# Subtraction Without Borrowing: Subtraction by Addition:

In our Math Tip last week, we showed one method of avoiding borrowing in subtraction. This is important not only to facilitate mental calculation but more importantly, it will minimize possible errors due to striking out or rewriting the figures in the minuend.

We will now present the second technique which we discussed in our book “25 Math Short Cuts” to avoid borrowing. We call this “Subtraction by Addition”.

The term may seem to be self-contradictory but it really means that we will simplify subtraction by using the 10’s complement of the subtrahend.

Let us take last week’s problem as an example: **12,432 – 9,789.**

**12, 432 + 211 = 12, 643**

** 9,789 + 211 = 10, 000**

**12,643 – 10,000 = 2, 643**

We can see that by adding **211**, which is the **10’s complement of** the subtrahend to the minuend **12, 432** will give **12, 643**, while **9,789** and **211** will complete a **whole 10,000 **when combined. The subtraction will then transform into an easy **12, 643 – 10,000 = 2, 643**.

This can easily be proven algebraically.

Let m – minuend

s – subtrahend

b – base

t – ten’s complement of the subtrahend

d – difference

we have **s = b – t** and

**d = m – s = m – (b – t)**

**= m – b + t**

**= m + t – b = (m + t) – b**

Since we can easily determine the 10’s complement of a number using the sutra **All from 9 and the Last from 10**, the subtrahend can be far from the base, which is the case with this week’s example: **54,725 – 35, 947.** This subtraction will require 4 regroupings or borrowing using the conventional method.

Now instead of performing that subtraction we added the 10’s complement of the **35, 947** which is **64, 053** to the minuend, to get the sum **118, 778** and we can deduct the base, **100, 000** by simply striking out the leading **1** in the answer.

This technique is particularly useful in computations like the following:

** + 89, 764 89, 764**

**-54, 367 45, 633**

**+76, 362 76, 362**

** -25,748 74, 252**

** -42,746 57, 254**

**343, 265 – 300,000 = 43, 265**

We can see that we replaced the numbers to be subtracted in the left group by their complements in the right group. So we can proceed to addition column by column in the right group. Since we replaced to three numbers by their complements, we will subtract 3 imes their bases or 3 x 100,000 = 300,000.

Try to solve the following exercises using subtraction by addition:

- 342 – 289 =
- 734 – 497 =
- 4, 125 – 3, 879 =
- 5, 245 – 4, 868 =
- 8, 124 – 3, 766 =
- 9, 232 – 5, 768 =
- 73, 514 – 27, 647 =
- 92, 465 – 38, 878 =
- 86, 485 – 23, 776 – 43, 554 =
- 88, 543 – 65, 374 + 43, 923 – 48, 778 =

Here are the answers to last week’s practice exercises:

- 222 – 198 = 24
- 334 – 289 = 45
- 715 – 476 = 239
- 1, 123 – 987 = 136
- 5, 323 – 889 = 4, 434
- 6,145 – 798 = 5, 347
- 23,456 – 9, 897 = 13, 559
- 55, 432 – 49,678 = 5, 754
- 53, 321 – 48, 787 = 4, 534
- 1,222,111 – 777,889 = 444,222