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# Decimal Rounding and Division

## Use digit following required place value for rounding

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Decimal Rounding and Division

Vincent is looking at a cupcake recipe. The recipe makes a dozen cupcakes, but he only wants to make 4 cupcakes. He knows that 4 out of 12 is also one-third. If he divides the ingredients in the recipe by 3, he will have enough ingredients to make just 4 cupcakes. The recipe calls for 215 grams of sugar and his kitchen scale only measures grams to the tenths place. How much sugar should Vincent measure out to make a third of the recipe?

In this concept, you will learn to divide and round decimals.

### Rounding Decimals When Dividing

You can divide decimals by whole numbers and use zero placeholders to find the most accurate quotient. Some quotients are so long that it is difficult to decipher the value of the decimal. Rounding the decimal number to a decimal place value would make the number easier to evaluate.

Remember to look at the number to the right of the place value you are rounding to. If the number is less than 5, round the number down. If the number is equal to or greater than 5, round the number up.

Round 2.1046891425 to the nearest hundredths place. The number to the right of the hundredths place is 4. Round the number down.

2.10468914252.10\begin{align*}\begin{array}{rcl} && 2.1\underline{0}46891425 \approx 2.10\\ && \quad \quad \uparrow \end{array}\end{align*}

Round 2.1046891425 to the nearest ten-thousandths place. The number to the right of the ten-thousandths place is 8. Round the number up.

2.10468914252.1047\begin{align*}\begin{array}{rcl} && 2.104\underline{6}891425 \approx 2.1047\\ && \qquad \quad \uparrow \end{array}\end{align*}

Here is a decimal division problem. Round the quotient to the nearest ten-thousandths place.

1.265÷4\begin{align*}1.265 \div 4\end{align*}

First, divide to find the quotient. Remember to bring up the decimal point in the same location as the dividend.

4)1.26500¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯   0.31625 12  06 425 24  10     8    20 20 0\begin{align*}\begin{array}{rcl} && \overset{\ \ \ 0.31625}{4 \overline{ ) {1.265{\color{red}00}}}}\\ && \ \underline{-12}\\ && \quad \ \ 06\\ && \quad \ \underline{-4}\\ && \quad \quad 25 \\ && \quad \ \underline{-24}\\ && \quad \quad\ \ 1{\color{red}0} \\ && \quad \ \ \ \ \ \underline{-8}\\ && \quad \quad \ \ \ \ 2{\color{red}0} \\ && \quad \quad \ \underline{-20}\\ && \qquad \quad \ 0 \end{array}\end{align*}

Then, take the quotient and round it to the nearest ten-thousandths place. Look at the digit to the right of the ten-thousandths; it is 5. Round the number up.

0.316250.3163\begin{align*}\begin{array}{rcl} &&0.316\underline{2}5 \approx 0.3163\\ && \qquad \quad \uparrow \end{array}\end{align*}

The quotient of 1.265 divided by 4 rounded to the nearest ten-thousandths place is 0.3163.

Keep in mind some decimal quotients can be quite long.

Here is a division problem with a very long quotient. Round the quotient to thousandths.

13.87÷7=1.98142857142857131.98142857142857131.981 \begin{align*}\begin{array}{rcl} && 13.87 \div 7 = 1.9814285714285713\\ && 1.98\underline{1}4285714285713 \approx 1.981\\ && \qquad \ \uparrow \end{array}\end{align*}

The quotient of 13.87 divided by 7, rounded to the thousandths is 1.981

If you know you will round the quotient, you can stop dividing after you’ve found the digit to the right of the place value you are rounding to. Take for example the problem above. Stop dividing once you’ve found the digit in the ten-thousandths place. The digits that come afterwards will not affect the rounding of the decimal number.

7)13.8700¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯   1.9814  7 68  63   5756 10    7   30    282\begin{align*}\begin{array}{rcl} && \overset{\ \ \ 1.98 \underline{1}4}{7 \overline{ ) {13.8700}}}\\ && \ \ \underline{-7}\\ && \quad \ 68\\ && \ \ \underline{-63}\\ && \quad \ \ \ 57 \\ && \quad \underline{-56}\\ && \quad \quad \ 10 \\ && \quad \ \ \ \ \underline{-7}\\ && \quad \quad \ \ \ 30 \\ && \quad \ \ \ \ \underline{-28}\\ && \qquad \quad 2 \end{array}\end{align*}

Round 1.9814 to the thousandths place.

1.98141.981  \begin{align*}\begin{array}{rcl} && 1.98\underline{1}4 \approx 1.981 \\ && \qquad \ \ \uparrow \end{array}\end{align*}

The answer is the same, 1.981.

### Examples

#### Example 1

Earlier, you were given a problem about Vincent making cupcakes.

He is trying to make a third of a cupcake recipe by dividing the ingredients by 3. Since his kitchen scale only measures to a tenths of a gram, find a third of 215 grams of sugar, rounded to the tenths place.

First, divide to find the quotient. You can stop at the hundredths place.

3)215.00¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯   71.66 21  05     3    20 18  20   18   2\begin{align*}\begin{array}{rcl} && \overset{\ \ \ 71. \underline{6}6}{3 \overline{ ) {215.{\color{red}00}}}}\\ && \ \underline{-21}\\ && \quad \ \ 05\\ && \ \ \ \ \ \underline{-3}\\ && \quad \ \ \ \ 2{\color{red}0} \\ && \quad \ \underline{-18}\\ && \quad \quad \ \ 2{\color{red}0} \\ && \quad \ \ \ \underline{-18}\\ && \qquad \ \ \ 2 \end{array}\end{align*}

Then, round the quotient to the tenths place.

71.6671.7   \begin{align*}\begin{array}{rcl} && 71.\underline{6}6 \approx 71.7\\ && \quad \ \ \ \uparrow \end{array}\end{align*}

#### Example 2

Find the quotient and round it to the nearest thousandth.

0.45622÷4\begin{align*}0.45622 \div 4\end{align*}

First, divide to find the quotient. Add zero placeholders if necessary. Remember to bring up the decimal point in the same location as the dividend. Divide all the way or you can stop dividing once you’ve found the digit in the ten-thousandths place.

4)0.456220¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯   0.114055   4  05 4 16  16   02  0 22  20   20200\begin{align*}\begin{array}{rcl} && \overset{\ \ \ 0.114055}{4 \overline{ ) {0.45622{\color{red}0}}}}\\ && \ \ \ \underline{-4}\\ && \quad \ \ 05\\ && \quad \ \underline{-4}\\ && \quad \quad \ 16 \\ && \quad \ \ \underline{-16}\\ && \quad \quad \ \ \ 02 \\ && \quad \quad \ \ \underline{-0}\\ && \quad \quad \quad \ 22 \\ && \quad \quad \ \ \underline{-20}\\ && \quad \quad \quad \ \ \ 2{\color{red}0} \\ && \quad \quad \quad \underline{-20}\\ && \qquad \qquad 0 \end{array}\end{align*}

OR

4)0.45622¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯   0.1140    4   05  4 16  16   02  0    2\begin{align*}\begin{array}{rcl} && \overset{\ \ \ 0.1140}{4 \overline{ ) {0.45622}}}\\ && \ \ \ \ \underline{-4}\\ && \quad \ \ \ 05\\ && \quad \ \ \underline{-4}\\ && \quad \quad \ 16 \\ && \quad \ \ \underline{-16}\\ && \quad \quad \ \ \ 02 \\ && \quad \quad \ \ \underline{-0}\\ && \qquad \ \ \ \ 2 \end{array}\end{align*}

Then, round the quotient to the thousandths place. Look at the digit to the right in the ten-thousandths place; it is 0. Round the number down.

0.1140550.114OR0.11400.114  \begin{align*}\begin{array}{rcl} && 0.11\underline{4}055 \approx 0.114 \quad \text{OR} \quad 0.11\underline{4}0 \approx 0.114\\ && \qquad \ \uparrow \qquad \qquad \qquad \qquad \qquad \ \uparrow \end{array}\end{align*}

The quotient of 0.45622 divided by 4 rounded to the thousandths place is 0.114. Both methods will bring you to the same conclusion. The method on the right may be quicker if you only need a rounded quotient.

Find the quotient rounded to the nearest thousandths for the following problems.

#### Example 3

0.51296÷2=\begin{align*}0.51296 \div 2 = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

First, divide to find the quotient.

2)0.51296¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯   0.25648   4  11   10    12 12  09     8    16     16     0\begin{align*}\begin{array}{rcl} && \overset{\ \ \ 0.25648}{2 \overline{ ) {0.51296}}}\\ && \ \ \ \underline{-4}\\ && \quad \ \ 11\\ && \ \ \ \underline{-10}\\ && \quad \ \ \ \ 12 \\ && \quad \ \underline{-12}\\ && \quad \quad \ \ 09 \\ && \quad \ \ \ \ \ \underline{-8}\\ && \quad \quad \ \ \ \ 16\\ && \quad \ \ \ \ \ \underline{-16}\\ && \qquad \ \ \ \ \ 0 \end{array}\end{align*}

Then, round the quotient to the thousandths place.

0.256480.256  \begin{align*}\begin{array}{rcl} && 0.25\underline{6}48\approx 0.256\\ && \qquad \ \ \uparrow \end{array}\end{align*}

The quotient of 0.51296 divided by 2 is 0.256.

#### Example 4

10.0767÷3=\begin{align*}10.0767 \div 3 = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

First, divide to find the quotient.

3)10.0767¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯   3.3589  9 10    9   1715     26  24   27    27    0\begin{align*}\begin{array}{rcl} && \overset{\ \ \ 3.3589}{3 \overline{ ) {10.0767}}}\\ && \ \ \underline{-9}\\ && \quad \ 10\\ && \ \ \ \ \underline{-9}\\ && \quad \ \ \ 17 \\ && \quad \underline{-15}\\ && \quad \ \ \ \ \ 26 \\ && \quad \ \ \underline{-24}\\ && \quad \quad \ \ \ 27\\ && \quad \ \ \ \ \underline{-27}\\ && \qquad \ \ \ \ 0 \end{array}\end{align*}

Then, round the quotient to the thousandths place.

3.35893.359 \begin{align*}\begin{array}{rcl} && 3.35\underline{8}9 \approx 3.359\\ && \qquad \ \uparrow \end{array}\end{align*}

The quotient of 10.0767 divided by 3 is 3.359.

#### Example 5

0.48684÷2=\begin{align*}0.48684 \div 2 = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

First, divide to find the quotient. You can stop at the ten-thousandths place.

2)0.48684¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯   0.2434   4  08 8    06   6  08 804\begin{align*}\begin{array}{rcl} && \overset{\ \ \ 0.24 \underline{3}4}{2 \overline{ ) {0.48684}}}\\ && \ \ \ \underline{-4}\\ && \quad \ \ 08\\ && \quad \ \underline{-8}\\ && \quad \ \ \ \ 06 \\ && \quad \ \ \ \underline{-6}\\ && \quad \quad \ \ 08 \\ && \quad \quad \ \underline{-8}\\ && \qquad \quad 04 \end{array}\end{align*}

Then, round the quotient to the thousandths place.

0.24340.243 \begin{align*}\begin{array}{rcl} && 0.24\underline{3}4 \approx 0.243\\ && \qquad \ \uparrow \end{array}\end{align*}

The quotient of 0.48684 divided by 2 rounded to the thousandths is 0.243.

### Review

Find each quotient rounded to the nearest thousandths.

1. 0.54686÷2\begin{align*}0.54686 \div 2\end{align*}
2. 0.84684÷2\begin{align*}0.84684 \div 2\end{align*}
3. 0.154586÷2\begin{align*}0.154586 \div 2\end{align*}
4. 0.34689÷3\begin{align*}0.34689 \div 3\end{align*}
5. 0.994683÷3\begin{align*}0.994683 \div 3\end{align*}
6. 0.154685÷5\begin{align*}0.154685 \div 5\end{align*}
7. 0.546860÷5\begin{align*}0.546860 \div 5\end{align*}
8. 0.25465÷5\begin{align*}0.25465 \div 5\end{align*}
9. 0.789003÷3\begin{align*}0.789003 \div 3\end{align*}
10. 0.18905÷5\begin{align*}0.18905 \div 5\end{align*}
11. 0.27799÷9\begin{align*}0.27799 \div 9\end{align*}
12. 0.354680÷10\begin{align*}0.354680 \div 10\end{align*}
13. 0.454686÷6\begin{align*}0.454686 \div 6\end{align*}
14. 0.954542÷2\begin{align*}0.954542 \div 2\end{align*}
15. 0.8546812÷4\begin{align*}0.8546812 \div 4\end{align*}

To see the Review answers, open this PDF file and look for section 4.12.

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Color Highlighted Text Notes

### Vocabulary Language: English

Divide

To divide is split evenly into groups. The result of a division operation is a quotient.

Dividend

In a division problem, the dividend is the number or expression that is being divided.

divisor

In a division problem, the divisor is the number or expression that is being divided into the dividend. For example: In the expression $152 \div 6$, 6 is the divisor and 152 is the dividend.

Quotient

The quotient is the result after two amounts have been divided.