Step by step
The question
Question: A chemist needs 15 litres of a 60% acid solution.
She mixes a 50% acid solution with an 80% acid solution.
How many litres of each does she use?
1 Find what we know
What do we know?
The mix: 15 litres that is 60% acid.
It is made from a 50% and a 80% solution.
What do we want?
How many litres of each.
2 Name the unknown
We don't know how many litres of the 80% solution.
So we call it x.
The rest is the 50% solution.
15 litres in all, take away x: 15 − x.
Let's put it in a table: how much, the acid part as a decimal, and the acid.
80% is 0.8.
50% is 0.5.
Acid = how much times the part: 0.8x.
0.5 times all of 15 − x: 0.5(15 − x).
We keep 15 − x together in brackets.
The mix: 15 litres at 0.6.
15 times 0.6 is 9 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 0.8x + 0.5(15 − x) = 9.
Now we solve it.
Careful!
Many people take half of each.
But half and half gives (80 + 50) ÷ 2 = 65%, not 60%.
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 × 0.8x = 8x.
10 × 0.5(15 − x) = 5(15 − x).
10 × 9 = 90.
Write the answers.
No more decimals!
5 Open the brackets
The 5 in front of the bracket multiplies every part inside.
So we write 5 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.
8x − 5x = 3x.
Now we have 3x + 75 = 90.
We want x by itself.
Two things are stuck to x: times 3 and + 75.
The + 75 is on the outside, so it comes off first.
Then the times 3.
7 Remove the + 75
To remove + 75, we take away 75.
We do it on both sides, so they stay equal.
+ 75 and − 75 make zero.
Cross them out.
90 − 75 = 15.
Now we have 3x = 15.
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.
15 divided by 3 is 5.
So x = 5!
9 Answer the question
The 50% solution: 15 − 5 = 10.
So 5 litres of the 80% solution and 10 litres of the 50% solution.
10 Check
Let's check in the story.
5 litres and 10 litres make 15 litres, like the story says.
The acid: 4 plus 5 is 9 litres.
That is 60% of 15, 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.