Step by step
The question
Question: Lentils cost $3 per kg and walnuts cost $15 per kg.
A shop mixes them to make 12 kg of soup mix that costs $6 per kg.
How many kg of each does it use?
1 Find what we know
What do we know?
Lentils: $3 per kg.
Walnuts: $15 per kg.
They make 12 kg of soup mix, at $6 per kg.
What do we want?
How many kg of walnuts, and how many of lentils.
2 Name the unknown
We don't know how many kg of walnuts.
So we call it x.
The rest is lentils.
12 kg in all, take away the x kg of walnuts: 12 − x.
Let's put it in a table: how much, the price, and the cost.
Walnuts: x kg at $15.
Lentils: 12 − x kg at $3.
Cost = how much times the price.
15 times x is 15x.
3 times all of 12 − x is 3(12 − x).
We keep 12 − x together in brackets.
The mix: 12 kg at $6.
12 times 6 is $72.
3 Write the equation
In words: the walnuts cost plus the lentils cost makes the cost of the mix.
So the equation is 15x + 3(12 − x) = 72.
Now we solve it.
Careful!
Many people take half of each.
But half and half costs (3 + 15) ÷ 2 = $9 per kg, not $6.
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.
15x − 3x = 12x.
Now we have 12x + 36 = 72.
We want x by itself.
Two things are stuck to x: times 12 and + 36.
The + 36 is on the outside, so it comes off first.
Then the times 12.
6 Remove the + 36
To remove + 36, we take away 36.
We do it on both sides, so they stay equal.
+ 36 and − 36 make zero.
Cross them out.
72 − 36 = 36.
Now we have 12x = 36.
7 Remove the 12
12x means 12 times x.
We want just x.
To remove times 12, we divide by 12.
We do it on both sides, so they stay equal.
12 divided by 12 is 1.
So just x is left.
36 divided by 12 is 3.
So x = 3!
8 Answer the question
Lentils: 12 − 3 = 9.
So 3 kg of walnuts and 9 kg of lentils.
9 Check
Let's check in the story.
3 kg and 9 kg make 12 kg, like the story says.
The cost: $45 plus $27 is $72.
That is 12 kg at $6, like the story says.
Does it make sense?
The mix costs between the two prices, $3 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.