins.
2 Name the unknown
We know the 5-cent coins from the 25-cent coins.
So we start with the 25-cent coins.
We call how many there are x.
6 more: x + 6.
We work in cents, so every number is whole.
$1 is 100 cents.
So $1.50 is 150 cents.
Careful!
There are 6 more 5-cent coins than 25-cent coins.
So the + 6 goes with the 5-cent coins, not the 25-cent coins.
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: x + 6 of them, 5 cents each.
Money = how many times the value of one coin.
25 times x is 25x.
5 times all of x + 6 is 5(x + 6).
We keep x + 6 together in brackets.
The total: 150 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(x + 6) = 150.
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.
25x + 5x = 30x.
Now we have 30x + 30 = 150.
We want x by itself.
Two things are stuck to x: times 30 and + 30.
The + 30 is on the outside, so it comes off first.
Then the times 30.
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.
150 − 30 = 120.
Now we have 30x = 120.
7 Remove the 30
30x means 30 times x.
We want just x.
To remove times 30, we divide by 30.
We do it on both sides, so they stay equal.
30 divided by 30 is 1.
So just x is left.
120 divided by 30 is 4.
So x = 4!
8 Answer the question
5-cent coins: x + 6.
4 + 6 = 10.
So 4 coins of 25 cents, and 10 coins of 5 cents.
9 Check
Let's check in the story.
10 is 6 more than 4, like the story says.
The money: 100 cents plus 50 cents makes 150 cents.
150 cents is $1.50, 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.