Step by step
The question
Question: A cook needs 20 litres of a 30% sugar solution.
He mixes a 10% sugar solution with a 50% sugar solution.
How many litres of each does he use?
1 Find what we know
What do we know?
The mix: 20 litres that is 30% sugar.
It is made from a 10% and a 50% solution.
What do we want?
How many litres of each.
2 Name the unknown
We don't know how many litres of the 50% solution.
So we call it x.
The rest is the 10% solution.
20 litres in all, take away x: 20 − x.
Let's put it in a table: how much, the sugar part as a decimal, and the sugar.
50% is 0.5.
10% is 0.1.
Sugar = how much times the part: 0.5x.
0.1 times all of 20 − x: 0.1(20 − x).
We keep 20 − x together in brackets.
The mix: 20 litres at 0.3.
20 times 0.3 is 6 litres of sugar.
3 Write the equation
In words: the sugar from the two solutions makes the sugar in the mix.
So the equation is 0.5x + 0.1(20 − 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 10.
We do it on both sides, so they stay equal.
Times 10 moves the point 1 place to the right.
10 × 0.5x = 5x.
10 × 0.1(20 − x) = (20 − x).
10 × 6 = 60.
Write the answers.
No more decimals!
5 Open the brackets
A bracket with a plus or nothing in front can just be removed.
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.
5x − x = 4x.
Now we have 4x + 20 = 60.
We want x by itself.
Two things are stuck to x: times 4 and + 20.
The + 20 is on the outside, so it comes off first.
Then the times 4.
7 Remove the + 20
To remove + 20, we take away 20.
We do it on both sides, so they stay equal.
+ 20 and − 20 make zero.
Cross them out.
60 − 20 = 40.
Now we have 4x = 40.
8 Remove the 4
4x means 4 times x.
We want just x.
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4 divided by 4 is 1.
So just x is left.
40 divided by 4 is 10.
So x = 10!
9 Answer the question
The 10% solution: 20 − 10 = 10.
So 10 litres of the 50% solution and 10 litres of the 10% solution.
10 Check
Let's check in the story.
10 litres and 10 litres make 20 litres, like the story says.
The sugar: 5 plus 1 is 6 litres.
That is 30% of 20, 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.