Step by step
The question
Question: Kai has 24 coins.
Some are 20-cent coins and the rest are 50-cent coins.
The coins are worth $7.80.
How many of each coin does he have?
1 Find what we know
What do we know?
24 coins in all.
Two kinds of coins: 20 cents and 50 cents each.
All the coins are worth $7.80.
What do we want?
How many 50-cent coins, and how many 20-cent coins.
2 Name the unknown
We don't know how many 50-cent coins.
So we call it x.
The rest are 20-cent coins.
24 in all, take away the x 50-cent coins: 24 − x.
For example: if 10 are 50-cent coins, then 24 − 10 = 14 are 20-cent coins.
So we need only one letter.
We work in cents, so every number is whole.
$1 is 100 cents.
So $7.80 is 780 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.
20-cent coins: 24 − x of them, 20 cents each.
Money = how many times the value of one coin.
50 times x is 50x.
20 times all of 24 − x is 20(24 − x).
We keep 24 − x together in brackets.
The total row: 24 coins in all, and 780 cents in all.
3 Write the equation
In words: the 50-cent money plus the 20-cent money makes all the money.
So the equation is 50x + 20(24 − x) = 780.
Now we solve it.
Careful!
Don't mix dollars and cents.
The coins are counted in cents.
So all the money is 780 cents, not 7.80.
4 Open the brackets
The 20 in front of the bracket multiplies every part inside.
So we write 20 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 − 20x = 30x.
Now we have 30x + 480 = 780.
We want x by itself.
Two things are stuck to x: times 30 and + 480.
The + 480 is on the outside, so it comes off first.
Then the times 30.
6 Remove the + 480
To remove + 480, we take away 480.
We do it on both sides, so they stay equal.
+ 480 and − 480 make zero.
Cross them out.
780 − 480 = 300.
Now we have 30x = 300.
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.
300 divided by 30 is 10.
So x = 10!
8 Answer the question
20-cent coins: 24 − x.
24 − 10 = 14.
So 10 coins of 50 cents, and 14 coins of 20 cents.
9 Check
Let's check in the story.
10 + 14 = 24 coins, like the story says.
The money: 500 cents plus 280 cents makes 780 cents.
780 cents is $7.80, 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.