Step by step
The question
Question: Sweets cost $5 per kg and chocolates cost $11 per kg.
A shop mixes them to make 6 kg of party mix that costs $9 per kg.
How many kg of each does it use?
1 Find what we know
What do we know?
Sweets: $5 per kg.
Chocolates: $11 per kg.
They make 6 kg of party mix, at $9 per kg.
What do we want?
How many kg of chocolates, and how many of sweets.
2 Name the unknown
We don't know how many kg of chocolates.
So we call it x.
The rest is sweets.
6 kg in all, take away the x kg of chocolates: 6 − x.
Let's put it in a table: how much, the price, and the cost.
Chocolates: x kg at $11.
Sweets: 6 − x kg at $5.
Cost = how much times the price.
11 times x is 11x.
5 times all of 6 − x is 5(6 − x).
We keep 6 − x together in brackets.
The mix: 6 kg at $9.
6 times 9 is $54.
3 Write the equation
In words: the chocolates cost plus the sweets cost makes the cost of the mix.
So the equation is 11x + 5(6 − x) = 54.
Now we solve it.
Careful!
Many people take half of each.
But half and half costs (5 + 11) ÷ 2 = $8 per kg, not $9.
4 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.
5 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.
11x − 5x = 6x.
Now we have 6x + 30 = 54.
We want x by itself.
Two things are stuck to x: times 6 and + 30.
The + 30 is on the outside, so it comes off first.
Then the times 6.
6 Remove the + 30
To remove + 30, we take away 30.
We do it on both sides, so they stay equal.
+ 30 and − 30 make zero.
Cross them out.
54 − 30 = 24.
Now we have 6x = 24.
7 Remove the 6
6x means 6 times x.
We want just x.
To remove times 6, we divide by 6.
We do it on both sides, so they stay equal.
6 divided by 6 is 1.
So just x is left.
24 divided by 6 is 4.
So x = 4!
8 Answer the question
Sweets: 6 − 4 = 2.
So 4 kg of chocolates and 2 kg of sweets.
9 Check
Let's check in the story.
4 kg and 2 kg make 6 kg, like the story says.
The cost: $44 plus $10 is $54.
That is 6 kg at $9, like the story says.
Does it make sense?
The mix costs between the two prices, $5 and $11.
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.