Step by step
The question
Question: A nurse needs 12 litres of a 15% salt solution.
She mixes a 10% salt solution with a 30% salt solution.
How many litres of each does she use?
1 Find what we know
What do we know?
The mix: 12 litres that is 15% 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.
12 litres in all, take away x: 12 − x.
Let's put it in a table: how much, the salt part as a decimal, and the salt.
30% is 0.3.
10% is 0.1.
Salt = how much times the part: 0.3x.
0.1 times all of 12 − x: 0.1(12 − x).
We keep 12 − x together in brackets.
The mix: 12 litres at 0.15.
12 times 0.15 is 1.8 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(12 − x) = 1.8.
Now we solve it.
Careful!
Many people take half of each.
But half and half gives (30 + 10) ÷ 2 = 20%, not 15%.
4 Remove the decimals
Decimals are hard to work with.
Let's get rid of them.
We multiply every part by 10.
We do it on both sides, so they stay equal.
Times 10 moves the point 1 place to the right.
10 × 0.3x = 3x.
10 × 0.1(12 − x) = (12 − x).
10 × 1.8 = 18.
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 + 12 = 18.
We want x by itself.
Two things are stuck to x: times 2 and + 12.
The + 12 is on the outside, so it comes off first.
Then the times 2.
7 Remove the + 12
To remove + 12, we take away 12.
We do it on both sides, so they stay equal.
+ 12 and − 12 make zero.
Cross them out.
18 − 12 = 6.
Now we have 2x = 6.
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.
6 divided by 2 is 3.
So x = 3!
9 Answer the question
The 10% solution: 12 − 3 = 9.
So 3 litres of the 30% solution and 9 litres of the 10% solution.
10 Check
Let's check in the story.
3 litres and 9 litres make 12 litres, like the story says.
The salt: 0.9 plus 0.9 is 1.8 litres.
That is 15% of 12, like the story says.
Does it make sense?
The mix is between the two solutions.
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.