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Contributor: Katie Schnabel. Lesson ID: 14194
Have you ever looked in the mirror while holding another mirror? Your reflection goes on and on forever! Discover what forever looks like with long division and repeating decimals.
Picture this: a room filled with mirrors on every wall, reflecting and reflecting until it seems to go on forever.
Mind-bending!
Now, imagine this concept in a math problem. Some decimal division problems work the same way—creating numbers that just keep going.
Dive in and see how these endless patterns connect to dividing decimals with remainders!
When you think about it, forever is a very long time. But don't think too hard—it could take forever!
Take a look at how a division problem can represent....forever.
Set up a long division problem: 0.1 divided by 0.3.
| 0. | 3 | 0. | 1 |
To make dividing easier, shift the decimal in the divisor to the right until it’s a whole number. Here, you only need to move it one place.
Remember, if you shift the decimal in the divisor, you have to do the same for the dividend. So, move the decimal one place to the right for 0.1 as well.
| 0 | 3. | 0 | 1. | ||||
| → | → |
Now, divide as you normally would.
It can't, so you must add a zero to make it 10.
| 0 | 3. | 0 | 1. | 0 |
That's right! 3 times, because 3 x 3 is 9.
Place a 3 on top and write 9 underneath to subtract.
| 0. | 3 | ||||||
| 0 | 3. | 0 | 1. | 0 | |||
| - | 9 | ||||||
| 1 |
You know that 10 - 9 equals 1. Because 3 cannot fit into 1, you must bring down another zero.
| 0. | 3 | 3 | |||||
| 0 | 3. | 0 | 1. | 0 | 0 | ||
| - | 9 | ↓ | |||||
| 1 | 0 |
Once again, 3 fits into 10 three times, giving you another 3 up top.
| 0. | 3 | 3 | 3 | ||||
| 0 | 3. | 0 | 1. | 0 | 0 | 0 | |
| - | 9 | ↓ | |||||
| 1 | 0 | ↓ | |||||
| - | 9 | ↓ | |||||
| 1 | 0 |
Yes! This pattern repeats. You’ll continue with 10 - 9 = 1 again and again, leading to 3 after 3 after 3 in the answer.
This means the answer is 0.33333 repeating.
When a quotient has a repeating decimal, use a line over the repeating digit.
| 0. | 3 |
Try another one: 0.555 divided by 0.27.
| 0. | 2 | 7 | 0. | 5 | 5 | 5 |
As before, make the divisor a whole number by moving the decimal two places to the right, and do the same to the dividend.
| 0 | 2 | 7. | 0 | 5 | 5. | 5 | ||||
| → | → | → | → |
Now divide as usual.
Think of 27 as close to 25 and 55 as close to 50. Start with 27 x 2 and see how that fits.
| 2 | 7 | ||
| x | 2 | ||
| 5 | 4 |
27 x 2 is 54, so that is as close as possible. Place the 2 up top and 54 below to subtract.
| 2 | ||||||||||
| 0 | 2 | 7. | 0 | 5 | 5. | 5 | ||||
| - | 5 | 4 | ||||||||
| 1 |
Bring down the next number to get 15.
Yes! Zero times. So put a 0 up top and 0 underneath because 27 x 0 is zero. Then, subtract.
| 2. | 0 | |||||||||
| 0 | 2 | 7. | 0 | 5 | 5. | 5 | ||||
| - | 5 | 4 | ↓ | |||||||
| 1 | 5 | |||||||||
| - | 0 | |||||||||
| 1 | 5 |
Now, 15 - 0 is still 15, but you can bring down another digit to make it 150.
| 2. | 0 | |||||||||
| 0 | 2 | 7. | 0 | 5 | 5. | 5 | 0 | |||
| - | 5 | 4 | ↓ | |||||||
| 1 | 5 | ↓ | ||||||||
| - | 0 | ↓ | ||||||||
| 1 | 5 | 0 |
Estimate again: since 27 is near 25, try multiplying by 5.
| 2 | 7 | ||
| x | 5 | ||
| 1 | 3 | 5 |
You got it! 135. Place 5 up top, 135 below, and subtract.
| 2. | 0 | 5 | ||||||||
| 0 | 2 | 7. | 0 | 5 | 5. | 5 | 0 | |||
| - | 5 | 4 | ||||||||
| 1 | 5 | |||||||||
| - | 0 | |||||||||
| 1 | 5 | 0 | ||||||||
| - | 1 | 3 | 5 | |||||||
| 1 | 5 |
You are left with 15, so you must bring down another zero. This gives you 150 again.....there seems to be a pattern arising. Keep going and see.
You know that 27 goes into 150 five times, so you can place a 5 on top again and 135 underneath to subtract.
| 2. | 0 | 5 | 5 | |||||||
| 0 | 2 | 7. | 0 | 5 | 5. | 5 | 0 | 0 | ||
| - | 5 | 4 | ↓ | |||||||
| 1 | 5 | ↓ | ||||||||
| - | 0 | ↓ | ||||||||
| 1 | 5 | 0 | ↓ | |||||||
| - | 1 | 3 | 5 | ↓ | ||||||
| 1 | 5 | 0 | ||||||||
| - | 1 | 3 | 5 | |||||||
| 1 | 5 |
This repeating pattern of 5s means the answer is 2.05, with the 5 repeating. So, write it with a line over the 5.
| 2. | 0 | 5 |
Awesome job! Something else needs repeating: Great job! Great job! Great job!
Head to the Got It? section for fun pract
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