Write 0.777... as a fraction Easy

The question

Question: write 0.777 repeating as a fraction.

Look at the digits.

The 7 repeats for ever.

1 Call it x

We do not know the fraction yet.

So we call the number x: x = 0.777 repeating.

Careful!

It is not 7/10.

That is 0.7, which stops.

Ours goes on for ever.

2 Move the point

1 digit repeats.

So we multiply both sides by 10.

That moves the point 1 place to the right.

3 Take away

Now we take the first one away from the new one.

Both sides, so they stay equal.

After the point, both go on with 7 for ever.

The tails are exactly the same, so they make zero.

Cross them out.

10x minus x is 9x.

7 minus 0 is 7.

4 Remove the 9

9x means 9 times x.

We want just x.

To remove times 9, we divide by 9.

9 divided by 9 is 1.

So just x is left.

So x = 7/9.

So 0.777 repeating is 7/9.

5 Check

Let's check: 7 divided by 9.

We get 0.777, and the digits go on the same way.

That is 0.777 repeating.

So the answer is right!

Let's look back.

First: call it x.

Then: move the point.

Then: take away.

Then: remove the 9.

Last: check.

▶ Watch it solved

Write 0.272727... as a fraction Medium

The question

Question: write 0.2727 repeating as a fraction.

Look at the digits.

The block 2 7 repeats for ever.

1 Call it x

We do not know the fraction yet.

So we call the number x: x = 0.2727 repeating.

Careful!

It is not 27/100.

That is 0.27, which stops.

Ours goes on for ever.

2 Move the point

2 digits repeat.

So we multiply both sides by 100.

That moves the point 2 places to the right, one whole block.

3 Take away

Now we take the first one away from the new one.

Both sides, so they stay equal.

After the point, both go on with 2 7 for ever.

The tails are exactly the same, so they make zero.

Cross them out.

100x minus x is 99x.

27 minus 0 is 27.

4 Remove the 99

99x means 99 times x.

We want just x.

To remove times 99, we divide by 99.

99 divided by 99 is 1.

So just x is left.

27 and 99 can both be divided by 9.

We can make 27/99 simpler.

It is the same as 3/11.

So x = 3/11.

So 0.2727 repeating is 3/11.

5 Check

Let's check: 3 divided by 11.

We get 0.2727, and the digits go on the same way.

That is 0.2727 repeating.

So the answer is right!

Let's look back.

First: call it x.

Then: move the point.

Then: take away.

Then: remove the 99.

Last: check.

▶ Watch it solved

Write 0.1666... as a fraction Harder

The question

Question: write 0.1666 repeating as a fraction.

Look at the digits.

1 comes once.

Then 6 repeats for ever.

1 Call it x

We do not know the fraction yet.

So we call the number x: x = 0.1666 repeating.

Careful!

It is not 16/100.

That is 0.16, which stops.

Ours goes on for ever.

2 Move the point

First we move the point past the 1, which does not repeat.

Times 10 moves the point 1 place to the right.

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

Then one block more: times 100 moves it 2 places.

Now both numbers have the same tail after the point.

3 Take away

Now we take the smaller one away from the bigger one.

Both sides, so they stay equal.

After the point, both go on with 6 for ever.

The tails are exactly the same, so they make zero.

Cross them out.

100x minus 10x is 90x.

16 minus 1 is 15.

4 Remove the 90

90x means 90 times x.

We want just x.

To remove times 90, we divide by 90.

90 divided by 90 is 1.

So just x is left.

15 and 90 can both be divided by 15.

We can make 15/90 simpler.

It is the same as 1/6.

So x = 1/6.

So 0.1666 repeating is 1/6.

5 Check

Let's check: 1 divided by 6.

We get 0.1666, and the digits go on the same way.

That is 0.1666 repeating.

So the answer is right!

Let's look back.

First: call it x.

Then: move the point.

Then: take away.

Then: remove the 90.

Last: check.

▶ Watch it solved