Step by step
The question
Question: Kim has 12 litres of a 45% acid solution.
How many litres of a 15% acid solution must she add to make a 35% acid solution?
1 Find what we know
What do we know?
We have 12 litres of a 45% solution.
The mix must be 35% acid.
What do we want?
How many litres of the 15% solution.
2 Name the unknown
We don't know how many litres of the 15% solution.
So we call it x.
The mix has both: x litres and 12 litres.
So it is x + 12 litres.
Careful!
The mix has the 15% solution in it too.
So it is x + 12 litres, not 12 litres.
Let's put it in a table: how much, the acid part as a decimal, and the acid.
15% is 15/100: 0.15.
45% is 0.45.
Acid = how much times the part.
x times 0.15 is 0.15x.
12 times 0.45 is 5.4.
The mix: x + 12 litres at 0.35.
Its acid is 0.35(x + 12): all of x + 12 is times 0.35.
3 Write the equation
In words: the acid from the two solutions makes the acid in the mix.
So the equation is 0.15x + 5.4 = 0.35(x + 12).
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 100.
We do it on both sides, so they stay equal.
Times 100 moves the point 2 places to the right.
100 × 0.15x = 15x.
100 × 5.4 = 540.
100 × 0.35(x + 12) = 35(x + 12).
Write the answers.
No more decimals!
5 Open the brackets
The 35 in front of the bracket multiplies every part inside.
So we write 35 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 15x from the left side.
To remove 15x, we take away 15x.
We do it on both sides, so they stay equal.
15x and − 15x make zero.
Cross them out.
35x − 15x = 20x.
Now we have 540 = 20x + 420.
Let's swap the two sides, so x is on the left.
We want x by itself.
Two things are stuck to x: times 20 and + 420.
The + 420 is on the outside, so it comes off first.
Then the times 20.
7 Remove the + 420
To remove + 420, we take away 420.
We do it on both sides, so they stay equal.
+ 420 and − 420 make zero.
Cross them out.
540 − 420 = 120.
Now we have 20x = 120.
8 Remove the 20
20x means 20 times x.
We want just x.
To remove times 20, we divide by 20.
We do it on both sides, so they stay equal.
20 divided by 20 is 1.
So just x is left.
120 divided by 20 is 6.
So x = 6!
9 Answer the question
So we need 6 litres of the 15% solution.
10 Check
Let's check in the story.
The acid: 0.9 plus 5.4 is 6.3 litres.
35% of the 18 litres of mix is 6.3 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.