Step by step
The question
Question: A jar has 50 coins.
Some are 10-cent coins and the rest are 5-cent coins.
The coins are worth $3.70.
How many of each coin are in the jar?
1 Find what we know
What do we know?
50 coins in all.
Two kinds of coins: 10 cents and 5 cents each.
All the coins are worth $3.70.
What do we want?
How many 10-cent coins, and how many 5-cent coins.
2 Name the unknown
We don't know how many 10-cent coins.
So we call it x.
The rest are 5-cent coins.
50 in all, take away the x 10-cent coins: 50 − x.
For example: if 10 are 10-cent coins, then 50 − 10 = 40 are 5-cent coins.
So we need only one letter.
We work in cents, so every number is whole.
$1 is 100 cents.
So $3.70 is 370 cents.
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: 50 − x of them, 5 cents each.
Money = how many times the value of one coin.
10 times x is 10x.
5 times all of 50 − x is 5(50 − x).
We keep 50 − x together in brackets.
The total row: 50 coins in all, and 370 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(50 − x) = 370.
Now we solve it.
Careful!
Don't mix dollars and cents.
The coins are counted in cents.
So all the money is 370 cents, not 3.70.
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 − 5x = 5x.
Now we have 5x + 250 = 370.
We want x by itself.
Two things are stuck to x: times 5 and + 250.
The + 250 is on the outside, so it comes off first.
Then the times 5.
6 Remove the + 250
To remove + 250, we take away 250.
We do it on both sides, so they stay equal.
+ 250 and − 250 make zero.
Cross them out.
370 − 250 = 120.
Now we have 5x = 120.
7 Remove the 5
5x means 5 times x.
We want just x.
To remove times 5, we divide by 5.
We do it on both sides, so they stay equal.
5 divided by 5 is 1.
So just x is left.
120 divided by 5 is 24.
So x = 24!
8 Answer the question
5-cent coins: 50 − x.
50 − 24 = 26.
So 24 coins of 10 cents, and 26 coins of 5 cents.
9 Check
Let's check in the story.
24 + 26 = 50 coins, like the story says.
The money: 240 cents plus 130 cents makes 370 cents.
370 cents is $3.70, 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.