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## Mental Division: Decimal Accuracy

Published on Sunday, July 29, 2012 in ,

### Introduction

While there are many mental math sites and videos freely available on the internet, very few seem to focus on division.

That's likely because division rarely comes out nice and neat, like addition, subtraction, and multiplication do. You have to deal with “leftovers” in the forms of remainders, fractions, or decimals.

Fractions are often seen as the most human-friendly way of expressing left-over numbers. Consequently, being able to give exact decimals, especially for unusual fractions such as 5/7 or 13/15, seems like an amazing feat to most people.

This tutorial will focus solely on division problems whose results are less than 1. For example, you'll learn how to deal with 5 ÷ 7 (or the equivalent fraction 5/7), but not how to deal with, say, 16 ÷ 7 (or 16/7).

Fortunately, there are numerous patterns of which we can take advantage, that will help make the task much easier than many people would suspect. You can start learning about these patterns in the next section.

### Fractions to Memorize

When dealing with smaller divisors, such as those from 2 to 11, it's quite useful to have the decimal equivalents of all such problems less than 1 memorized.

You probably already know all the answers for dividing by 2 through 4 already:

1/2=.5

1/3=.333... (the dots are used to refer to the endless repetition of digits)
2/3=.666...

1/4=.25
2/4=1/2=.5
3/4=.75

Dividing by 5 is very easy. You just double the numerator, and put a decimal point in front of it:

1/5=.2
2/5=.4
3/5=.6
4/5=.8

You also already know how to divide most of the numbers by 6, as well:

2/6=1/3=.333...
3/6=1/2=.5
4/6=2/3=.666...

All you have to know is two more 6ths:

1/6=.1666...
5/6=.8333...

I'll come back to the 7th, but for now, I'm going to cover the 8ths, since they're not much more difficult than 4ths. The trick here is to multiply the numerator by 125, and stick the decimal point in front of it:

1/8=.125
2/8=1/4=.250
3/8=.375
4/8=1/2=.500
5/8=.625
6/8=3/4=.750
7/8=.875

The last one almost everyone is familiar with is how to express 10ths:

1/10=.1
2/10=.2
3/10=.3
4/10=.4
5/10=.5
6/10=.6
7/10=.7
8/10=.8
9/10=.9

### New Patterns

To be able to memorize every fraction/division problem up to dividing by 11, then, all most people need to learn is how to handle 7ths, 9ths, and 11ths.

9ths are especially easy, as you simply start with the decimal point, and repeat the numerator endlessly:

1/9=.111...
2/9=.222...
3/9=.333...
4/9=.444...
5/9=.555...
6/9=.666...
7/9=.777...
8/9=.888...

If you know your multiples of 9 up to 10, then you can handle 11ths. Simply multiply the numerator by 9, and express that as a 2-digit number (9 × 1 = “09”). Repeat these 2 digits endlessly, and you've got your 11ths:

1/11=.090909...
2/11=.181818...
3/11=.272727...
4/11=.363636...
5/11=.454545...
6/11=.545454...
7/11=.636363...
8/11=.727272...
9/11=.818181...
10/11=.909090...

Finally, there's the 7ths. The 7ths have a very unusual pattern, starting with 1 ÷ 7:

1/7=.142857142857142857...

Notice that same sequence of numbers, 142857, repeats over and over again. To make things even easier, every 7th has this same repeating pattern. The only thing that changes is which number comes immediately after the decimal point. Here are the 7ths:

1/7=.142857142857142857...
2/7=.2857142857142857...
3/7=.42857142857142857...
4/7=.57142857142857...
5/7=.7142857142857...
6/7=.857142857142857...

To determine which number comes first, simply multiply the numerator by 14, and note the digit in the 10s place. That will be the digit that comes first. For example, with 3/7ths, you'd do 3 × 14 = 42, and since 4 is the digit in the tens place, you'd start the pattern with 4, giving you .42857142857... and so on.

Once you're comfortable with these basic memorized patterns, you're ready to move full speed ahead to the next section, where you'll learn to divide by numbers near 100.

### Dividing by 100

Dividing a 2-digit number by 100 is easy, of course. Take the numerator, place the decimal point in front of it, and you're done. 46/100? .46! 67/100? .67!

### Dividing by 99

Surprisingly, dividing a 2-digit number by 99 is almost as easy as dividing by 100. The only difference is that the numerator repeats without end. 46/99? .464646...! 67/99? It's .676767...!

Think of 99 as being like 100, but since the 9s repeat, so does the numerator.

### Dividing by 90

The technique for dividing any number from 1 to 89 by 90 takes us back to grade school math. Remember learning long division and getting answers like “3 remainder 7”? That's the type of thinking you'll need here.

To divide a 2-digit number by 90, just divide the numerator by 9, but work out the answer in the “x remainder y” format. Once you know that, x will be the first digit after the decimal point, and y repeats after that.

For example, let's work out 58 ÷ 90. Ask yourself, what is 58 divided by 9? Assuming your know your multiplication tables, you should think of it as 6 remainder 4. That means the decimal equivalent of 58 ÷ 90 is .6444....

Remember this trick for 90 by thinking of the way you divided by 9 when you had 0 knowledge of fractions.

### Dividing by 91

To dividing by 91, you'll first need to know how to multiply any number from 1 through 90 by 11. Below is a video to give you a good quick tutorial if you're not already familiar with the technique (If your browser supports Flash, here's another excellent tutorial). There's also a page where you can practice.

To divide by 91, you start by multiplying the numerator by 11, and then subtract 1. This number will likely be 3 digits, but if it's only 2 digits, place a 0 in front of it (76, for example, becomes 076). Put a decimal point in front of this 3 digit number, and you've got the first 3 digits of your answer.

To get the next 3 digits of the answer, subtract each of the digits from the previous step from 9. At this point, the number will repeat with the same 6 digits forever.

As a full example, let's work out 44 ÷ 91. We multiply 44 by 11 to get 484, and subtract 1, so we have 483. The first 3 digits of the answer, then are .483. Next, we work out 9 (always) - 4 (first digit) to get 5. 9 - 8 (second digit) = 1 and 9 - 3 (third digit) = 6, so the next 3 digits are these answers, 5, 1, and 6. Putting that all together, and repeating those same 6 digits, the decimal comes to .483516483516....

If you think of the 6 digits in the answer as two 3-digit numbers that always add up to 999, (Such as 483 and 516 in our previous example) it makes this easier. Since 91 ends in 1 just like 11, it's easy to remember that the technique for dividing by 91 involves multiplying by 11.

### Dividing by 98

98 has a particularly surprising pattern. To start, you begin with the numerator, adding a 0 in front if it's a 1-digit number, and keep doubling the number. For example, with 6 ÷ 98, we start with 06 after the decimal point and keep doubling:
`06 12 24 48`
Checking with Wolfram|Alpha, we see that 6 ÷ 98 does indeed start with .06122448!

There are two more rules you need to know for this technique. First, anytime your doubling sequence gives you a number of 49 or more, you need to add 1 before continuing. Second, when you continue the sequence, double that modified number, but only give the last two digits. Continues the sequence as if these last two digits were just another two digits in the sequence, remembering to follow both of these rules.

To make this clearer, let's work out 16 ÷ 98. We start the series as before:
`16 32 64`
Stop! That 64 is over 49, so we need to add 1, making it a 65:
`16 32 65`
From there, we double 65 to get 130, but remember the second rule, that we only include the final two digits, the 30:
`16 32 65 30`
From here, we continue as if 30 were just another 2-digit number in the sequence. The next number would be 60, which is more than 49, so we add 1:
`16 32 65 30 61`
Sure enough, 16 ÷ 98 works out to be .1632653061 (and beyond).

Once you get the hang of these rules, you can carry out the decimal equivalent as far as you like. Remember the add 1 rule even when you're starting with a number equal to or more than 49. For 78 ÷ 98, you'd start by adding 1 to 78, and continue from there:
`79 59 18 36 73`
Do you follow the pattern there? If so, then you can mentally work out that 78 ÷ 98 is .79591836734693877551020408 and even go beyond that if you like!

Since 98 is 2 away from 100, remembering this slightly unusual doubling sequence shouldn't be a problem.

Take some time and master these specific numbers. When you're ready to move on, I'll show you how to use what you already know to automatically handle even more division problems.

### Reduce

As with many things in life, simplifying before working on the problem is the first thing you should do. If you get a problem such as 25 ÷ 35, or a number like 25/35ths, reduce the problem by dividing both parts by the same number, and keep doing this until you make the two numbers as small as possible. When you realize this boils down to 5/7, you can use the 7ths technique to realize that this works out to .7142857....

Here's a quick refresher course in reducing division problems and fractions:

Being familiar with the quick divisibility tests for the numbers 2, 3, 4, 5, 6, 9, and 10 will help greatly here.

### Enlarge

Attempting to reduce the fraction or division problem should always be your first step, but sometimes you just can't reduce the problem to dividing to 2 through 11. This is where knowing how to divide by numbers near 100 comes in handy. See if you can make the problem larger, into one you're already familiar with.

Let's say you want to know the decimal equivalent of 11/25ths. You don't have a technique for handling 25ths, but if you scale both parts of the problem up so it reads as 44/100, then it's easily apparent that the answer is .44.

What about 13/15ths? Multiply both numbers by 6, and you get 78/90ths, which you should easily work out to .86666... using the strategies you've learned.

Similarly, dividing by 13 can be turned into dividing by 91 by multiplying times 7 (because 91= 13 × 7) and dividing by 33 becomes dividing by 99 just by multiplying by 3. You can also divide by 49 by scaling up by 2, which turns it into a problem of dividing by 98.

If you need to divide by 14, and you can't reduce it to 7ths, you can multiply both numbers by 7 to turn it into a problem of dividing by 98. 11/14ths won't reduce to 7ths, but will scale up to 77/98ths, or .7857142857....

### Reduce AND Enlarge

Again, reducing should always be the first step, but sometimes enlarging the problem after reducing can help. For example, here's how to handle 24 ÷ 39. Both numbers are divisible by 3, so the same result can be obtained by reducing it to 8/13. We can then scale this problem up to 56/91, and work out that it's .6153846....

Using this approach, you'll be able to handle any problem involving dividing by 13, 14, 15, 18, 20, 25, 30, 45, 49, and 50. Often, you'll be able to handle many other numbers, as well. 26/28ths is the same as 13/14ths, and thus 91/98ths.

### Estimate

Sometimes, however, you're stuck with a division problem or fraction such as 32/43rds. That won't reduce to anything, and scaling the 43rds up to 86ths or 129ths doesn't bring up any familiar divisors.

If you look closely, however, you can see that it's quite close to 33/44, or 3/4, so you can estimate that it's roughly .75. 32/43 works out to roughly .7441860, so you can see that this is a good guess.

Most lessons in estimating fractions, such as this video, focus on rounding to 0, ½, or 1, but if you can round to the nearest 3rd, 4ths and 5ths, your estimated answers will improve greatly.

I hope you've found this tutorial useful and enjoyable!

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