Step by step
The question
Question: Bran costs $3 per kg and nuts cost $13 per kg.
A shop mixes them to make 5 kg of cereal that costs $7 per kg.
How many kg of each does it use?
1 Find what we know
What do we know?
Bran: $3 per kg.
Nuts: $13 per kg.
They make 5 kg of cereal, at $7 per kg.
What do we want?
How many kg of nuts, and how many of bran.
2 Name the unknown
We don't know how many kg of nuts.
So we call it x.
The rest is bran.
5 kg in all, take away the x kg of nuts: 5 − x.
Let's put it in a table: how much, the price, and the cost.
Nuts: x kg at $13.
Bran: 5 − x kg at $3.
Cost = how much times the price.
13 times x is 13x.
3 times all of 5 − x is 3(5 − x).
We keep 5 − x together in brackets.
The mix: 5 kg at $7.
5 times 7 is $35.
3 Write the equation
In words: the nuts cost plus the bran cost makes the cost of the mix.
So the equation is 13x + 3(5 − x) = 35.
Now we solve it.
Careful!
Many people take half of each.
But half and half costs (3 + 13) ÷ 2 = $8 per kg, not $7.
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 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.
13x − 3x = 10x.
Now we have 10x + 15 = 35.
We want x by itself.
Two things are stuck to x: times 10 and + 15.
The + 15 is on the outside, so it comes off first.
Then the times 10.
6 Remove the + 15
To remove + 15, we take away 15.
We do it on both sides, so they stay equal.
+ 15 and − 15 make zero.
Cross them out.
35 − 15 = 20.
Now we have 10x = 20.
7 Remove the 10
10x means 10 times x.
We want just x.
To remove times 10, we divide by 10.
We do it on both sides, so they stay equal.
10 divided by 10 is 1.
So just x is left.
20 divided by 10 is 2.
So x = 2!
8 Answer the question
Bran: 5 − 2 = 3.
So 2 kg of nuts and 3 kg of bran.
9 Check
Let's check in the story.
2 kg and 3 kg make 5 kg, like the story says.
The cost: $26 plus $9 is $35.
That is 5 kg at $7, like the story says.
Does it make sense?
The mix costs between the two prices, $3 and $13.
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.