Step by step
The question
Question: Peanuts cost $5 per kg and pecans cost $15 per kg.
A shop mixes them to make 25 kg of nut mix that costs $9 per kg.
How many kg of each does it use?
1 Find what we know
What do we know?
Peanuts: $5 per kg.
Pecans: $15 per kg.
They make 25 kg of nut mix, at $9 per kg.
What do we want?
How many kg of pecans, and how many of peanuts.
2 Name the unknown
We don't know how many kg of pecans.
So we call it x.
The rest is peanuts.
25 kg in all, take away the x kg of pecans: 25 − x.
Let's put it in a table: how much, the price, and the cost.
Pecans: x kg at $15.
Peanuts: 25 − x kg at $5.
Cost = how much times the price.
15 times x is 15x.
5 times all of 25 − x is 5(25 − x).
We keep 25 − x together in brackets.
The mix: 25 kg at $9.
25 times 9 is $225.
3 Write the equation
In words: the pecans cost plus the peanuts cost makes the cost of the mix.
So the equation is 15x + 5(25 − x) = 225.
Now we solve it.
Careful!
Many people take half of each.
But half and half costs (5 + 15) ÷ 2 = $10 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.
15x − 5x = 10x.
Now we have 10x + 125 = 225.
We want x by itself.
Two things are stuck to x: times 10 and + 125.
The + 125 is on the outside, so it comes off first.
Then the times 10.
6 Remove the + 125
To remove + 125, we take away 125.
We do it on both sides, so they stay equal.
+ 125 and − 125 make zero.
Cross them out.
225 − 125 = 100.
Now we have 10x = 100.
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.
100 divided by 10 is 10.
So x = 10!
8 Answer the question
Peanuts: 25 − 10 = 15.
So 10 kg of pecans and 15 kg of peanuts.
9 Check
Let's check in the story.
10 kg and 15 kg make 25 kg, like the story says.
The cost: $150 plus $75 is $225.
That is 25 kg at $9, like the story says.
Does it make sense?
The mix costs between the two prices, $5 and $15.
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.