Step by step
The question
Question: A box has 10-cent coins and 25-cent coins.
There are 5 fewer 25-cent coins than 10-cent coins.
The coins are worth $2.95.
How many of each coin are in the box?
1 Find what we know
What do we know?
Two kinds of coins: 10 cents and 25 cents each.
There are 5 fewer 25-cent coins than 10-cent coins.
All the coins are worth $2.95.
What do we want?
How many 10-cent coins, and how many 25-cent coins.
2 Name the unknown
We know the 25-cent coins from the 10-cent coins.
So we start with the 10-cent coins.
We call how many there are x.
5 fewer: x − 5.
We work in cents, so every number is whole.
$1 is 100 cents.
So $2.95 is 295 cents.
Careful!
There are 5 fewer 25-cent coins than 10-cent coins.
So the − 5 goes with the 25-cent coins, not the 10-cent coins.
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.
25-cent coins: x − 5 of them, 25 cents each.
Money = how many times the value of one coin.
10 times x is 10x.
25 times all of x − 5 is 25(x − 5).
We keep x − 5 together in brackets.
The total: 295 cents in all.
3 Write the equation
In words: the 10-cent money plus the 25-cent money makes all the money.
So the equation is 10x + 25(x − 5) = 295.
Now we solve it.
4 Open the brackets
The 25 in front of the bracket multiplies every part inside.
So we write 25 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 + 25x = 35x.
Now we have 35x − 125 = 295.
We want x by itself.
Two things are stuck to x: times 35 and − 125.
The − 125 is on the outside, so it comes off first.
Then the times 35.
6 Remove the − 125
To remove − 125, we add 125.
We do it on both sides, so they stay equal.
− 125 and + 125 make zero.
Cross them out.
295 + 125 = 420.
Now we have 35x = 420.
7 Remove the 35
35x means 35 times x.
We want just x.
To remove times 35, we divide by 35.
We do it on both sides, so they stay equal.
35 divided by 35 is 1.
So just x is left.
420 divided by 35 is 12.
So x = 12!
8 Answer the question
25-cent coins: x − 5.
12 − 5 = 7.
So 12 coins of 10 cents, and 7 coins of 25 cents.
9 Check
Let's check in the story.
7 is 5 fewer than 12, like the story says.
The money: 120 cents plus 175 cents makes 295 cents.
295 cents is $2.95, 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.