Step by step
The question
Question: Ella has 15 litres of an 8% sugar solution.
How many litres of a 20% sugar solution must she add to make a 12% sugar solution?
1 Find what we know
What do we know?
We have 15 litres of a 8% solution.
The mix must be 12% sugar.
What do we want?
How many litres of the 20% solution.
2 Name the unknown
We don't know how many litres of the 20% solution.
So we call it x.
The mix has both: x litres and 15 litres.
So it is x + 15 litres.
Careful!
The mix has the 20% solution in it too.
So it is x + 15 litres, not 15 litres.
Let's put it in a table: how much, the sugar part as a decimal, and the sugar.
20% is 20/100: 0.2.
8% is 8/100: 0.08.
Sugar = how much times the part.
x times 0.2 is 0.2x.
15 times 0.08 is 1.2.
The mix: x + 15 litres at 0.12.
Its sugar is 0.12(x + 15): all of x + 15 is times 0.12.
3 Write the equation
In words: the sugar from the two solutions makes the sugar in the mix.
So the equation is 0.2x + 1.2 = 0.12(x + 15).
Now we solve it.
4 Remove the decimals
Decimals are hard to work with.
Let's get rid of them.
0.12 has 2 digits after the point.
Times 100 moves the point 2 places to the right.
Then every number is a whole number.
We multiply every part by 100.
We do it on both sides, so they stay equal.
100 × 0.2x = 20x.
100 × 1.2 = 120.
100 × 0.12(x + 15) = 12(x + 15).
Write the answers.
No more decimals!
5 Open the brackets
The 12 in front of the bracket multiplies every part inside.
So we write 12 times each part.
12 times x is 12x.
12 times 15 is 180.
6 Get x on one side
There is x on both sides.
We want x on one side only.
The numbers in front of x are 12 and 20.
We remove the smaller one, 12x.
Then the x part that is left is plus.
A plus x part is easier to work with.
So we remove the 12x from the right side.
To remove 12x, we take away 12x.
We do it on both sides, so they stay equal.
12x and − 12x make zero.
Cross them out.
20x − 12x = 8x.
Now we have 8x + 120 = 180.
We want x by itself.
Two things are stuck to x: times 8 and + 120.
First x was times 8.
Then + 120.
To take them off, we go backwards.
First we remove the + 120.
Then the times 8.
7 Remove the + 120
To remove + 120, we take away 120.
We do it on both sides, so they stay equal.
+ 120 and − 120 make zero.
Cross them out.
180 − 120 = 60.
Now we have 8x = 60.
8 Remove the 8
8x means 8 times x.
We want just x.
To remove times 8, we divide by 8.
We do it on both sides, so they stay equal.
8 divided by 8 is 1.
So just x is left.
60 divided by 8 is 7.5.
So x = 7.5.
9 Answer the question
15/2 is 7.5.
So we need 7.5 litres of the 20% solution.
10 Check
Let's check in the story.
The sugar: 1.5 plus 1.2 is 2.7 litres.
12% of the 22.5 litres of mix is 2.7 too, like the story says.
Does it make sense?
12% is nearer to 8%.
So there is more of the 8% solution: 15 litres is more than 7.5 litres.
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.