Step by step
The question
Question: A lab needs 60 litres of a 45% salt solution.
It mixes a 30% salt solution with a 90% salt solution.
How many litres of each does it use?
1 Find what we know
What do we know?
The mix: 60 litres that is 45% salt.
It is made from a 30% and a 90% solution.
What do we want?
How many litres of each.
2 Name the unknown
We don't know how many litres of the 90% solution.
So we call it x.
The rest is the 30% solution.
60 litres in all, take away x: 60 − x.
Let's put it in a table: how much, the salt part as a decimal, and the salt.
90% is 0.9.
30% is 0.3.
Salt = how much times the part: 0.9x.
0.3 times all of 60 − x: 0.3(60 − x).
We keep 60 − x together in brackets.
The mix: 60 litres at 0.45.
60 times 0.45 is 27 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 0.9x + 0.3(60 − x) = 27.
Now we solve it.
Careful!
Many people take half of each.
But half and half gives (90 + 30) ÷ 2 = 60%, not 45%.
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 × 0.9x = 9x.
10 × 0.3(60 − x) = 3(60 − x).
10 × 27 = 270.
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 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.
9x − 3x = 6x.
Now we have 6x + 180 = 270.
We want x by itself.
Two things are stuck to x: times 6 and + 180.
The + 180 is on the outside, so it comes off first.
Then the times 6.
7 Remove the + 180
To remove + 180, we take away 180.
We do it on both sides, so they stay equal.
+ 180 and − 180 make zero.
Cross them out.
270 − 180 = 90.
Now we have 6x = 90.
8 Remove the 6
6x means 6 times x.
We want just x.
To remove times 6, we divide by 6.
We do it on both sides, so they stay equal.
6 divided by 6 is 1.
So just x is left.
90 divided by 6 is 15.
So x = 15!
9 Answer the question
The 30% solution: 60 − 15 = 45.
So 15 litres of the 90% solution and 45 litres of the 30% solution.
10 Check
Let's check in the story.
15 litres and 45 litres make 60 litres, like the story says.
The salt: 13.5 plus 13.5 is 27 litres.
That is 45% of 60, 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.