Lindsey made $415.85 babysitting over the summer. She went to the bank to deposit her money into a savings account. At the bank she learned that she can expect to earn 2% interest on her money in one year. She does the calculations and finds that 2% of 415.85 is 8.317. Assuming Lindsey doesn’t add any more money to her savings account, how can she figure out how much money will be in her account after one year?

In this concept, you will learn how to add decimals with and without rounding.

### Adding Decimals

The **decimal system** lets us represent numbers that are less than 1 and other numbers that exist between whole numbers. In a decimal number, the **decimal point** divides the whole part of the number from the fractional part of the number.

To add decimal numbers:

- Arrange the numbers vertically so that the decimal points line up.
- Add zeros for any missing spaces so that each number has the same number of digits to the right of the decimal point.
- Add from right to left, carrying just as if you were adding whole numbers.
- Insert a decimal point into your answer so that it lines up with the decimal points of the original numbers.

Here is an example.

Find the sum of 1.87 and 2.3.

First, arrange the numbers vertically so that the decimal points line up.

\begin{align*}\begin{array}{rcl}
& \ \ 1.87\\
& \underline{+ 2.3 \;\;}
\end{array}\end{align*}

Next, add a zero after the 3 in the second number so that both numbers have two digits to the right of the decimal point.

Now, add from right to left. Start by adding 7 and 0 to make 7. Then 8 and 3 to make 11. At this point you will carry the 1 and add it along with the 1 and the 2 to make 4. Insert a decimal point into your answer directly under the decimal points in the original numbers.

\begin{align*}1.87\\
\underline{+ 2.30}\\
4.17\end{align*}

The answer is \begin{align*}1.87+2.3=4.17\end{align*}

Here is another example.

First, arrange the numbers vertically so that the decimal points line up.

Next, add two zeros after the 5 in the first number so that both numbers have three digits to the right of the decimal point.

Now, add from right to left. Start by adding 0 and 7 to make 7. Then 0 and 3 to make 3. Then 5 and 1 to make 6. Finally add 4 and 2 to make 6. Insert a decimal point into your answer directly under the decimal points in the original numbers.

The answer is

.Sometimes you will want to round numbers before adding them. Recall that there are four steps for rounding numbers.

- Identify the place you want to round to and find that digit within your number.
- Look at the digit to the right of the place you want to round to.
- If the digit to the right is between 5 and 9, increase the digit in the place you want to round to by 1. If the digit to the right is between 0 and 4, keep the digit in the place you want to round to the same. NOTE: If the digit in the place you want to round to is a 9 and you need to increase it by 1, change it to zero and increase the digit to its left by 1.
- Replace all digits after the place you are rounding to with zeros. If these digits occur to the right of the decimal point, it is not necessary to write the zeros.

Here is an example.

Round each number to the nearest hundredth and then find the sum.

First, round each number to the nearest hundredth. Remember that the hundredths place is the second digit to the right of the decimal point. Look at each digit in the hundredths place and then the digit to the right of that to decide if you will need to round up the digit in the hundredths place or keep it the same.

- 9.199 rounds to 9.20
- 11.353 rounds to 11.35
- 10.648 rounds to 10.65

Now, find the sum of the three rounded numbers. Start by arranging the numbers vertically so that the decimal points line up.

Next, add from right to left. Start by adding 0, 5, and 5 to make 10. Carry the 1 and add it with the 2, 3, and 6 to make 12. Once again carry the 1 and add it with the 9 and 1 to make 11. Carry the 1 one more time and add it with the 1 and 1 to make 3. Insert a decimal point into your answer directly under the decimal points in the original numbers.

The answer is

.### Examples

#### Example 1

Earlier, you were given a problem about Lindsey and her savings account.

She deposited $415.85 into the account and expects to earn $8.317 in interest in one year. She wants to figure out how much money she will have in her account after one year.

First, round both numbers to the nearest cent, which is the nearest hundredth. $415.85 is already to the nearest hundredth, but $8.317 needs to be rounded.

- 8.317 rounds to 8.32

Now, find the sum of the two rounded numbers. Start by arranging the numbers vertically so that the decimal points line up.

Next, add from right to left. Start by adding 5 and 2 to make 7. Then add 8 and 3 to make 11. Carry the 1 and add it to 5 and 8 to make 14. Carry the 1 and add it to 41 to make 42. Insert a decimal point into your answer directly under the decimal points in the original numbers.

The answer is

, so Lindsey can expect to have $424.17 in her account after one year.#### Example 2

Round each value to the nearest tenth and then find the sum.

First, round each number to the nearest tenth. Remember that the tenths place is the first digit to the right of the decimal point. Look at each digit in the tenths place and then the digit to the right of that to decide if you will need to round up the digit in the tenths place or keep it the same.

- 4.56 rounds to 4.6
- 3.47 rounds to 3.5
- 2.04 rounds to 2.0
- 6.21 rounds to 6.2

Now, find the sum of the four rounded numbers. Start by arranging the numbers vertically so that the decimal points line up.

Next, add from right to left. Start by adding 6, 5, 0, and 2 to make 13. Carry the 1 and add it with the 4, 3, 2, and 6 to make 16. Insert a decimal point into your answer directly under the decimal points in the original numbers.

The answer is

.#### Example 3

First, arrange the numbers vertically so that the decimal points line up.

Next, add one zero after the 2 in the second number so that all numbers have two digits to the right of the decimal point.

Now, add from right to left. Start by adding 6, 0 and 9 to make 15. Carry the 1 and add it with the 5, 2, and 8 to make 16. Once again carry the 1 and add it with the 4, 1, and 7 to make 13. Carry the 1 one more time and add it with the 3 to make 4. Insert a decimal point into your answer directly under the decimal points in the original numbers.

The answer is \begin{align*}4.56+1.20+37.89=43.65\end{align*}.

#### Example 4

First, arrange the numbers vertically so that the decimal points line up.

Next, add one zero after the 2 in the first number and one zero after the 4 in the third number so that all numbers have two digits to the right of the decimal point.

Now, add from right to left. Start by adding 0, 8, and 0 to make 8. Then add 2, 7, and 4 to make 13. Carry the 1 and add it with the 4, 6, and 3 to make 14. Once again carry the 1 and add it with the 1, 5, and 12 to make 19. Insert a decimal point into your answer directly under the decimal points in the original numbers.

The answer is

.#### Example 5

\begin{align*}189.34+123.5+7.2=\underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

First, arrange the numbers vertically so that the decimal points line up.

Next, add one zero after the 5 in the second number and one zero after the 2 in the third number so that all numbers have two digits to the right of the decimal point.

Now, add from right to left. Start by adding 4, 0, and 0 to make 4. Then add 3, 5, and 2 to make 10. Carry the 1 and add it with the 9, 3, and 7 to make 20. Carry the 2 and add it with the 8 and 2 to make 12. Carry the 1 and add it with the 1 and 1 to make 3. Insert a decimal point into your answer directly under the decimal points in the original numbers.

The answer is

.### Review

Find each sum.

- \begin{align*}29.451+711.36\end{align*}
- \begin{align*}0.956+0.9885\end{align*}
- \begin{align*}13.69+42.23\end{align*}
- \begin{align*}58.744+12.06+8.8\end{align*}
- \begin{align*}67.90+1.2+3.456\end{align*}

Round to the nearest tenth, then find the sum.

- \begin{align*}88.95+23.451\end{align*}
- \begin{align*}690.026+831.163\end{align*}
- \begin{align*}3.27+7.818\end{align*}
- \begin{align*}0.99+0.909+0.048\end{align*}
- \begin{align*}5.67+3.45+1.23\end{align*}

Round to the nearest hundredth, then find the sum.

- \begin{align*}511.51109+99.0986\end{align*}
- \begin{align*}0.744+7.005+0.7071\end{align*}
- \begin{align*}32.368+303.409\end{align*}
- \begin{align*}1.0262+1.0242\end{align*}
- \begin{align*}1.309+3.4590\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.4.