Step by step
The question
Question: A lab needs 100 litres of an 18% salt solution.
It mixes a 10% salt solution with a 30% salt solution.
How many litres of each does it use?
1 Find what we know
What do we know?
The mix: 100 litres that is 18% salt.
It is made from a 10% and a 30% solution.
What do we want?
How many litres of each.
2 Name the unknown
We don't know how many litres of the 30% solution.
So we call it x.
The rest is the 10% solution.
100 litres in all, take away x: 100 − x.
Let's put it in a table: how much, the salt part as a decimal, and the salt.
30% is 30/100: 0.3.
10% is 10/100: 0.1.
Salt = how much times the part: 0.3x.
0.1 times all of 100 − x: 0.1(100 − x).
We keep 100 − x together in brackets.
The mix: 100 litres at 0.18.
100 times 0.18 is 18 litres of salt.
3 Write the equation
In words: the salt from the two solutions makes the salt in the mix.
So the equation is 0.3x + 0.1(100 − x) = 18.
Now we solve it.
Careful!
Many people take half of each.
But half and half gives (30 + 10) ÷ 2 = 20%, not 18%.
4 Remove the decimals
Decimals are hard to work with.
Let's get rid of them.
0.3 has 1 digit after the point.
Times 10 moves the point 1 place to the right.
Then every number is a whole number.
We multiply every part by 10.
We do it on both sides, so they stay equal.
10 × 0.3x = 3x.
10 × 0.1(100 − x) = (100 − x).
10 × 18 = 180.
Write the answers.
No more decimals!
5 Open the brackets
A bracket with a plus or nothing in front can just be removed.
6 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.
3x − x = 2x.
Now we have 2x + 100 = 180.
We want x by itself.
Two things are stuck to x: times 2 and + 100.
First x was times 2.
Then + 100.
To take them off, we go backwards.
First we remove the + 100.
Then the times 2.
7 Remove the + 100
To remove + 100, we take away 100.
We do it on both sides, so they stay equal.
+ 100 and − 100 make zero.
Cross them out.
180 − 100 = 80.
Now we have 2x = 80.
8 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
80 divided by 2 is 40.
So x = 40.
9 Answer the question
The 10% solution: 100 − 40 = 60.
So 40 litres of the 30% solution and 60 litres of the 10% solution.
10 Check
Let's check in the story.
40 litres and 60 litres make 100 litres, like the story says.
The salt: 12 plus 6 is 18 litres.
That is 18% of 100, like the story says.
Does it make sense?
18% is nearer to 10%.
So there is more of the 10% solution: 60 litres is more than 40 litres.
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.