Step by step
The question
Question: A chemist needs 10 litres of a 40% acid solution.
She mixes a 25% acid solution with a 50% acid solution.
How many litres of each does she use?
1 Find what we know
What do we know?
The mix: 10 litres that is 40% acid.
It is made from a 25% and a 50% solution.
What do we want?
How many litres of each.
2 Name the unknown
We don't know how many litres of the 50% solution.
So we call it x.
The rest is the 25% solution.
10 litres in all, take away x: 10 − x.
Let's put it in a table: how much, the acid part as a decimal, and the acid.
50% is 0.5.
25% is 0.25.
Acid = how much times the part: 0.5x.
0.25 times all of 10 − x: 0.25(10 − x).
We keep 10 − x together in brackets.
The mix: 10 litres at 0.4.
10 times 0.4 is 4 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 0.5x + 0.25(10 − x) = 4.
Now we solve it.
Careful!
Many people take half of each.
But half and half gives (50 + 25) ÷ 2 = 37.5%, not 40%.
4 Remove the decimals
Decimals are hard to work with.
Let's get rid of them.
We multiply every part by 100.
We do it on both sides, so they stay equal.
Times 100 moves the point 2 places to the right.
100 × 0.5x = 50x.
100 × 0.25(10 − x) = 25(10 − x).
100 × 4 = 400.
Write the answers.
No more decimals!
5 Open the brackets
The 25 in front of the bracket multiplies every part inside.
So we write 25 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.
50x − 25x = 25x.
Now we have 25x + 250 = 400.
We want x by itself.
Two things are stuck to x: times 25 and + 250.
The + 250 is on the outside, so it comes off first.
Then the times 25.
7 Remove the + 250
To remove + 250, we take away 250.
We do it on both sides, so they stay equal.
+ 250 and − 250 make zero.
Cross them out.
400 − 250 = 150.
Now we have 25x = 150.
8 Remove the 25
25x means 25 times x.
We want just x.
To remove times 25, we divide by 25.
We do it on both sides, so they stay equal.
25 divided by 25 is 1.
So just x is left.
150 divided by 25 is 6.
So x = 6!
9 Answer the question
The 25% solution: 10 − 6 = 4.
So 6 litres of the 50% solution and 4 litres of the 25% solution.
10 Check
Let's check in the story.
6 litres and 4 litres make 10 litres, like the story says.
The acid: 3 plus 1 is 4 litres.
That is 40% of 10, 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.