Step by step
The question
Question: A box has 25-cent coins and 50-cent coins.
There are 2 fewer 50-cent coins than 25-cent coins.
The coins are worth $4.25.
How many of each coin are in the box?
1 Find what we know
What do we know?
Two kinds of coins: 25 cents and 50 cents each.
There are 2 fewer 50-cent coins than 25-cent coins.
All the coins are worth $4.25.
What do we want?
How many 25-cent coins, and how many 50-cent coins.
2 Name the unknown
We know the 50-cent coins from the 25-cent coins.
So we start with the 25-cent coins.
We call how many there are x.
2 fewer: x − 2.
We work in cents, so every number is whole.
$1 is 100 cents.
So $4.25 is 425 cents.
Careful!
There are 2 fewer 50-cent coins than 25-cent coins.
So the − 2 goes with the 50-cent coins, 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.
50-cent coins: x − 2 of them, 50 cents each.
Money = how many times the value of one coin.
25 times x is 25x.
50 times all of x − 2 is 50(x − 2).
We keep x − 2 together in brackets.
The total: 425 cents in all.
3 Write the equation
In words: the 25-cent money plus the 50-cent money makes all the money.
So the equation is 25x + 50(x − 2) = 425.
Now we solve it.
4 Open the brackets
The 50 in front of the bracket multiplies every part inside.
So we write 50 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 + 50x = 75x.
Now we have 75x − 100 = 425.
We want x by itself.
Two things are stuck to x: times 75 and − 100.
The − 100 is on the outside, so it comes off first.
Then the times 75.
6 Remove the − 100
To remove − 100, we add 100.
We do it on both sides, so they stay equal.
− 100 and + 100 make zero.
Cross them out.
425 + 100 = 525.
Now we have 75x = 525.
7 Remove the 75
75x means 75 times x.
We want just x.
To remove times 75, we divide by 75.
We do it on both sides, so they stay equal.
75 divided by 75 is 1.
So just x is left.
525 divided by 75 is 7.
So x = 7!
8 Answer the question
50-cent coins: x − 2.
7 − 2 = 5.
So 7 coins of 25 cents, and 5 coins of 50 cents.
9 Check
Let's check in the story.
5 is 2 fewer than 7, like the story says.
The money: 175 cents plus 250 cents makes 425 cents.
425 cents is $4.25, 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.