Step by step
The question
Question: Rice costs $2 per kg and beans cost $4 per kg.
A shop mixes them to make 10 kg of mix that costs $5 per kg.
How many kg of each does it use?
1 Find what we know
What do we know?
Rice: $2 per kg.
Beans: $4 per kg.
They make 10 kg of mix, at $5 per kg.
What do we want?
How many kg of beans, and how many of rice.
2 Name the unknown
We don't know how many kg of beans.
So we call it x.
The rest is rice.
10 kg in all, take away the x kg of beans: 10 − x.
Let's put it in a table: how much, the price, and the cost.
Beans: x kg at $4.
Rice: 10 − x kg at $2.
Cost = how much times the price.
4 times x is 4x.
2 times all of 10 − x is 2(10 − x).
We keep 10 − x together in brackets.
The mix: 10 kg at $5.
10 times 5 is $50.
3 Write the equation
In words: the beans cost plus the rice cost makes the cost of the mix.
So the equation is 4x + 2(10 − x) = 50.
Now we solve it.
4 Open the brackets
The 2 in front of the bracket multiplies every part inside.
So we write 2 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.
4x − 2x = 2x.
Now we have 2x + 20 = 50.
We want x by itself.
Two things are stuck to x: times 2 and + 20.
The + 20 is on the outside, so it comes off first.
Then the times 2.
6 Remove the + 20
To remove + 20, we take away 20.
We do it on both sides, so they stay equal.
+ 20 and − 20 make zero.
Cross them out.
50 − 20 = 30.
Now we have 2x = 30.
7 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
30 divided by 2 is 15.
So x = 15!
8 Answer the question
x = 15 does not work.
A mix always costs between the two prices.
So there is no answer.
Let's look back.
First: find what we know.
Then: name the unknown.
Then: write the equation.
Then: solve it.
Last: answer the question.