Step by step
The question
Question: Flour costs $1.50 per kg and sugar costs $3 per kg.
A baker mixes them to make 10 kg of mix that costs $2.10 per kg.
How many kg of each does it use?
1 Find what we know
What do we know?
Flour: $1.50 per kg.
Sugar: $3 per kg.
They make 10 kg of mix, at $2.10 per kg.
What do we want?
How many kg of sugar, and how many of flour.
2 Name the unknown
We don't know how many kg of sugar.
So we call it x.
The rest is flour.
10 kg in all, take away the x kg of sugar: 10 − x.
Let's put it in a table: how much, the price, and the cost.
Sugar: x kg at $3.
Flour: 10 − x kg at $1.50.
Cost = how much times the price.
3 times x is 3x.
1.5 times all of 10 − x is 1.5(10 − x).
We keep 10 − x together in brackets.
The mix: 10 kg at $2.10.
10 times 2.1 is $21.
3 Write the equation
In words: the sugar cost plus the flour cost makes the cost of the mix.
So the equation is 3x + 1.5(10 − x) = 21.
Now we solve it.
Careful!
Many people take half of each.
But half and half costs (1.5 + 3) ÷ 2 = $2.25 per kg, not $2.10.
4 Remove the decimals
Decimals are hard to work with.
Let's get rid of them.
We multiply every part by 10.
We do it on both sides, so they stay equal.
Times 10 moves the point 1 place to the right.
10 × 3x = 30x.
10 × 1.5(10 − x) = 15(10 − x).
10 × 21 = 210.
Write the answers.
No more decimals!
5 Open the brackets
The 15 in front of the bracket multiplies every part inside.
So we write 15 times each part.
Work out each multiplication.
6 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.
30x − 15x = 15x.
Now we have 15x + 150 = 210.
We want x by itself.
Two things are stuck to x: times 15 and + 150.
The + 150 is on the outside, so it comes off first.
Then the times 15.
7 Remove the + 150
To remove + 150, we take away 150.
We do it on both sides, so they stay equal.
+ 150 and − 150 make zero.
Cross them out.
210 − 150 = 60.
Now we have 15x = 60.
8 Remove the 15
15x means 15 times x.
We want just x.
To remove times 15, we divide by 15.
We do it on both sides, so they stay equal.
15 divided by 15 is 1.
So just x is left.
60 divided by 15 is 4.
So x = 4!
9 Answer the question
Flour: 10 − 4 = 6.
So 4 kg of sugar and 6 kg of flour.
10 Check
Let's check in the story.
4 kg and 6 kg make 10 kg, like the story says.
The cost: $12 plus $9 is $21.
That is 10 kg at $2.10, like the story says.
Does it make sense?
The mix costs between the two prices, $1.50 and $3.
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.