Peanuts cost $4 per kg and cashews cost $10 per kg. A shop mixes… Easy

The question

Question: Peanuts cost $4 per kg and cashews cost $10 per kg.

A shop mixes them to make 12 kg of nut mix that costs $6 per kg.

How many kg of each does it use?

1 Find what we know

What do we know?

Peanuts: $4 per kg.

Cashews: $10 per kg.

They make 12 kg of nut mix, at $6 per kg.

What do we want?

How many kg of cashews, and how many of peanuts.

2 Name the unknown

We don't know how many kg of cashews.

So we call it x.

The rest is peanuts.

12 kg in all, take away the x kg of cashews: 12 − x.

Let's put it in a table: how much, the price, and the cost.

Cashews: x kg at $10.

Peanuts: 12 − x kg at $4.

Cost = how much times the price.

10 times x is 10x.

4 times all of 12 − x is 4(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 cashews cost plus the peanuts cost makes the cost of the mix.

So the equation is 10x + 4(12 − x) = 72.

Now we solve it.

Careful!

Many people take half of each.

But half and half costs (4 + 10) ÷ 2 = $7 per kg, not $6.

4 Open the brackets

The 4 in front of the bracket multiplies every part inside.

So we write 4 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.

10x − 4x = 6x.

Now we have 6x + 48 = 72.

We want x by itself.

Two things are stuck to x: times 6 and + 48.

The + 48 is on the outside, so it comes off first.

Then the times 6.

6 Remove the + 48

To remove + 48, we take away 48.

We do it on both sides, so they stay equal.

+ 48 and − 48 make zero.

Cross them out.

72 − 48 = 24.

Now we have 6x = 24.

7 Remove the 6

6x means 6 times x.

We want just x.

To remove times 6, we divide by 6.

We do it on both sides, so they stay equal.

6 divided by 6 is 1.

So just x is left.

24 divided by 6 is 4.

So x = 4!

8 Answer the question

Peanuts: 12 − 4 = 8.

So 4 kg of cashews and 8 kg of peanuts.

9 Check

Let's check in the story.

4 kg and 8 kg make 12 kg, like the story says.

The cost: $40 plus $32 is $72.

That is 12 kg at $6, like the story says.

Does it make sense?

The mix costs between the two prices, $4 and $10.

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.

▶ Watch it solved

Lentils cost $3 per kg and walnuts cost $15 per kg. A shop mixes… Medium

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.

▶ Watch it solved

Raisins cost $2 per kg and nuts cost $6 per kg. How many kg of… Harder

The question

Question: Raisins cost $2 per kg and nuts cost $6 per kg.

How many kg of raisins must be mixed with 4 kg of nuts to make a mix that costs $3 per kg?

1 Find what we know

What do we know?

Raisins: $2 per kg.

Nuts: $6 per kg.

We use 4 kg of nuts.

The mix costs $3 per kg.

What do we want?

How many kg of raisins.

2 Name the unknown

We don't know how many kg of raisins.

So we call it x.

The mix has both: x kg of raisins and 4 kg of nuts.

So it is x + 4 kg.

Careful!

The mix has the raisins too.

So it weighs x + 4 kg, not 4 kg.

Let's put it in a table: how much, the price, and the cost.

Raisins: x kg at $2.

Nuts: 4 kg at $6.

Cost = how much times the price.

2 times x is 2x.

4 times 6 is $24.

The mix: x + 4 kg at $3.

Its cost is 3(x + 4): all of x + 4 is times 3.

3 Write the equation

In words: the raisins cost plus the nuts cost makes the cost of the mix.

So the equation is 2x + 24 = 3(x + 4).

Now we solve it.

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 Get x on one side

There is x on both sides.

We want x on one side only.

So we remove the 2x from the left side.

To remove 2x, we take away 2x.

We do it on both sides, so they stay equal.

2x and − 2x make zero.

Cross them out.

3x − 2x = x.

Now we have 24 = x + 12.

Let's swap the two sides, so x is on the left.

6 Remove the + 12

We want x by itself.

The + 12 is in the way.

To remove + 12, we take away 12.

We do it on both sides, so they stay equal.

+ 12 and − 12 make zero.

Cross them out.

24 − 12 = 12.

So x = 12!

7 Answer the question

So we need 12 kg of raisins.

8 Check

Let's check in the story.

The cost: $24 plus $24 is $48.

The mix: 16 kg at $3 is $48.

The same cost, like the story says.

Does it make sense?

The mix costs between the two prices, $2 and $6.

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.

▶ Watch it solved