Step by step
The question
Question: Sara has 5-cent coins and 10-cent coins.
She has 3 times as many 5-cent coins as 10-cent coins.
The coins are worth $1.25.
How many of each coin does she have?
1 Find what we know
What do we know?
Two kinds of coins: 5 cents and 10 cents each.
There are 3 times as many 5-cent coins as 10-cent coins.
All the coins are worth $1.25.
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.
There are 3 times as many 5-cent coins as 10-cent coins.
3 times x: 3x.
We work in cents, so every number is whole.
$1 is 100 cents.
So $1.25 is 125 cents.
Careful!
There are 3 times as many 5-cent coins as 10-cent coins.
So the 5-cent coins are 3x, 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: 3x of them, 5 cents each.
Money = how many times the value of one coin.
10 times x is 10x.
5 times all of 3x is 5(3x).
We keep 3x together in brackets.
The total: 125 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(3x) = 125.
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 + 15x = 25x.
Now we have 25x = 125.
6 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.
125 divided by 25 is 5.
So x = 5!
7 Answer the question
5-cent coins: 3x.
3 × 5 = 15.
So 5 coins of 10 cents, and 15 coins of 5 cents.
8 Check
Let's check in the story.
3 times 5 is 15: 3 times as many, like the story says.
The money: 50 cents plus 75 cents makes 125 cents.
125 cents is $1.25, 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.