Step by step
The question
Question: A lab needs 30 litres of a 25% acid solution.
It mixes a 10% acid solution with a 40% acid solution.
How many litres of each does it use?
1 Find what we know
What do we know?
The mix: 30 litres that is 25% acid.
It is made from a 10% and a 40% solution.
What do we want?
How many litres of each.
2 Name the unknown
We don't know how many litres of the 40% solution.
So we call it x.
The rest is the 10% solution.
30 litres in all, take away x: 30 − x.
Let's put it in a table: how much, the acid part as a decimal, and the acid.
40% is 0.4.
10% is 0.1.
Acid = how much times the part: 0.4x.
0.1 times all of 30 − x: 0.1(30 − x).
We keep 30 − x together in brackets.
The mix: 30 litres at 0.25.
30 times 0.25 is 7.5 litres of acid.
3 Write the equation
In words: the acid from the two solutions makes the acid in the mix.
So the equation is 0.4x + 0.1(30 − x) = 7.5.
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 × 0.4x = 4x.
10 × 0.1(30 − x) = (30 − x).
10 × 7.5 = 75.
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.
4x − x = 3x.
Now we have 3x + 30 = 75.
We want x by itself.
Two things are stuck to x: times 3 and + 30.
The + 30 is on the outside, so it comes off first.
Then the times 3.
7 Remove the + 30
To remove + 30, we take away 30.
We do it on both sides, so they stay equal.
+ 30 and − 30 make zero.
Cross them out.
75 − 30 = 45.
Now we have 3x = 45.
8 Remove the 3
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just x is left.
45 divided by 3 is 15.
So x = 15!
9 Answer the question
The 10% solution: 30 − 15 = 15.
So 15 litres of the 40% solution and 15 litres of the 10% solution.
10 Check
Let's check in the story.
15 litres and 15 litres make 30 litres, like the story says.
The acid: 6 plus 1.5 is 7.5 litres.
That is 25% of 30, 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.