Step by step
The question
Question: A jar has 5-cent coins and 25-cent coins.
There are 4 times as many 5-cent coins as 25-cent coins.
The coins are worth $5.40.
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 25 cents each.
There are 4 times as many 5-cent coins as 25-cent coins.
All the coins are worth $5.40.
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.
There are 4 times as many 5-cent coins as 25-cent coins.
4 times x: 4x.
We work in cents, so every number is whole.
$1 is 100 cents.
So $5.40 is 540 cents.
Careful!
There are 4 times as many 5-cent coins as 25-cent coins.
So the 5-cent coins are 4x, 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: 4x of them, 5 cents each.
Money = how many times the value of one coin.
25 times x is 25x.
5 times all of 4x is 5(4x).
We keep 4x together in brackets.
The total: 540 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(4x) = 540.
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 + 20x = 45x.
Now we have 45x = 540.
6 Remove the 45
45x means 45 times x.
We want just x.
To remove times 45, we divide by 45.
We do it on both sides, so they stay equal.
45 divided by 45 is 1.
So just x is left.
540 divided by 45 is 12.
So x = 12!
7 Answer the question
5-cent coins: 4x.
4 × 12 = 48.
So 12 coins of 25 cents, and 48 coins of 5 cents.
8 Check
Let's check in the story.
4 times 12 is 48: 4 times as many, like the story says.
The money: 300 cents plus 240 cents makes 540 cents.
540 cents is $5.40, 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.