Step by step
The question
Question: Ana has 12 coins.
Some are 10-cent coins and the rest are 25-cent coins.
The coins are worth $1.20.
How many of each coin does she have?
1 Find what we know
What do we know?
12 coins in all.
Two kinds of coins: 10 cents and 25 cents each.
All the coins are worth $1.20.
What do we want?
How many 25-cent coins, and how many 10-cent coins.
2 Name the unknown
We don't know how many 25-cent coins.
So we call it x.
The rest are 10-cent coins.
12 in all, take away the x 25-cent coins: 12 − x.
For example: if 10 are 25-cent coins, then 12 − 10 = 2 are 10-cent coins.
So we need only one letter.
We work in cents, so every number is whole.
$1 is 100 cents.
So $1.20 is 120 cents.
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.
10-cent coins: 12 − x of them, 10 cents each.
Money = how many times the value of one coin.
25 times x is 25x.
10 times all of 12 − x is 10(12 − x).
We keep 12 − x together in brackets.
The total row: 12 coins in all, and 120 cents in all.
3 Write the equation
In words: the 25-cent money plus the 10-cent money makes all the money.
So the equation is 25x + 10(12 − x) = 120.
Now we solve it.
Careful!
Don't mix dollars and cents.
The coins are counted in cents.
So all the money is 120 cents, not 1.20.
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.
25x − 10x = 15x.
Now we have 15x + 120 = 120.
We want x by itself.
Two things are stuck to x: times 15 and + 120.
The + 120 is on the outside, so it comes off first.
Then the times 15.
6 Remove the + 120
To remove + 120, we take away 120.
We do it on both sides, so they stay equal.
+ 120 and − 120 make zero.
Cross them out.
120 − 120 = 0.
Now we have 15x = 0.
7 Remove the 15
15x means 15 times x.
We want just x.
To remove times 15, we divide by 15.
We do it on both sides, so they stay equal.
15 divided by 15 is 1.
So just x is left.
0 divided by 15 is 0.
So x = 0!
8 Answer the question
10-cent coins: 12 − x.
12 − 0 = 12.
So 0 coins of 25 cents, and 12 coins of 10 cents.
9 Check
Let's check in the story.
0 + 12 = 12 coins, like the story says.
The money: 0 cents plus 120 cents makes 120 cents.
120 cents is $1.20, 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.