Step by step
The question
Question: Sara has 5 litres of a 40% salt solution.
How many litres of a 10% salt solution must she add to make a 25% salt solution?
1 Find what we know
What do we know?
We have 5 litres of a 40% solution.
The mix must be 25% salt.
What do we want?
How many litres of the 10% solution.
2 Name the unknown
We don't know how many litres of the 10% solution.
So we call it x.
The mix has both: x litres and 5 litres.
So it is x + 5 litres.
Careful!
The mix has the 10% solution in it too.
So it is x + 5 litres, not 5 litres.
Let's put it in a table: how much, the salt part as a decimal, and the salt.
10% is 10/100: 0.1.
40% is 0.4.
Salt = how much times the part.
x times 0.1 is 0.1x.
5 times 0.4 is 2.
The mix: x + 5 litres at 0.25.
Its salt is 0.25(x + 5): all of x + 5 is times 0.25.
3 Write the equation
In words: the salt from the two solutions makes the salt in the mix.
So the equation is 0.1x + 2 = 0.25(x + 5).
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 × 0.1x = 10x.
100 × 2 = 200.
100 × 0.25(x + 5) = 25(x + 5).
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 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the 10x from the left side.
To remove 10x, we take away 10x.
We do it on both sides, so they stay equal.
10x and − 10x make zero.
Cross them out.
25x − 10x = 15x.
Now we have 200 = 15x + 125.
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 + 125.
The + 125 is on the outside, so it comes off first.
Then the times 15.
7 Remove the + 125
To remove + 125, we take away 125.
We do it on both sides, so they stay equal.
+ 125 and − 125 make zero.
Cross them out.
200 − 125 = 75.
Now we have 15x = 75.
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.
75 divided by 15 is 5.
So x = 5!
9 Answer the question
So we need 5 litres of the 10% solution.
10 Check
Let's check in the story.
The salt: 0.5 plus 2 is 2.5 litres.
25% of the 10 litres of mix is 2.5 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.