Step by step
The question
Question: A shop needs 50 litres of a 12% juice solution.
It mixes a 5% juice solution with a 15% juice solution.
How many litres of each does it use?
1 Find what we know
What do we know?
The mix: 50 litres that is 12% juice.
It is made from a 5% and a 15% solution.
What do we want?
How many litres of each.
2 Name the unknown
We don't know how many litres of the 15% solution.
So we call it x.
The rest is the 5% solution.
50 litres in all, take away x: 50 − x.
Let's put it in a table: how much, the juice part as a decimal, and the juice.
15% is 0.15.
5% is 0.05.
Juice = how much times the part: 0.15x.
0.05 times all of 50 − x: 0.05(50 − x).
We keep 50 − x together in brackets.
The mix: 50 litres at 0.12.
50 times 0.12 is 6 litres of juice.
3 Write the equation
In words: the juice from the two solutions makes the juice in the mix.
So the equation is 0.15x + 0.05(50 − x) = 6.
Now we solve it.
Careful!
Many people take half of each.
But half and half gives (15 + 5) ÷ 2 = 10%, not 12%.
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 × 0.05(50 − x) = 5(50 − x).
100 × 6 = 600.
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 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.
15x − 5x = 10x.
Now we have 10x + 250 = 600.
We want x by itself.
Two things are stuck to x: times 10 and + 250.
The + 250 is on the outside, so it comes off first.
Then the times 10.
7 Remove the + 250
To remove + 250, we take away 250.
We do it on both sides, so they stay equal.
+ 250 and − 250 make zero.
Cross them out.
600 − 250 = 350.
Now we have 10x = 350.
8 Remove the 10
10x means 10 times x.
We want just x.
To remove times 10, we divide by 10.
We do it on both sides, so they stay equal.
10 divided by 10 is 1.
So just x is left.
350 divided by 10 is 35.
So x = 35!
9 Answer the question
The 5% solution: 50 − 35 = 15.
So 35 litres of the 15% solution and 15 litres of the 5% solution.
10 Check
Let's check in the story.
35 litres and 15 litres make 50 litres, like the story says.
The juice: 5.25 plus 0.75 is 6 litres.
That is 12% of 50, 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.