Subtracting Whole Numbers

Subtraction is in some sense inverse of addition.

Let's start subtracting whole numbers.

Suppose you have $20 and your friend $40. Friend gives you $15. You obtain $15 and now you have $20+$15=$35. But you friend "loses" $15 and now has $40-$15=$25.

So, you can think about subtraction as a process during which you lose something.

Another example: suppose you have 9 apples. You've eaten two. How much apples do you have? Answer is 7, so 9-2=7.

In general, for two numbers `a` and `b` if you want to find their difference (subtract one from another) `c=b-a` then you actually want to find such number `c` that `b=a+c`.

Also, for `c=b-a`, `b` is called minuend, `a` is called subtrahend and `c` is called difference.

Thus, in 9-2=7, 9 is minuend, 2 is subtrahend and 7 is difference.

Technique for subtracting whole numbers is similar to technique for addition whole numbers.

Let's start from simple example.

Example 1 . Calculate 35-21.

Let's write numbers one under another:

$$$ \begin{array}{l@{\,}l@{\,}l} \ & \color{blue}{3}&\color{green}{5} \\ - & \color{blue}{2}&\color{green}{1} \\ \hline & \color{blue}{}&\color{green}{} \\ \end{array}$$$

Start from right, let's subtract 1 from 5. Answer is 4. Write it under 5 and 1.

Now subtract 2 from 3. Result is 1. Write 1 under 3 and 2.

$$$ \begin{array}{l@{\,}l@{\,}l} \ & \color{blue}{3}&\color{green}{5} \\ - & \color{blue}{2}&\color{green}{1} \\ \hline & \color{blue}{1}&\color{green}{4} \\ \end{array}$$$

So, 35-21=14.

Actually same applies for 3-digit numbers and in general for any numbers.

Example 2. Calculate 345-101.

Let's write numbers one under another:

$$$ \begin{array}{l@{\,}l@{\,}l@{\,}l} \ & \color{red}{3}&\color{blue}{4}&\color{green}{5} \\ - & \color{red}{1}& \color{blue}{0}&\color{green}{1} \\ \hline & \color{red}{}&\color{blue}{}&\color{green}{} \\ \end{array}$$$

Start from right, subtract 1 from 5. Result is 4. Write it under 5 and 1.

Now subtract 0 from 4. Result is 4. Write 4 under 4 and 0.

Finally, write 3-1=2 under 3 and 1.

$$$ \begin{array}{l@{\,}l@{\,}l@{\,}l} \ & \color{red}{3}&\color{blue}{4}& \color{green}{5} \\ - & \color{red}{1}& \color{blue}{0}&\color{green}{1} \\ \hline & \color{red}{2}&\color{blue}{4}&\color{green}{4} \\ \end{array}$$$

So, 345-101=244.

Let's do a harder example now.

Example 3. Calculate 946-197.

Let's write numbers one under another:

$$$ \begin{array}{l@{\,}l@{\,}l@{\,}l} \ & \color{red}{9}&\color{blue}{4}& \color{green}{8} \\ - & \color{red}{1}& \color{blue}{9}& \color{green}{7} \\ \hline & & &\color{green}{} \\ \end{array}$$$

Now, subtract 6 from 7. Oops! Looks like we subtracting bigger number from smaller. To avoid this, add 10 and 6 (result is 16) and remember that you borrow 1. Now subtract 7 from 16. Result is 9.

$$$ \begin{array}{l@{\,}l@{\,}l@{\,}l} \ & \color{red}{9}& \overset{\color{green}{-1}}{\color{blue}{4}}& \color{green}{8} \\ - & \color{red}{1}& \color{blue}{9}& \color{green}{7} \\ \hline & & & \color{green}{9} \\ \end{array}$$$

Now, subtract 9 from 4. Again we subtracting bigger number from smaller. To avoid this, add 10 and 4 (result is 14). Now subtract 9 from 14. Result is 5. And don't forget to subtract borrowed 1. Result is 4.

$$$ \begin{array}{l@{\,}l@{\,}l@{\,}l} \ & \overset{\color{blue}{-1}}{\color{red}{9}}& \color{blue}{4}& \color{green}{8} \\ - & \color{red}{1}& \color{blue}{9}& \color{green}{7} \\ \hline & & \color{blue}{4} & \color{green}{9} \\ \end{array}$$$

Finally subtract 1 from 9 and also subtract remembered 1. Result is 7.

$$$ \begin{array}{l@{\,}l@{\,}l@{\,}l} \ & \color{red}{9}&\color{blue}{4}& \color{green}{8} \\ - & \color{red}{1}& \color{blue}{9}& \color{green}{7} \\ \hline & \color{red}{7}& \color{blue}{4}& \color{green}{9} \\ \end{array}$$$

So, 946-197=749. You can make sure that 946=749+197.

Next example shows how to subtract numbers that have different number of digits.

If two numbers have different number of digits then take number that has smaller number of digits and add zeros in front of it until number of digits will be same.

For example, suppose you need to subtract 23 from 5537. We take 2-digit number 23 and place zeros in front of it: 23 becomes 0023. Now we can subtract 0023 from 5537.

Example 4. Calculate 345-56.

Here we add one zero in front of 56: 056.

Now, we can subtract 056 from 345 using standard technique.

We will do this example a bit faster.

$$$ \begin{array}{l@{\,}l@{\,}l@{\,}l} \ & \overset{\color{blue}{-1}}{\color{red}{3}}& \overset{\color{green}{-1}}{\color{blue}{4}}& \color{green}{5} \\ - & \color{red}{0}& \color{blue}{5}& \color{green}{6} \\ \hline & \color{red}{2}& \color{blue}{8}& \color{green}{9} \\ \end{array}$$$

So, 345-56=289.

Last example is about subtracting more than two numbers.

Example 5. Find 501-47-368.

There is no really difference between subtracting just two numbers.

First we write 0 in front of 47: 047.

$$$ \begin{array}[l@{\,}l@{\,}l@{\,}l] \ & \color{red}{5}& \color{blue}{0}& \color{green}{1} \\ - & \color{red}{0}& \color{blue}{4}& \color{green}{7} \\ - & \color{red}{3}&\color{blue}{6}& \color{green}{8} \\\hline & & & \\ \end{array}$$$

We subtract 7 from 1. Again we need to add 10 and remember that we borrow 1: 1+10-7=4.

$$$ \begin{array}{l@{\,}l@{\,}l@{\,}l} & \color{red}{5} & \overset{\color{green}{-1}}{\color{blue}{0}} & \color{green}{1} \\ - & \color{red}{0} & \color{blue}{4} & \color{green}{7\color{purple}{(4)}} \\ - & \color{red}{3}&\color{blue}{6}&\color{green}{8} \\\hline & & &\color{green}{} \\ \end{array}$$$

Now, continue subtracting: subtract 8 from 4. Again we add 10 and borrow 1. But since we already have reserved 1 total we borrowed 2. Now, calculate 4+10-8=6.

$$$ \begin{array}{l@{\,}l@{\,}l@{\,}l} & \color{red}{5} & \overset{\color{green}{-2}}{\color{blue}{0}} & \color{green}{1} \\ - & \color{red}{0} & \color{blue}{4} & \color{green}{7} \\ - & \color{red}{3}&\color{blue}{6}&\color{green}{8} \\\hline & & &\color{green}{6} \\ \end{array}$$$

Next, work with second column. Subtract 4 from 0. Again we add 10 and borrow 1: 0+10-4=6.

$$$ \begin{array}{l@{\,}l@{\,}l@{\,}l} & \overset{\color{blue}{-1}}{\color{red}{5}} & \overset{\color{green}{-2}}{\color{blue}{0}} & \color{green}{1} \\ - & \color{red}{0} & \color{blue}{4\color{purple}{(6)}} & \color{green}{7} \\ - & \color{red}{3}&\color{blue}{6}&\color{green}{8} \\\hline & & &\color{green}{6} \\ \end{array}$$$

Continue subtracting in the second column: subtract 6 from 6. Result is 0. Now subtract borrowed 2 - again we need to borrow 1 (so we have two borrowed 1s): 0+10-2=8.

$$$ \begin{array}{l@{\,}l@{\,}l@{\,}l} & \overset{\color{blue}{-2}}{\color{red}{5}} & \color{blue}{0} & \color{green}{1} \\ - & \color{red}{0} & \color{blue}{6} & \color{green}{7} \\ - & \color{red}{3}&\color{blue}{6}&\color{green}{8} \\\hline & & &\color{green}{6} \\ \end{array}$$$

Finally, work with left-most column. Subtract 3 from 5. Result is 2. And subtract borrowed 2: 2-2=0.

$$$ \begin{array}{l@{\,}l@{\,}l@{\,}l} & \color{red}{5} & \color{blue}{0} & \color{green}{1} \\ - & \color{red}{0} & \color{blue}{4} & \color{green}{7} \\ - & \color{red}{3}&\color{blue}{6}&\color{green}{8} \\\hline &\color{red}{0} &\color{blue}{8} &\color{green}{6} \\ \end{array}$$$

We obtained result 086, but this is just 86.

So, 501-47-368=86.

If this technique is hard for you, there is always easier method. Do it step-by-step. First subtract 47 from 501 to get 454 and then subtract 368 from 454 to get 86. This two-step procedure is more clear.

Now, it is your turn. Take pen and paper and solve following problems.

Exercise 1. Find 58-31.

Answer: 47.

Exercise 2. Find 91-37.

Answer: 54.

Exercise 3. Find 523-86.

Answer: 437.

Exercise 4. Find 900-567-228.

Answer: 105.

Exercise 5. Find 8745-51-99-856-1785.

Answer: 5954.