Step by step
The question
Question: Tom has 8 litres of a 70% acid solution.
How many litres of a 30% acid solution must he add to make a 50% acid solution?
1 Find what we know
What do we know?
We have 8 litres of a 70% solution.
The mix must be 50% acid.
What do we want?
How many litres of the 30% solution.
2 Name the unknown
We don't know how many litres of the 30% solution.
So we call it x.
The mix has both: x litres and 8 litres.
So it is x + 8 litres.
Careful!
The mix has the 30% solution in it too.
So it is x + 8 litres, not 8 litres.
Let's put it in a table: how much, the acid part as a decimal, and the acid.
30% is 30/100: 0.3.
70% is 0.7.
Acid = how much times the part.
x times 0.3 is 0.3x.
8 times 0.7 is 5.6.
The mix: x + 8 litres at 0.5.
Its acid is 0.5(x + 8): all of x + 8 is times 0.5.
3 Write the equation
In words: the acid from the two solutions makes the acid in the mix.
So the equation is 0.3x + 5.6 = 0.5(x + 8).
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 × 0.3x = 3x.
10 × 5.6 = 56.
10 × 0.5(x + 8) = 5(x + 8).
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 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the 3x from the left side.
To remove 3x, we take away 3x.
We do it on both sides, so they stay equal.
3x and − 3x make zero.
Cross them out.
5x − 3x = 2x.
Now we have 56 = 2x + 40.
Let's swap the two sides, so x is on the left.
We want x by itself.
Two things are stuck to x: times 2 and + 40.
The + 40 is on the outside, so it comes off first.
Then the times 2.
7 Remove the + 40
To remove + 40, we take away 40.
We do it on both sides, so they stay equal.
+ 40 and − 40 make zero.
Cross them out.
56 − 40 = 16.
Now we have 2x = 16.
8 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
16 divided by 2 is 8.
So x = 8!
9 Answer the question
So we need 8 litres of the 30% solution.
10 Check
Let's check in the story.
The acid: 2.4 plus 5.6 is 8 litres.
50% of the 16 litres of mix is 8 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.