1.2 – Adding and Subtracting Whole Numbers

Adding and Subtracting Whole numbers

Addition is the process of combining two numbers. When there is an addition, the two numbers being added are called the addends. The result of the addition is the sum. In the equation

$7+4=11$

7 and 4 are the addends and 11 is the sum. It is assumed that a basic knowledge of addition has been acquired and addition doesn’t need to be defined here on a number line. If desired, the student can read this material in a college arithmetic manual, or find a number of resources available online.

Commutative Property of Addition

One will notice that the addition of 7 and 4 results in 11: $7 + 4 = 11$ Switching the order of addition gives the same result: $4 + 7 = 11$ . That is to say that the order of the addends does not change the result of our addition. This property is known as the Commutative Property of Addition. It says in general:

The Commutative Property of Addition
$\begin{array}{ll}\text{If}& a+b=c \\ \text{then} & b+a=c\end{array}$

or more generally, $a+b = b+a$ . This will be used a great deal in future problems. For now, we will examine a widely preferred method of adding two numbers that is useful when adding two larger numbers together. To add two numbers together, place one on top of the other, lining up their place values. Beginning in the one’s-place, add the digits together, writing the result below in the one’s-place of your result. Move to the ten’s place, then the hundreds, and so on. Do this until every place value with digits has been combined.

Add: $1,254 + 3,402$
To add these numbers together, we will use a columnar addition. That is, we will put stack the two numbers on top of each other, and add each column, or place value together, starting in the ones column: $\begin{array}{r}1,25{\color{red}4}\\+3,40{\color{red}2}\\ \hline {\phantom{4,65}{\color{red}6}\end{array}$ Then the tens: $\begin{array}{r}1,2{\color{red}5}4\\+3,4{\color{red}0}2\\ \hline \phantom{4,6}{\color{red}5}6\end{array}$ Then the hundreds: $\begin{array}{r}1,{\color{red}2}54\\+3,{\color{red}4}02\\ \hline \phantom{4,}{\color{red}{6}56\end{array}$ And finally, the thousands: $\begin{array}{r}{\color{red}1},254\\+{\color{red}3},402\\ \hline {\color{red}4},656\end{array}$

Add: $1,254 + 3,402$

To add these numbers together, we will use a columnar addition. That is, we will put stack the two numbers on top of each other, and add each column, or place value together, starting in the ones column:

$\begin{array}{r}1,25{\color{red}4}\\+3,40{\color{red}2}\\ \hline {\phantom{4,65}{\color{red}6}\end{array}$

Then the tens:

$\begin{array}{r}1,2{\color{red}5}4\\+3,4{\color{red}0}2\\ \hline \phantom{4,6}{\color{red}5}6\end{array}$

Then the hundreds:

$\begin{array}{r}1,{\color{red}2}54\\+3,{\color{red}4}02\\ \hline \phantom{4,}{\color{red}{6}56\end{array}$

And finally, the thousands:

$\begin{array}{r}{\color{red}1},254\\+{\color{red}3},402\\ \hline {\color{red}4},656\end{array}$

Sometimes, when adding digits in a column, the result is larger than 9. If that is the case, we “carry” the tens place of the result to the next column in the addition:

Add: $4,264 + 318$
Starting with the ones, notice that 4+8=12, so the 2 is written in the ones place of the sum and the 1 is carried to the next column: $\begin{array}{r}{\color{red}1}\phantom{r} \\ 4,26{\color{red}4} \\ + \ 31{\color{red}8} \\ \hline \phantom{4,58}{\color{red}2}\end{array}$ Then the tens are added, including the carried 1: $\begin{array}{r}{\color{red}1}\phantom{r} \\ 4,2{\color{red}6}4 \\ + \ 3{\color{red}1}8 \\ \hline \phantom{4,5}{\color{red}8}2\end{array}$ Continuing through the rest of the addition: $\begin{array}{r}1\phantom{r} \\ 4,{\color{red}2}64 \\ + \ {\color{red}3}18 \\ \hline 4,{\color{red}5}82\end{array}$ Note that there is no digit in the thousands place in the number 318. We treat it as a zero: $\begin{array}{r}1\phantom{r} \\ {\color{red}4},264 \\ + \ {\color{red}0},318 \\ \hline {\color{red}4},582\end{array}$

Add: $4,264 + 318$

Starting with the ones, notice that 4+8=12, so the 2 is written in the ones place of the sum and the 1 is carried to the next column:

$\begin{array}{r}{\color{red}1}\phantom{r} \\ 4,26{\color{red}4} \\ + \ 31{\color{red}8} \\ \hline \phantom{4,58}{\color{red}2}\end{array}$ Then the tens are added, including the carried 1:

$\begin{array}{r}{\color{red}1}\phantom{r} \\ 4,2{\color{red}6}4 \\ + \ 3{\color{red}1}8 \\ \hline \phantom{4,5}{\color{red}8}2\end{array}$ Continuing through the rest of the addition:

$\begin{array}{r}1\phantom{r} \\ 4,{\color{red}2}64 \\ + \ {\color{red}3}18 \\ \hline 4,{\color{red}5}82\end{array}$ Note that there is no digit in the thousands place in the number 318. We treat it as a zero:

$\begin{array}{r}1\phantom{r} \\ {\color{red}4},264 \\ + \ {\color{red}0},318 \\ \hline {\color{red}4},582\end{array}$

Note that the Commutative Property of Addition allows us to add these numbers stacked in either order:

$\begin{array}{r}{\color{red}1}\phantom{r} \\ 318\\ + 4,264\\ \hline 4,582\end{array}$ When more than two addends are present, addition can be completed in the same manner by stacking the numbers:

Add: $125 + 319 + 215$
Ones, carrying the excess: $\begin{array}{r}{\color{red}1}\phantom{r} \\ 12{\color{red}5}\\ 31{\color{red}9}\\ + 21{\color{red}5}\\ \hline \phantom{65}{\color{red}9}\end{array}$ Tens: $\begin{array}{r}{\color{red}1}\phantom{r} \\ 1{\color{red}2}5\\ 3{\color{red}1}9\\ + 2{\color{red}1}5\\ \hline \phantom{6}{\color{red}5}9\end{array}$ Hundreds: $\begin{array}{r}1\phantom{r} \\ {\color{red}1}25\\ {\color{red}3}19\\ + {\color{red}2}15\\ \hline {\color{red}6}59\end{array}$

Add: $125 + 319 + 215$

Ones, carrying the excess:

$\begin{array}{r}{\color{red}1}\phantom{r} \\ 12{\color{red}5}\\ 31{\color{red}9}\\ + 21{\color{red}5}\\ \hline \phantom{65}{\color{red}9}\end{array}$ Tens:

$\begin{array}{r}{\color{red}1}\phantom{r} \\ 1{\color{red}2}5\\ 3{\color{red}1}9\\ + 2{\color{red}1}5\\ \hline \phantom{6}{\color{red}5}9\end{array}$ Hundreds:

$\begin{array}{r}1\phantom{r} \\ {\color{red}1}25\\ {\color{red}3}19\\ + {\color{red}2}15\\ \hline {\color{red}6}59\end{array}$

Associative Property of Addition

A second useful property of addition is the Associative Property of Addition. When adding more than two numbers it does not matter which two are added first.

The Associative Property of Addition
The order of addition of three (or more) addends does not matter: $(a+b)+c = a+(b+c)$

Add: $25+19+10$ by grouping in two ways.
Grouping the first two addends and then adding the third: $(25+19)+10$ $44 +10$ $54$ Yields the same result as grouping the second two and then adding the first: $25+(19+10)$ $25+29$ $54$

Solving Addition Problems and Perimeter

The following are some examples of word problems and perimeter, an application of addition.

John decided one day to drive to his grandmother’s house in Florida. He drove for 7 hours the first day and an additional 9 hours the second. How many hours did John drive altogether?
This is an addition problem. It requires two amounts of time to be combined. Adding the times together yields $7+9=16$ . So John drove 16 hours total.

The perimeter of a shape is the total distance around its edges. It is the sum of all of its sides.

Joe wants to build a fence around an his garden. His garden is a rectangle with a width of 15 ft and a length of 7 ft. How much fencing does Joe need?
Drawing the garden is useful to visualize the problem. The rectangle has two sides of length 7 feet, and two sides of length 15 feet. So the perimeter is $7+7+15+15=44$ feet

Subtracting Whole Numbers

Subtraction is the process of taking one number away from another. For example, If I have ten apples and you take 4 of them from me, I now have 6 apples. This is a subtraction problem, denoted by

$10-6=4$ In this problem, the original number of apples I have is called the minuend, while the number of apples taken is referred to as the subtrahend. The result of the subtraction is called the difference. In every subtraction problem, the minuend minus the subtrahend is the difference.

To find the difference between two numbers, they are placed into a column similar to addition. Then the bottom number is taken away from the top column by column, starting in the ones place.

Subtract: $497 - 36$
Starting withy the ones: $\begin{align}49{\color{red}7}\\ \underline{- 3{\color{red}6}}\\ 46{\color{red}1} \end{align}$ Then the tens: $\begin{align}4{\color{red}9}7\\ \underline{- {\color{red}3}6}\\ 4{\color{red}6}1 \end{align}$ And finally the hundreds, noting that the hundreds value in the subtrahend is zero: $\begin{align}{\color{red}4}97\\ \underline{- {\color{red}0}36}\\ {\color{red}4}61 \end{align}$

Subtract: $497 - 36$

Starting withy the ones:

$\begin{align*}49{\color{red}7}\\ \underline{- 3{\color{red}6}}\\ 46{\color{red}1} \end{align*}$

Then the tens:

$\begin{align*}4{\color{red}9}7\\ \underline{- {\color{red}3}6}\\ 4{\color{red}6}1 \end{align*}$

And finally the hundreds, noting that the hundreds value in the subtrahend is zero:

$\begin{align*}{\color{red}4}97\\ \underline{- {\color{red}0}36}\\ {\color{red}4}61 \end{align*}$

If the lower number in a particular place value is larger than the upper number, one must “borrow” from the next place value in order to complete the subtraction.

Subtract: $2,684-1,218$
Note that one can not take 8 from 4 in the ones-place. So ten is borrowed from the tens place and a 14 is written in the ones place. Because we took from the 8, it is reduced to a 7: $\begin{align}{\color{red}\scriptscriptstyle{\ \ 7\ 14}}\\2,6\cancel{8}\cancel{4}\ \\\underline{-1,21{\color{red}8}}\ \\\phantom{1,46}{\color{red}6}\ \end{align}$ Moving to the tens place, the subtraction continues: $\begin{align}{\scriptscriptstyle{\ \ {\color{red}7}\ 14}}\\2,6\cancel{8}\cancel{4}\ \\\underline{-1,2{\color{red}1}8}\ \\1,4{\color{red}6}6\ \end{align}$ The hundreds place: $\begin{align}{\scriptscriptstyle{\ \ 7\ 14}}\\2,{\color{red}6}\cancel{8}\cancel{4}\ \\\underline{-1,{\color{red}2}18}\ \\1,{\color{red}4}66\ \end{align}$ And finally, the thousands: $\begin{align}{\color{red}\scriptscriptstyle{\ \ 7\ 14}}\\{\color{red}2},6\cancel{8}\cancel{4}\ \\\underline{-{\color{red}1},218}\ \\{\color{red}1},466\ \end{align}$

Subtract: $2,684-1,218$

Note that one can not take 8 from 4 in the ones-place. So ten is borrowed from the tens place and a 14 is written in the ones place. Because we took from the 8, it is reduced to a 7:

$\begin{align*}{\color{red}\scriptscriptstyle{\ \ 7\ 14}}\\2,6\cancel{8}\cancel{4}\ \\\underline{-1,21{\color{red}8}}\ \\\phantom{1,46}{\color{red}6}\ \end{align*}$

Moving to the tens place, the subtraction continues:

$\begin{align*}{\scriptscriptstyle{\ \ {\color{red}7}\ 14}}\\2,6\cancel{8}\cancel{4}\ \\\underline{-1,2{\color{red}1}8}\ \\1,4{\color{red}6}6\ \end{align*}$

The hundreds place:

$\begin{align*}{\scriptscriptstyle{\ \ 7\ 14}}\\2,{\color{red}6}\cancel{8}\cancel{4}\ \\\underline{-1,{\color{red}2}18}\ \\1,{\color{red}4}66\ \end{align*}$

And finally, the thousands:

$\begin{align*}{\color{red}\scriptscriptstyle{\ \ 7\ 14}}\\{\color{red}2},6\cancel{8}\cancel{4}\ \\\underline{-{\color{red}1},218}\ \\{\color{red}1},466\ \end{align*}$

Suppose a farmer has 3,429 chickens. One day, a wolf comes and eats 573 of them. How many are left?
This is a subtraction problem, so we line up our numbers and subtract: $\begin{align} \phantom{\color{red}\scriptscriptstyle{2\ 13 \ \ \ }}\\ \phantom{\color{red}\scriptscriptstyle{ \cancel{3} \ 12 \ }}\\ 3,429\\ \underline{- \ \ 573}\\ \phantom{2,856} \end{align}$ The ones column subtraction is executed without any issue: $\begin{align} \phantom{\color{red}\scriptscriptstyle{2\ 13 \ \ \ }}\\ \phantom{\color{red}\scriptscriptstyle{ \cancel{3} \ 12 \ }}\\ 3,42{\color{red}9}\\ \underline{- \ \ 57{\color{red}3}}\\ \phantom{2,85}{\color{red}{6}} \end{align}$ To subtract 7 from 2 in the tens column, it is necessary to borrow from the hundreds column. This changes the 2 in the tens column to 12 and the 4 in the hundreds column to 3. Then the tens column subtraction may be completed. $\begin{align} \phantom{\color{red}\scriptscriptstyle{2\ 13 \ \ \ }}\\ {\scriptscriptstyle{ 3 \ {\color{red}12} \ }}\\ 3,\cancel{4}\cancel{2}9\\ \underline{- \ \ 5{\color{red}7}3}\\ \phantom{2,8}{\color{red}5}6 \end{align}$ The hundreds column will also require a borrow from the thousands column. The 3 becomes a 13 and reduces the thousands column from 3 to 2. $\begin{align} {\color{red}\scriptscriptstyle{2\ 13 \ \ \ }}\\ {\color{red}\scriptscriptstyle{ \cancel{3}} \ 12 \ }\\ \cancel{3},\cancel{4}\cancel{2}9\\ \underline{- \ \ {\color{red}5}73}\\ \phantom{2,}{\color{red}8}56 \end{align}$ And finally the thousands: $\begin{align} {\scriptscriptstyle{{\color{red}2}\ 13 \ \ \ }}\\ {\scriptscriptstyle{ \cancel{3} \ 12 \ }}\\ \cancel{3},\cancel{4}\cancel{2}9\\ \underline{- \ \ {\color{red}0,}573}\\ {\color{red}2},856 \end{align}$

Suppose a farmer has 3,429 chickens. One day, a wolf comes and eats 573 of them. How many are left?

This is a subtraction problem, so we line up our numbers and subtract:

$\begin{align*} \phantom{\color{red}\scriptscriptstyle{2\ 13 \ \ \ }}\\ \phantom{\color{red}\scriptscriptstyle{ \cancel{3} \ 12 \ }}\\ 3,429\\ \underline{- \ \ 573}\\ \phantom{2,856} \end{align*}$

The ones column subtraction is executed without any issue:

$\begin{align*} \phantom{\color{red}\scriptscriptstyle{2\ 13 \ \ \ }}\\ \phantom{\color{red}\scriptscriptstyle{ \cancel{3} \ 12 \ }}\\ 3,42{\color{red}9}\\ \underline{- \ \ 57{\color{red}3}}\\ \phantom{2,85}{\color{red}{6}} \end{align*}$

To subtract 7 from 2 in the tens column, it is necessary to borrow from the hundreds column. This changes the 2 in the tens column to 12 and the 4 in the hundreds column to 3. Then the tens column subtraction may be completed.

$\begin{align*} \phantom{\color{red}\scriptscriptstyle{2\ 13 \ \ \ }}\\ {\scriptscriptstyle{ 3 \ {\color{red}12} \ }}\\ 3,\cancel{4}\cancel{2}9\\ \underline{- \ \ 5{\color{red}7}3}\\ \phantom{2,8}{\color{red}5}6 \end{align*}$

The hundreds column will also require a borrow from the thousands column. The 3 becomes a 13 and reduces the thousands column from 3 to 2.

$\begin{align*} {\color{red}\scriptscriptstyle{2\ 13 \ \ \ }}\\ {\color{red}\scriptscriptstyle{ \cancel{3}} \ 12 \ }\\ \cancel{3},\cancel{4}\cancel{2}9\\ \underline{- \ \ {\color{red}5}73}\\ \phantom{2,}{\color{red}8}56 \end{align*}$

And finally the thousands:

$\begin{align*} {\scriptscriptstyle{{\color{red}2}\ 13 \ \ \ }}\\ {\scriptscriptstyle{ \cancel{3} \ 12 \ }}\\ \cancel{3},\cancel{4}\cancel{2}9\\ \underline{- \ \ {\color{red}0,}573}\\ {\color{red}2},856 \end{align*}$

George’s checkbook states that he has $525. If his rent is $450, how much does he have left over after paying rent?
The question asks to find the difference between 525 and 450: $\begin{align} {\color{red}\scriptscriptstyle{4 \ 12 \ }}\\ \cancel{5}\cancel{2}5\\ \underline{- 450}\\ 75 \end{align}$

George’s checkbook states that he has $525. If his rent is $450, how much does he have left over after paying rent?

The question asks to find the difference between 525 and 450:

$\begin{align*} {\color{red}\scriptscriptstyle{4 \ 12 \ }}\\ \cancel{5}\cancel{2}5\\ \underline{- 450}\\ 75 \end{align*}$

PreCalculus I