Step by step
The question
Question: Omar has 6 litres of a 75% acid solution.
How many litres of a 25% acid solution must he add to make a 40% acid solution?
1 Find what we know
What do we know?
We have 6 litres of a 75% solution.
The mix must be 40% acid.
What do we want?
How many litres of the 25% solution.
2 Name the unknown
We don't know how many litres of the 25% solution.
So we call it x.
The mix has both: x litres and 6 litres.
So it is x + 6 litres.
Careful!
The mix has the 25% solution in it too.
So it is x + 6 litres, not 6 litres.
Let's put it in a table: how much, the acid part as a decimal, and the acid.
25% is 25/100: 0.25.
75% is 0.75.
Acid = how much times the part.
x times 0.25 is 0.25x.
6 times 0.75 is 4.5.
The mix: x + 6 litres at 0.4.
Its acid is 0.4(x + 6): all of x + 6 is times 0.4.
3 Write the equation
In words: the acid from the two solutions makes the acid in the mix.
So the equation is 0.25x + 4.5 = 0.4(x + 6).
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 × 0.25x = 25x.
100 × 4.5 = 450.
100 × 0.4(x + 6) = 40(x + 6).
Write the answers.
No more decimals!
5 Open the brackets
The 40 in front of the bracket multiplies every part inside.
So we write 40 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 25x from the left side.
To remove 25x, we take away 25x.
We do it on both sides, so they stay equal.
25x and − 25x make zero.
Cross them out.
40x − 25x = 15x.
Now we have 450 = 15x + 240.
Let's swap the two sides, so x is on the left.
We want x by itself.
Two things are stuck to x: times 15 and + 240.
The + 240 is on the outside, so it comes off first.
Then the times 15.
7 Remove the + 240
To remove + 240, we take away 240.
We do it on both sides, so they stay equal.
+ 240 and − 240 make zero.
Cross them out.
450 − 240 = 210.
Now we have 15x = 210.
8 Remove the 15
15x means 15 times x.
We want just x.
To remove times 15, we divide by 15.
We do it on both sides, so they stay equal.
15 divided by 15 is 1.
So just x is left.
210 divided by 15 is 14.
So x = 14!
9 Answer the question
So we need 14 litres of the 25% solution.
10 Check
Let's check in the story.
The acid: 3.5 plus 4.5 is 8 litres.
40% of the 20 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.