To be able to accurately determine the amount of change is a highly practical skill to have. When you buy something from a grocery store, dine in a restaurant or doing any transaction that involves cash, you would most likely receive changes. However, many people find calculating change without using a calculator very challenging. The good news is that there is a simple way of doing it. In this tutorial, I’ll teach you an easy and **doable** way to calculate change in your head. For example, you need to pay $39.47 and you paid $100, so how much should be the change?

The problem looks daunting even if you calculate it on paper since you need to do several borrowings to solve it. Fortunately, using the method that I’m about to show you, you don’t have to worry about borrowing at all!

## The Basics

You only need to understand one concept to do this trick. This simple concept is called **complement**. You may not have remembered it, but you have most likely learned this during your elementary days. I won’t go technical here since complement is just a simple concept. For now, we just need to classify complement into two types, **9’s complement** and **10’s** **complement**.

### 9’s complement

When dealing with complements, we do it one digit at a time. If you need to find the 9’s complement of a number, it means that you need to think of another number that makes the number equals to 9 when you add them together. For example, the complement of 3 is 6 since 3 + 6 = 9 and the complement of 9 is 0 since 9 + 0 = 9.

### 10’s Complement

This is similar to the 9’s complement but instead, you need to think of another number that makes a total of 10 if it is added to the number that you need to find the 10’s complement of. For example, the 10’s complement of 2 is 8 (2 + 8 = 10) and the 10’s complement of 5 is also 5 (5 + 5 = 10).

The following is the list of the 9’s and 10’s complements of 0 to 9.

**Number 9’s Comp. 10’s Comp.**

0 9 10

1 8 9

2 7 8

3 6 7

4 5 6

5 4 5

6 3 4

7 2 3

8 1 2

9 0 1

## The Trick

### For $1, $10, $100 & $1000

Here is the general rule for these denominations;

*Find the 9’s complement(s) of the first digit(s) of the purchase price and the 10’s complement of the last digit of the purchase price. The numbers that you would get would be the amount of change.*

To start off simple, suppose you bought something cost 64 cents and you paid $1.00. So, how much is the change. The problem would look like:

1.00 – 0.64

**Step 1:** Get the 9’s complement of the first digit which is **3** (6 + 3 = 9). So the first part of the answer is, 0.3 _

**Step 2: **Get the 10’s complement of the last digit which is **6** (4 + 6 = 10). The last part of the answer is 6.

0.36

Therefore, $1.00 – 64 cents = **36 cents**. As you can see, you don’t have to do any borrowing at all!

### Thought Pr**ocess**

$1.00 – 64 cents

1. 6 + ? = 9 → 3 → 0.3 _

2. 4 + ? = 10 → 6 → 0.36

So, $1.00 – 64 cents = **36 cents**

Now, suppose that you need to pay $3.79 and you paid $10.00, calculate your change. The method is still the same, you just need to get the 9’s complements of the first two digits and the 10’s complement of the last digit (remember the rule?).

### Thought Pr**ocess**

$10.00 – $3.79

1. 3 + ? = 9 → 6 → 6. _ _ → 7 + ? = 9 → 2 → 6.2 _

2. 9 + ? = 10 → 1 → 6.21

$10.00 – $3.79 = **$6.21**

For $100: If your purchases cost $59.38 and you paid $100.00, calculate the change. The method is still the same but now, you need to get the 9’s complements of the first three digits and the 10’s complement of the last digit.

### Thought Pr**ocess**

$100.00 – $59.38

1. 5 + ? = 9 → 4 → 4 _._ _ → 9 + ? = 0 → 0 → 40._ _ → 3 + ? = 9 → 6 → 40.6 _

2. 8 + ? = 10 → 2 → 40.62

$100.00 – $59.38 = **40.62**

For $1000: You need to pay $427.15 and you paid $1000.00, calculate the change. Still the same method…

### Thought Pr**ocess**

$1000.00 – $427.15

1. 4 + ? = 9 → 5 → 5 _ _._ _ → 2 + ? = 9 → 7 → 57 _._ _ → 7 + ? = 9 → 2 → 572._ _ → 1 + ? = 9 → 8 → 572.8 _

2. 5 + ? = 10 → 5 → 572.85

$1000.00 – $427.15 = **$572.85**

There are also some special cases that you should know.

**First Special Case:** Calculate, $1000.00 – $430.00.

(**Note: **If there are decimals after 430.00 such as 430.42, then use the method discussed above.)

Treat this problem like 100.00 – 43.00 then just add a zero to your answer.

### Thought Pr**ocess**

$1000 – $430. Treat it as 100 – 43

1. 4 + ? = 9 → 5 → 5 _

2. 3 + ? = 10 → 7 → 57

3. 570

$1000.00 – $430.00 = **$570**

**Note: **This method also applies to problems like 100.00 – 40.00.

**Second Special Case: **Calculate $1000.00 – $78.41. Since the subtrahend only contains two digits (excluding the digits after the decimal), you need to treat 78.41 as 078.41.

### Thought Pr**ocess**

$1000.00 – $078.41

1. 0 + ? = 9 → 9 → 9 _ _._ _ → 7 + ? = 9 → 2 → 92 _._ _ → 8 + ? = 9 → 1 → 921._ _ → 4 + ? = 9 → 5 → 921.5 _

2. 1 + ? = 10 → 9 → 921.59

$1000.00 – $078.41 = **$921.59**

**Note:** This method also applies to problem like 100.00 – 8.54.

### For $5, $20, $50 & $500

For denominations such as $20 and $50, the calculation has a very minor difference but the method is still relatively the same. Aside from the general rule mentioned above, there is one more additional minor rule that you need to keep in mind.

**You need to get the n-complement of the digit, where n is one less than the first digit of the number you are subtracting from.**

For example, you need to subtract** $3.48** from** $5.00,** then you need to get the **4’s** complement (**5 – 1 = 4)** of the first digit of **$3.48**. Afterwards, process as before.

### Thought Pr**ocess**

$5.00 – $3.48

1. 3 + ? = 4 → 1 → 1._ _

2. 4 + ? = 9 → 5 → 1.5 _

3. 8 + ? = 10 → 2 → 1.52

$5.00 – $3.48 = **$1.52**

**Another Example: **You need to pay $12.53 and you paid $20.00, calculate your change. For $20.00, you need to find the 1’s complement of the first digit.

### Thought Pr**ocess**

$20.00 – $12.53

1. 1 + ? = 1 → 0 → 0 _._ _

2. 2 + ? = 9 → 7 → 07._ _ → 5 + ? = 9 → 4 → 07.4 _

3. 3 + ? = 10 → 7 → 07.47

$20.00 – $12.53 = **$7.47**

For $50: Same as $5.00 (with 4’s complement). Example: $50.00 – $27.08

### Thought Pr**ocess**

$50.00 – $27.08

1. 2 + ? = 4 → 2 → 2 _._ _

2. 7 + ? = 9 → 2 → 22._ _ → 0 + ? = 9 → 9 → 22.9 _

3. 8 + ? = 10 → 2 → 22.92

$50.00 – $27.08 = **22.92**

For $500.00: Same as $5.00 and $50.00 (with 4’s complement).

**Example:** $500.00 – $84.76

### Thought Pr**ocess**

$500.00 – $084.76

1. 0 + ? = 4 → 4 → 4 _ _._ _

2. 8 + ? = 9 → 1 → 41 _._ _ → 4 + ? = 9 → 5 → 415._ _ → 7 + ? = 9 → 2 → 415.2 _

3. 6 + ? = 10 → 4 → 415.24

$500.00 – $084.76 = **$415.24**

**An Arbitrary Example: **What if you need to pay **$62.96** and you paid **$80.00**. Calculate the change. In this case, you need to find the **7’s** complement (**8 – 1**) of the first digit of the purchase price.

### Thought Pr**ocess**

$80.00 – $62.96

1. 6 + ? = 7 → 1 → 1 _._ _

2. 2 + ? = 9 → 7 → 17._ _ → 9 + ? = 9 → 0 → 17.0 _

3. 6 + ? = 10 → 4 → 17.04

$80.00 – $62.96 =** $17.04**

**Final Remarks**

I hope that you understood my explanations and eventually be able to do all the calculations mentally. I intentionally provide many examples to try to cover all the possible problems that you might encounter. So, don’t overwhelm yourself and don’t learn them all in one go.

You should first practice this technique using pen and paper. As you calculate each of the digit’s complement, write the digits on the paper to not strain your memory. With some practice, you will be abler to execute the calculations easier and will be able to do all the work mentally.

Lastly, I once again stress the fact that this is a very useful and practical skill and you can really use it in your everyday life.

If you have any questions, feel free to ask me in the comments section below.

In elementary school (1953) we had several “exchange” visits with kids from the School for Blind. They taught us this methodology and I’ve used it since almost without actually thinking of the process.

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Interesting.

I only included a “thought process” section as a guide. However, for this to be truly practical, the mental calculations should be instantaneous and executed without hesitation.

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It actually works better if you do it the other way round. I have just bought a PC this morning and I explained your method (with the reversing) to the seller. The PC was £82, I paid £100 cash and received £27 in change which I was quite happy with.

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Yes, it’s a good idea. Use it together with Abbott & Costello’s 2 tens for a 5 then you would have lots of free money.

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I think it’s a very good habit to have; mentally calculating your change at shops etc

My Dad was a Maths teacher, so we were taught mental arithmetic at an early age.

My quick method for subtraction is always to make the number you are subtracting ‘easy’. So for your first example of 100.00 – 39.47, I subtract 40, then add on the rest to balance it. 🙂

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There are many methods for mental subtraction though I think that this is the quickest method for subtraction from numbers with several zeros.

The method that you have mentioned is flexible once you become familiar with it. For example, 126 – 42 can be solved with 126 – 40 – 2 or 130 – 40 – 2 – 4, depending on which one you’re more comfortable with.

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Yes, I think you’re right with your method, especially trying to teach someone who is maybe not so good at maths. That’s why I’m not a Maths teacher! 😀

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Reminds me of the Trachtenberg System. My grandmother used it a lot, but I never quite caught on.

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I personally use some of the techniques offered by the Tratchtenberg system in my lightning calculation routine back when I was still performing magic professionally. But to perform lightning calculations under fire, you need a lot more than the Tratchtenberg.

Thanks for reading.

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Heh. So you know the cube root trick? Determine the root of any cubed two-digit number without a calculator.

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Yes, I use the mnemonics of Kellar (with some modifications) for this. I’ve also made my mnemonics for 4th and 5th power of 2-digit numbers.

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Wow. That’s a post I’d like to see.

This guy has some stuff that you might enjoy: trigonometry-is-my-bitchDOTtumblr.

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It’s a shame that it’s in Rumblr since I don’t like it’s interface. I’ll still check it out nonetheless. Thanks.

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Way back when in elementary school, I was one of the lucky chosen few who were taught the Base 10 system. I was fairly decent at math then and got straight A’s. These days, however, I need a calculator to add two 3-digit numbers! I don’t know if that’s old age or laziness, probably a bit of both. In any event, I found your blog when you “liked” one of my posts, and I’m glad I did. Your blog is unique, and I’m looking forward to expanding my somewhat limited mind by reading your posts.

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Perhaps you mean changing base 10 to other bases since we are using base 10 :).

Calculators and computers have made calculations much easier. I don’t remember a day during this last few years when I didn’t use Mathematica or Matlab. However, I believe that mental math is still important not just for its utility but it’s a good brain exercise as well.

Thank you and I’m glad to hear that.

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Ah, I’m sure you’re right. That was 50 years ago, so who knows what base they were trying to teach? I just recall that it didn’t make a whole lot of sense to me at the time. I agree that mental math is still very important, but most of us are just too lazy to use it now that everyone has a calculator on the computer or smart phone.

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I’m not sure why, but most of the textbooks that I’ve seen seem to make a big deal out of the procedure for converting base 10 to, say, base 2 or 16. They give cumbersome procedures and complicated explanations. This caused a lot of confusion among the students.

However, the reality is that the method for converting one number base to other bases is simple and elementary students should not have any problem understanding it.

Anyway, if you aren’t a mathematician, scientist or programmer, it’s safe to say that you won’t have much use for it.

Yeah, in a world where people want things to be done quickly and efficiently (aka almost not exerting any work), it’s no surprise. In many cases, I could have manually solve the equations, but I use the computer anyway since writing them all require a lot of time.

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Thankfully I can add in my head … but I so enjoyed reading this post!

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