Step by step
The question
Question: Raisins cost $2 per kg and nuts cost $6 per kg.
How many kg of raisins must be mixed with 4 kg of nuts to make a mix that costs $3 per kg?
1 Find what we know
What do we know?
Raisins: $2 per kg.
Nuts: $6 per kg.
We use 4 kg of nuts.
The mix costs $3 per kg.
What do we want?
How many kg of raisins.
2 Name the unknown
We don't know how many kg of raisins.
So we call it x.
The mix has both: x kg of raisins and 4 kg of nuts.
So it is x + 4 kg.
Careful!
The mix has the raisins too.
So it weighs x + 4 kg, not 4 kg.
Let's put it in a table: how much, the price, and the cost.
Raisins: x kg at $2.
Nuts: 4 kg at $6.
Cost = how much times the price.
2 times x is 2x.
4 times 6 is $24.
The mix: x + 4 kg at $3.
Its cost is 3(x + 4): all of x + 4 is times 3.
3 Write the equation
In words: the raisins cost plus the nuts cost makes the cost of the mix.
So the equation is 2x + 24 = 3(x + 4).
Now we solve it.
4 Open the brackets
The 3 in front of the bracket multiplies every part inside.
So we write 3 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 2x from the left side.
To remove 2x, we take away 2x.
We do it on both sides, so they stay equal.
2x and − 2x make zero.
Cross them out.
3x − 2x = x.
Now we have 24 = x + 12.
Let's swap the two sides, so x is on the left.
6 Remove the + 12
We want x by itself.
The + 12 is in the way.
To remove + 12, we take away 12.
We do it on both sides, so they stay equal.
+ 12 and − 12 make zero.
Cross them out.
24 − 12 = 12.
So x = 12!
7 Answer the question
So we need 12 kg of raisins.
8 Check
Let's check in the story.
The cost: $24 plus $24 is $48.
The mix: 16 kg at $3 is $48.
The same cost, like the story says.
Does it make sense?
The mix costs between the two prices, $2 and $6.
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.