Mia has 6 litres of a 50% salt solution. How many litres of a 20%… Easy

The question

Question: Mia has 6 litres of a 50% salt solution.

How many litres of a 20% salt solution must she add to make a 30% salt solution?

1 Find what we know

What do we know?

We have 6 litres of a 50% solution.

The mix must be 30% salt.

What do we want?

How many litres of the 20% solution.

2 Name the unknown

We don't know how many litres of the 20% solution.

So we call it x.

The mix has both: x litres and 6 litres.

So it is x + 6 litres.

Careful!

The mix has the 20% solution in it too.

So it is x + 6 litres, not 6 litres.

Let's put it in a table: how much, the salt part as a decimal, and the salt.

20% is 20/100: 0.2.

50% is 0.5.

Salt = how much times the part.

x times 0.2 is 0.2x.

6 times 0.5 is 3.

The mix: x + 6 litres at 0.3.

Its salt is 0.3(x + 6): all of x + 6 is times 0.3.

3 Write the equation

In words: the salt from the two solutions makes the salt in the mix.

So the equation is 0.2x + 3 = 0.3(x + 6).

Now we solve it.

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 × 0.2x = 2x.

10 × 3 = 30.

10 × 0.3(x + 6) = 3(x + 6).

Write the answers.

No more decimals!

5 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.

6 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 30 = x + 18.

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

7 Remove the + 18

We want x by itself.

The + 18 is in the way.

To remove + 18, we take away 18.

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

+ 18 and − 18 make zero.

Cross them out.

30 − 18 = 12.

So x = 12!

8 Answer the question

So we need 12 litres of the 20% solution.

9 Check

Let's check in the story.

The salt: 2.4 plus 3 is 5.4 litres.

30% of the 18 litres of mix is 5.4 too, like the story says.

Does it make sense?

The mix is between the two solutions.

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

A lab needs 30 litres of a 25% acid solution. It mixes a 10% acid… Medium

The question

Question: A lab needs 30 litres of a 25% acid solution.

It mixes a 10% acid solution with a 40% acid solution.

How many litres of each does it use?

1 Find what we know

What do we know?

The mix: 30 litres that is 25% acid.

It is made from a 10% and a 40% solution.

What do we want?

How many litres of each.

2 Name the unknown

We don't know how many litres of the 40% solution.

So we call it x.

The rest is the 10% solution.

30 litres in all, take away x: 30 − x.

Let's put it in a table: how much, the acid part as a decimal, and the acid.

40% is 0.4.

10% is 0.1.

Acid = how much times the part: 0.4x.

0.1 times all of 30 − x: 0.1(30 − x).

We keep 30 − x together in brackets.

The mix: 30 litres at 0.25.

30 times 0.25 is 7.5 litres of acid.

3 Write the equation

In words: the acid from the two solutions makes the acid in the mix.

So the equation is 0.4x + 0.1(30 − x) = 7.5.

Now we solve it.

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 × 0.4x = 4x.

10 × 0.1(30 − x) = (30 − x).

10 × 7.5 = 75.

Write the answers.

No more decimals!

5 Open the brackets

A bracket with a plus or nothing in front can just be removed.

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.

4x − x = 3x.

Now we have 3x + 30 = 75.

We want x by itself.

Two things are stuck to x: times 3 and + 30.

The + 30 is on the outside, so it comes off first.

Then the times 3.

7 Remove the + 30

To remove + 30, we take away 30.

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

+ 30 and − 30 make zero.

Cross them out.

75 − 30 = 45.

Now we have 3x = 45.

8 Remove the 3

3x means 3 times x.

We want just x.

To remove times 3, we divide by 3.

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

3 divided by 3 is 1.

So just x is left.

45 divided by 3 is 15.

So x = 15!

9 Answer the question

The 10% solution: 30 − 15 = 15.

So 15 litres of the 40% solution and 15 litres of the 10% solution.

10 Check

Let's check in the story.

15 litres and 15 litres make 30 litres, like the story says.

The acid: 6 plus 1.5 is 7.5 litres.

That is 25% of 30, like the story says.

Does it make sense?

The mix is between the two solutions.

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

A lab needs 100 litres of a 18% salt solution. It mixes a 10% salt… Harder

The question

Question: A lab needs 100 litres of a 18% salt solution.

It mixes a 10% salt solution with a 30% salt solution.

How many litres of each does it use?

1 Find what we know

What do we know?

The mix: 100 litres that is 18% salt.

It is made from a 10% and a 30% solution.

What do we want?

How many litres of each.

2 Name the unknown

We don't know how many litres of the 30% solution.

So we call it x.

The rest is the 10% solution.

100 litres in all, take away x: 100 − x.

Let's put it in a table: how much, the salt part as a decimal, and the salt.

30% is 0.3.

10% is 0.1.

Salt = how much times the part: 0.3x.

0.1 times all of 100 − x: 0.1(100 − x).

We keep 100 − x together in brackets.

The mix: 100 litres at 0.18.

100 times 0.18 is 18 litres of salt.

3 Write the equation

In words: the salt from the two solutions makes the salt in the mix.

So the equation is 0.3x + 0.1(100 − x) = 18.

Now we solve it.

Careful!

Many people take half of each.

But half and half gives (30 + 10) ÷ 2 = 20%, not 18%.

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 × 0.3x = 3x.

10 × 0.1(100 − x) = (100 − x).

10 × 18 = 180.

Write the answers.

No more decimals!

5 Open the brackets

A bracket with a plus or nothing in front can just be removed.

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.

3x − x = 2x.

Now we have 2x + 100 = 180.

We want x by itself.

Two things are stuck to x: times 2 and + 100.

The + 100 is on the outside, so it comes off first.

Then the times 2.

7 Remove the + 100

To remove + 100, we take away 100.

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

+ 100 and − 100 make zero.

Cross them out.

180 − 100 = 80.

Now we have 2x = 80.

8 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.

80 divided by 2 is 40.

So x = 40!

9 Answer the question

The 10% solution: 100 − 40 = 60.

So 40 litres of the 30% solution and 60 litres of the 10% solution.

10 Check

Let's check in the story.

40 litres and 60 litres make 100 litres, like the story says.

The salt: 12 plus 6 is 18 litres.

That is 18% of 100, like the story says.

Does it make sense?

The mix is between the two solutions.

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