Step by step
The question
Question: Apples cost $3 per kg and grapes cost $7 per kg.
A shop mixes them to make 8 kg of fruit salad that costs $4 per kg.
How many kg of each does it use?
1 Find what we know
What do we know?
Apples: $3 per kg.
Grapes: $7 per kg.
They make 8 kg of fruit salad, at $4 per kg.
What do we want?
How many kg of grapes, and how many of apples.
2 Name the unknown
We don't know how many kg of grapes.
So we call it x.
The rest is apples.
8 kg in all, take away the x kg of grapes: 8 − x.
Let's put it in a table: how much, the price, and the cost.
Grapes: x kg at $7.
Apples: 8 − x kg at $3.
Cost = how much times the price.
7 times x is 7x.
3 times all of 8 − x is 3(8 − x).
We keep 8 − x together in brackets.
The mix: 8 kg at $4.
8 times 4 is $32.
3 Write the equation
In words: the grapes cost plus the apples cost makes the cost of the mix.
So the equation is 7x + 3(8 − x) = 32.
Now we solve it.
Careful!
Many people take half of each.
But half and half costs (3 + 7) ÷ 2 = $5 per kg, not $4.
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.
7x − 3x = 4x.
Now we have 4x + 24 = 32.
We want x by itself.
Two things are stuck to x: times 4 and + 24.
The + 24 is on the outside, so it comes off first.
Then the times 4.
6 Remove the + 24
To remove + 24, we take away 24.
We do it on both sides, so they stay equal.
+ 24 and − 24 make zero.
Cross them out.
32 − 24 = 8.
Now we have 4x = 8.
7 Remove the 4
4x means 4 times x.
We want just x.
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4 divided by 4 is 1.
So just x is left.
8 divided by 4 is 2.
So x = 2!
8 Answer the question
Apples: 8 − 2 = 6.
So 2 kg of grapes and 6 kg of apples.
9 Check
Let's check in the story.
2 kg and 6 kg make 8 kg, like the story says.
The cost: $14 plus $18 is $32.
That is 8 kg at $4, like the story says.
Does it make sense?
The mix costs between the two prices, $3 and $7.
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.