Step by step
The question
Question: A cook needs 24 litres of a 25% sugar solution.
He mixes a 15% sugar solution with a 45% sugar solution.
How many litres of each does he use?
1 Find what we know
What do we know?
The mix: 24 litres that is 25% sugar.
It is made from a 15% and a 45% solution.
What do we want?
How many litres of each.
2 Name the unknown
We don't know how many litres of the 45% solution.
So we call it x.
The rest is the 15% solution.
24 litres in all, take away x: 24 − x.
Let's put it in a table: how much, the sugar part as a decimal, and the sugar.
45% is 0.45.
15% is 0.15.
Sugar = how much times the part: 0.45x.
0.15 times all of 24 − x: 0.15(24 − x).
We keep 24 − x together in brackets.
The mix: 24 litres at 0.25.
24 times 0.25 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 0.45x + 0.15(24 − x) = 6.
Now we solve it.
Careful!
Many people take half of each.
But half and half gives (45 + 15) ÷ 2 = 30%, not 25%.
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 × 0.45x = 45x.
100 × 0.15(24 − x) = 15(24 − x).
100 × 6 = 600.
Write the answers.
No more decimals!
5 Open the brackets
The 15 in front of the bracket multiplies every part inside.
So we write 15 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.
45x − 15x = 30x.
Now we have 30x + 360 = 600.
We want x by itself.
Two things are stuck to x: times 30 and + 360.
The + 360 is on the outside, so it comes off first.
Then the times 30.
7 Remove the + 360
To remove + 360, we take away 360.
We do it on both sides, so they stay equal.
+ 360 and − 360 make zero.
Cross them out.
600 − 360 = 240.
Now we have 30x = 240.
8 Remove the 30
30x means 30 times x.
We want just x.
To remove times 30, we divide by 30.
We do it on both sides, so they stay equal.
30 divided by 30 is 1.
So just x is left.
240 divided by 30 is 8.
So x = 8!
9 Answer the question
The 15% solution: 24 − 8 = 16.
So 8 litres of the 45% solution and 16 litres of the 15% solution.
10 Check
Let's check in the story.
8 litres and 16 litres make 24 litres, like the story says.
The sugar: 3.6 plus 2.4 is 6 litres.
That is 25% of 24, 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.