Step by step
The question
Question: Mia has 6 litres of a 50% salt solution.
How many litres of a 20% salt solution must she add to make a 30% salt solution?
1 Find what we know
What do we know?
We have 6 litres of a 50% solution.
The mix must be 30% salt.
What do we want?
How many litres of the 20% solution.
2 Name the unknown
We don't know how many litres of the 20% solution.
So we call it x.
The mix has both: x litres and 6 litres.
So it is x + 6 litres.
Careful!
The mix has the 20% solution in it too.
So it is x + 6 litres, not 6 litres.
Let's put it in a table: how much, the salt part as a decimal, and the salt.
20% is 20/100: 0.2.
50% is 0.5.
Salt = how much times the part.
x times 0.2 is 0.2x.
6 times 0.5 is 3.
The mix: x + 6 litres at 0.3.
Its salt is 0.3(x + 6): all of x + 6 is times 0.3.
3 Write the equation
In words: the salt from the two solutions makes the salt in the mix.
So the equation is 0.2x + 3 = 0.3(x + 6).
Now we solve it.
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 × 0.2x = 2x.
10 × 3 = 30.
10 × 0.3(x + 6) = 3(x + 6).
Write the answers.
No more decimals!
5 Open the brackets
The 3 in front of the bracket multiplies every part inside.
So we write 3 times each part.
Work out each multiplication.
6 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the 2x from the left side.
To remove 2x, we take away 2x.
We do it on both sides, so they stay equal.
2x and − 2x make zero.
Cross them out.
3x − 2x = x.
Now we have 30 = x + 18.
Let's swap the two sides, so x is on the left.
7 Remove the + 18
We want x by itself.
The + 18 is in the way.
To remove + 18, we take away 18.
We do it on both sides, so they stay equal.
+ 18 and − 18 make zero.
Cross them out.
30 − 18 = 12.
So x = 12!
8 Answer the question
So we need 12 litres of the 20% solution.
9 Check
Let's check in the story.
The salt: 2.4 plus 3 is 5.4 litres.
30% of the 18 litres of mix is 5.4 too, 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.