Step by step
The question
Question: Coffee costs $16 per kg and cocoa costs $6 per kg.
How many kg of cocoa must be mixed with 5 kg of coffee to make a mix that costs $10 per kg?
1 Find what we know
What do we know?
Coffee: $16 per kg.
Cocoa: $6 per kg.
We use 5 kg of coffee.
The mix costs $10 per kg.
What do we want?
How many kg of cocoa.
2 Name the unknown
We don't know how many kg of cocoa.
So we call it x.
The mix has both: x kg of cocoa and 5 kg of coffee.
So it is x + 5 kg.
Careful!
The mix has the cocoa too.
So it weighs x + 5 kg, not 5 kg.
Let's put it in a table: how much, the price, and the cost.
Cocoa: x kg at $6.
Coffee: 5 kg at $16.
Cost = how much times the price.
6 times x is 6x.
5 times 16 is $80.
The mix: x + 5 kg at $10.
Its cost is 10(x + 5): all of x + 5 is times 10.
3 Write the equation
In words: the cocoa cost plus the coffee cost makes the cost of the mix.
So the equation is 6x + 80 = 10(x + 5).
Now we solve it.
4 Open the brackets
The 10 in front of the bracket multiplies every part inside.
So we write 10 times each part.
Work out each multiplication.
5 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the 6x from the left side.
To remove 6x, we take away 6x.
We do it on both sides, so they stay equal.
6x and − 6x make zero.
Cross them out.
10x − 6x = 4x.
Now we have 80 = 4x + 50.
Let's swap the two sides, so x is on the left.
We want x by itself.
Two things are stuck to x: times 4 and + 50.
The + 50 is on the outside, so it comes off first.
Then the times 4.
6 Remove the + 50
To remove + 50, we take away 50.
We do it on both sides, so they stay equal.
+ 50 and − 50 make zero.
Cross them out.
80 − 50 = 30.
Now we have 4x = 30.
7 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.
We can make 30/4 simpler.
It is the same as 15/2.
So x = 15/2!
8 Answer the question
15/2 is 7.5.
So we need 7.5 kg of cocoa.
9 Check
Let's check in the story.
The cost: $45 plus $80 is $125.
The mix: 12.5 kg at $10 is $125.
The same cost, like the story says.
Does it make sense?
The mix costs between the two prices, $6 and $16.
That makes sense.
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.