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# Plain numbers

Drawing fact based conclusions is one of the most common weapons consultants use to impress clients (and justify their fees). Let us go through this example.

```Revenue per visit 1.2 USD
Targeted revenue per visit= 1.2*3.5=4.2 USD
Profit per visit 4.2*0.2=0.84 USD```

Hence if a visit costs 0.55 USD there will be a new profit of 0.29 USD which is significant for a website with thousands of daily visits.

The calculations above are quite simple, right? Anyone could do them. However, as a consultant, you need to be able to think about the problem and perform the calculations as you speak, in essentially no time at all.

## HOW to beat texas instruments

We put together all the tricks you can possibly imagine to impress anyone with your calculation speed. If you read these and practice using our Plain Number package, with more than 7000 calculations, no one will ever stop you.

### RULE #1: TWO STEPS BETTER THAN ONE

It is usually better to add numbers in two steps instead of one, rounding the first to the closest hundredth and subtracting the added number to the second. For example,

`268 + 64 = (268 + 32) + (64 - 32) = 300 + 32 = 332.`

In general, you can add what comes easy to you first and then add the rest, as in the following example:

`2074 + 254 = 2100 + (254 - 26) = 2100 + 228 = 2328.`

This trick may not work for everyone but it is useful to know.

### RULE #2: ONE DIGIT AT A TIME

When dealing with large numbers it is often easier to add a single digit at a time instead of trying to add the whole number.

```478+529
=478+500+20+9
=978+20+9
=998+9 = 1,007```

When adding 5 to a number whose last digit is greater than 5, first subtract 5 and then add 10. For example:

```18 + 5 = 23
18 - 5 = 13
13 + 10 = 23```

### RULE #4: averages

When adding a long list of numbers, estimate the average of the list ( the closer the average, the easier it gets), then just add the difference of each number and the average. Finally, simply multiply the average by the number of addends and add the result of the previous sum. Even though it sounds tedious and confusing, the following example shows that the method is very efficient.

So assume we need to compute the sum of 12 numbers:

`77 + 74 + 94 + 89 + 85 + 70 + 84 + 72 + 77 + 94 + 78 + 85.`

Assume that the average is 80.
If we add all the differences we get

`-3-6+14+9+5-10+4-8-3+14-2+5`

The two 5s cancel out with the 10, -3-6 and +9 cancel out as well and we are left with

`14+4-8-3+14-2=19`

So

```80*12=80*10+80*2
=800+160+19
960+19=979```

## SUBTRACTION

### RULE #1: subtracting 5

When subtracting 5 from a number whose last digit is less than 5, first subtract 10 and then add 5. For example:

```23-5 =
=23-10 = 13
=13+5 = 18```

### RULE #2: two steps

It is usually better to subtract numbers in two steps instead of one, rounding the first to the closest hundredth and subtracting the added number to the second. For example,

`268 - 85 = (268 - 68) - (85 - 68) = 200-17= 183 `

Just subtract what is obvious first and then the rest.
So let’s see another example:

`2074 - 254 = 2000 - (254 - 74) = 2000 + 180 = 1820.`

## Multiplication

While addition and subtraction are fairly straightforward even without tricks, knowing the following tricks can make a huge difference to your calculation speed for multiplication and division

### RULE #1: multiplying by 5

When multiplying by 5, multiply by 10 and then divide by 2.
For example,

`247×5 = 2470/2 = 1235`

### RULE #2: multiplying by 4

When multiplying by 4, multiply the number twice by two.
For example

`372×4 = 744×2 = 1488`

### RULE #3: MULTIPLYING BY 8

When multiplying by 8, multiply the number three times by two.
For example

```376×8 = 752×4
1,504*2 = 3008```

### RULE #4: MULTIPLYING BY 50

When multiplying by 50, multiply by 100 and divide by 2. Use operations with 4 instead. For example,

`824×50 = 82,400/2 = 41,200`

### RULE #5: MULTIPLYING BY 25

When multiplying by 25, multiply by 100 and divide by 4. For example:

`27×25 = 2700/4 = 1350/2 = 675`

### RULE #6: big x small

When multiplying a big number by a small one, multiply each digit and then add them up.
For example

```743*7
=700*7+40*7+3*7
=4900+280+21 = 5201```

### RULE #7: the magic rule of 11

If you are multiplying a 2 digit number by 11, just write the sum of its digits between its digits. If the sum is greater than 10, carry the 1 over to the left. For example:

```63×11 = 693 since 6 + 3 = 9
76×11 = 836 since 7 + 6 = 13```

### RULE #8: multiplying by 9

When multiplying by 9, multiply by 10 instead, and then subtract the other number. For example:

`38×9 = 380 - 38 = 342`

Another way of multiplying numbers by 9 is the following: For any 1 digit number a: 9*a=bc (where bc is not the product b*c but just the two digits), where b=a-1 and c=9-b.
So if we have for instance:

```9*7
= (7-1)*(9-7+1) = 63
```

## Division

### RULE #1: dividing by 10

When dividing by 5, multiply by 2 and then divide by 10. For example:

`247/5 = 494/10 = 49.4.`

### RULE #2: DIVIDING BY 4

When dividing by 4, divide by 2 twice.
For example,

`376/4 = 188/2 = 94`

### RULE #3: DIVIDING BY 8

When dividing by 8, divide by 2 three times. For example:

376/8 = 188/4 = 94/2 = 47

### RULE #4: DIVIDING BY 25

When dividing by 25, divide by 100 and multiply by 4. For example:

`3,000/25 = 3,000/100 = 30*4 = 120.`

## SQUARES

It is a good idea to memorise the first 25 squares. Even though it looks a lot, you know the first 10 for sure. Then 11 and 20 are easy. By practising you will soon remember the others as well:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 441 484 529 576 625

A quick and easy way to find the square of any two digits number ab is to :
add b to ab and get b+ab. Then square b and append the square to b+ab. If the square of the number is greater than 10, carry the one over to the left.  Let’s see how it works with 14.
Find 14+4=18. Then 4²=16. So appending the last digit and adding the previous one to 18 we get 196.

There is also an advanced technique to calculate squares for 26 to 50 but it is quite challenging to do mentally. We will go through it as it could still become useful.
If A is a square between 26 and 50, Subtract 25 from A to get x. Subtract x from 25 to get, say, a. Then A² = a² + 100x. For example, if A = 28, then x = 3 and a = 22.  Hence

`28² = 22² + 100*3 = 784`

### Squares of numbers from 51 through 99

If A is between 50 and 100, then A = 50 + x. Compute a = 50 - x. Then A² = a² + 200x. For example,
63² = 37² + 200×13 = 1369 + 2600 = 3969.

### ANY square

In order to find any square, we need to apply another formula which is again given without proof. To find the square of any number (a), find a number close to it whose square can be found relatively easily(b).
Then find the difference between the two numbers (a-b) and subtract the difference from a hence a-(b-a). Then the square of the number can be found using the following formula:

`b*(a-(b-a))+(a-b)^2`

## Percentages

### RULE #1: simplify

A key skill here is to be able to simplify percentages so that divisions and multiplications are simple and easy to make. Finding percentages like 10% and 50% is trivial, so one can approximate most percentages as multiple of 5 and find them as the sum of 10%s and 5%s. So 5% becomes 10% divided by 2, 20% is twice 10%, 25% is half of 50%, 60% is 50%+10%, 75% is 50%+25% which is half of 50%. With an afternoon of practice with these, you will be soon become a master.

### RULE #2: x% of y

Also, fact of the day, x% of y is the same as y% of x. That’s right weird as it seems, the two are interchangeable. In fact x% of y is simply x/100*y, which is equivalent to y/100*x. So, how is this useful? We can invert percentages so calculating one is easier than the other. For example 28% of 75 becomes much easier if we look at 75% of 28, which is just 3/4 i.e. 21.

### RULE #2: higher than 100%

On a final note, for percentages higher than 100% simply multiply the number by the percentage divided by 100. So 250% of 120 will simply be 120*2.5=300.

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