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# Whole Number Division

## Find quotients of multi-digit numbers.

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Whole Number Division

Jessica won a bet with her friends and received a lunchbox full of mints as her reward. Jessica counts 286 mints in total. She wants this supply to last until the end of the semester, which is 5 weeks away. If she only has mints on school days, how many can Jessica eat per day in order to make her supply last?

In this concept, you will learn how to divide whole numbers.

### Dividing Whole Numbers

The opposite operation of multiplication is division. To multiply means to add groups of matching things together, to divide means to split up into matching groups.

Let's look at an example.

72÷9=\begin{align*}72 \div 9 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

In this problem, 72 is the dividend - it is the number being divided. The divisor is the number of parts that the dividend is being split into, in this case, 9. The answer to a division problem is called the quotientOne way to complete this problem and find the quotient is to recall multiplication facts and work backwards.

To divide 72 by 9, start be asking "What number multiplied by 9 equals 72?"

9×8=72\begin{align*}9\times 8=72\end{align*}

If 8 groups of 9 equal 72, then of course 72 can be split into 8 groups of 9.

72÷9=8\begin{align*}72\div 9=8\end{align*}

The quotient is 8.

Here is another example.

15÷2=\begin{align*}15 \div 2 =\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

This is tricky because 15 is not an even number. This means it won't divide evenly. When this happens and you are using only whole numbers, will be a remainder.

Start by asking "What number mutliplied by 2 comes closest to 15, without going over?"

2×7=14\begin{align*}2\times 7=14\end{align*}

So, 7 groups of 2 comes closest to 15, with 1 left over. That is the remainder. Use "r" to show that there is a remainder.

15÷2=7 r 1\begin{align*}15\div 2=7 \ r \ 1\end{align*}

When dividing larger numbers, it may be easier to keep things organized with a division box.

8)825¯¯¯¯¯¯¯¯¯¯¯\begin{align*}8 \overline{)825 \;}\end{align*}

Here there is a one digit divisor, 8, and a three digit dividend, 825. You need to figure out how many times 8 goes into 825. To do this, divide the divisor (8) into each digit of the dividend.

8)825¯¯¯¯¯¯¯¯¯¯¯\begin{align*}8 \overline{)825 \;}\end{align*}

First, divide 8 into 8. Of course the answer is 1. Put the 1 on top of the division box above the 8.

8)825¯¯¯¯¯¯¯¯¯1802\begin{align*}& \overset{ 1}{8\overline{ ) 825}}\\ & \underline{-8} \Bigg \downarrow\\ & \quad 02\end{align*}

Next, multiply 1 by 8 and subtract the result from the dividend. Then bring down the next number in the dividend (the 2).

Next, look at the next digit in the dividend. There are no 8's in 2, so put a 0 in the answer, next to the 1.

8)825¯¯¯¯¯¯¯¯¯ 108 025\begin{align*}& \overset{\ 10}{8\overline{ ) 825}}\\ & \underline{-8} \;\; \Bigg \downarrow\\ & \quad \ 025\end{align*}

Because 8 wouldn't divide into 2, bring down the next number, 5, and use the two numbers together: 25

Next, look at the next digit in the dividend. There are three 8's in 25, with a remainder of 1. Add this into the answer.

8)825¯¯¯¯¯¯¯¯¯¯¯ 103 r1  8   025241\begin{align*}& \overset{\ \quad 1 \, 0 \, 3 \ \, r1}{8\overline{ ) 825 \;}}\\ & \ \ \underline{\, -8 \ \ }\\ & \ \quad 025\\ & \quad \underline{-24}\\ & \qquad 1\end{align*}

You can check your work by multiplying the answer by the divisor.

103 ×  8 824+r of 1=825\begin{align*}& \qquad 103\\ & \ \underline {\times \quad \ \ 8 \ }\\ & \qquad 824 + r \ \text{of} \ 1 = 825\end{align*}

You can apply these same steps to any division problem even if the divisor has two or three digits. Work through each value of the dividend with the value of the divisor. Then check your work by multiplying your answer by the divisor.

### Examples

#### Example 1

Earlier, you were given a problem about Jessica and her minty mother lode.

Jessica wants to make 286 mints last 5 school weeks.

First, figure out how many days Jessica needs to consider in her calculation, since she is only going to eat mints 5 days per week (on school days).

5×5=25\begin{align*}5\times 5=25\end{align*}

Next, Jessica needs to divide the large number of items by the small number of days.

286 mints÷25 days\begin{align*}286 \text{ mints} \div 25 \text{ days}\end{align*}

Perhaps it will be easier to use a division box

25)286¯¯¯¯¯¯¯¯¯¯¯\begin{align*}25\overline{ ) 286 \;}\end{align*}

01125) 286¯¯¯¯¯¯¯¯¯¯¯28    25  36   25 11 r11First, how many times does 25 go into 2? Put a 0 above the answer line.Next, bring down the 8.Next, how many times does 25 go into 28? Put a 1 above the answer line.2825=3, Then, bring down the 6.Finally, how many times does 25 go into 36? Put a 1 above the answer line.3625=11. This is the remainder.\begin{align*}_{0 \hspace{.4mm} 1 \hspace{.4 mm} 1} & _{\text{r11}}\\ 25\overline{ ) \ 286}& \quad \text{First, how many times does 25 go into 2? Put a 0 above the answer line.}\\ 28 \ \ \, & \quad \text{Next, bring down the 8.}\\ \underline{- \ \ 25} \ \ \, & \quad \text{Next, how many times does 25 go into 28? Put a 1 above the answer line.}\\ 36 \ & \quad 28 - 25 = 3, \ \text{Then, bring down the 6.}\\ \underline{ - \ \ 25} \ & \quad \text{Finally, how many times does 25 go into 36? Put a 1 above the answer line.}\\ 11 \ & \quad 36 - 25 = 11. \ \text{This is the remainder.}\\\end{align*}

The quotient is 11 remainder 11.

Jessica can have 11 mints per day and she will have 11 left at the end of the semester.

#### Example 2

Let’s look at a problem with a two-digit divisor.

12)2448¯¯¯¯¯¯¯¯¯¯¯¯02042404  48   48 0 First, note that 12 does not go into 2. Put a 0 above the answer line.How many times does 12 go into 24? Put a 2 above the answer line.Next, bring down the 4.How many times does 12 go into 4? Put a 0 above the answer line.Then, bring down the 8.Finally, how many times does 12 go into 48? Put a 4 above the answer line.There is no remainder.\begin{align*} \overset{\quad \, 0 \hspace{.3 mm}2 \hspace{.3 mm} 0 \hspace{.3 mm} 4}{12\overline{ ) 2448}}& \quad \text{First, note that 12 does not go into 2. Put a 0 above the answer line.}\\ \underline{-24 } \quad & \quad \text{How many times does 12 go into 24? Put a 2 above the answer line.}\\ 04 \ \ \, & \quad \text{Next, bring down the 4.}\\ & \quad \text{How many times does 12 go into 4? Put a 0 above the answer line.}\\ 48 \ & \quad \text{Then, bring down the 8.}\\ \underline{- \ \ 48} \ & \quad \text{Finally, how many times does 12 go into 48? Put a 4 above the answer line.}\\ 0 \ & \quad \text{There is no remainder.}\\\end{align*}

The quotient is 204.

You can check your work by multiplying: 204×12\begin{align*}204 \times 12\end{align*}.

204× 12408+ 20402448\begin{align*}204 & \\ \underline{\times \quad \ 12} & \\ 408 & \\ \underline{+ \quad \ 2040} & \\ 2448 & \\\end{align*}

#### Example 3

Find the quotient.

4)469¯¯¯¯¯¯¯¯¯¯¯\begin{align*}4\overline{ ) 469 \;}\end{align*} 1174 )469¯¯¯¯¯¯¯¯406    4  2  29   281 r1First, how many times does 4 go into 4? Put a 1 above the answer line.Next, bring down the 6.How many times does 4 go into 6? Put a 1 above the answer line.46=2Then bring down the 9. How many times does 4 go into 29? Put a 7 above the answer line.The one left over is the remainder.\begin{align*}_1 \hspace{.2 mm} _1 \hspace{.2 mm} _7 & \ _{\text{r1}}\\ 4 \ \big) \overline{ 469} & \\ \underline{-4 } \quad & \quad \text{First, how many times does 4 go into 4? Put a 1 above the answer line.}\\ 06 \ \ & \quad \text{Next, bring down the 6.}\\ \underline{- \ \ 4} \ \ & \quad \text{How many times does 4 go into 6? Put a 1 above the answer line.}\\ 2 \ \ & \quad 4 - 6 = 2\\ 29 & \quad \text{Then bring down the 9. How many times does 4 go into 29? Put a 7 above the answer line.}\\ \underline{- \ \ \ 28} & \\ 1 & \quad \text{The one left over is the remainder.}\\\end{align*}

The answer is 117 remainder 1.

#### Example 4

Find the quotient.

18)3678¯¯¯¯¯¯¯¯¯¯¯¯¯\begin{align*}18\overline{ ) 3678 \;}\end{align*}

020418) 3678¯¯¯¯¯¯¯¯¯¯¯¯¯36  3607  7  78  762r2First, how many times does 18 go into 3? Put a 0 above the answer line.Next, bring down the 6.Next, how many times does 18 go into 36? Put a 2 above the answer line.Then, bring down the 7.Then, how many times does 18 go into 7? Put a 0 above the answer line.Then, bring down the 8.Finally, how many times does 18 go into 78? Put a 4 above the answer line.7876=2. This is the remainder.\begin{align*}_{0 \hspace{.4mm} 2 \hspace{.4 mm} 0 \hspace{.4 mm} 4}& _{r2} \\ 18\overline{ ) \ 3678}& \quad \text{First, how many times does 18 go into 3? Put a 0 above the answer line.}\\ 36 \quad & \quad \text{Next, bring down the 6.}\\ \underline{- \ \ 36} \quad & \quad \text{Next, how many times does 18 go into 36? Put a 2 above the answer line.}\\ 07 \ \ \, & \quad \text{Then, bring down the 7.}\\ 7 \ \ \, & \quad \text{Then, how many times does 18 go into 7? Put a 0 above the answer line.}\\ 78 \, & \quad \text{Then, bring down the 8.}\\ \underline{ - \ \ 76} \, & \quad \text{Finally, how many times does 18 go into 78? Put a 4 above the answer line.}\\ 2 \, & \quad 78 - 76 = 2. \ \text{This is the remainder.}\\\end{align*}

The answer is 204 remainder 2.

#### Example 5

Find the quotient.

20)5020¯¯¯¯¯¯¯¯¯¯¯¯¯\begin{align*}20\overline{ ) 5020 \;}\end{align*}
025120) 5020¯¯¯¯¯¯¯¯¯¯¯¯¯50  40102    100  20   20 0 First, how many times does 20 go into 5? Put a 0 above the answer line.Next, bring down the 0.Next, how many times does 20 go into 50? Put a 2 above the answer line.5040=10 Bring down the 2.Next, how many times does 20 go into 102? Put a 5 above the answer line.100102=2. Then, bring down the 0.20 goes into 20 once.There is no remainder.\begin{align*}_{0 \hspace{.4mm} 2 \hspace{.4 mm} 5 \hspace{.4 mm} 1}& \\ 20\overline{ ) \ 5020}& \quad \text{First, how many times does 20 go into 5? Put a 0 above the answer line.}\\ 50 \quad & \quad \text{Next, bring down the 0.}\\ \underline{- \ \ 40} \quad & \quad \text{Next, how many times does 20 go into 50? Put a 2 above the answer line.}\\ 102 \ \ \, & \quad 50 - 40 = 10 \ \text{Bring down the 2.}\\ \underline{ - \ \ 100} \ \ \, & \quad \text{Next, how many times does 20 go into 102? Put a 5 above the answer line.}\\ 20 \ & \quad 100 - 102 = 2. \ \text{Then, bring down the 0.}\\ \underline{- \ \ 20} \ & \quad 20 \text{ goes into 20 once.}\\ 0 \ & \quad \text{There is no remainder.} \\\end{align*}

### Review

Find the quotient.

1. 12÷6=\begin{align*}12 \div 6 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
2. 13÷4=\begin{align*}13 \div 4 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
3. 132÷7=\begin{align*}132 \div 7 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
4. 124÷4=\begin{align*}124 \div 4 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
5. 130÷5=\begin{align*}130 \div 5 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
6. 216÷6=\begin{align*}216 \div 6 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
7. 1,161÷43=\begin{align*}1,161 \div 43 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
8. 400÷16=\begin{align*}400 \div 16 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
9. 1,827÷21=\begin{align*}1,827 \div 21 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
10. 1,244÷40=\begin{align*}1,244 \div 40 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
11. 248÷18=\begin{align*}248 \div 18 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
12. 3,264÷16=\begin{align*}3,264 \div 16 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
13. 4,440÷20=\begin{align*}4,440 \div 20 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
14. 7,380÷123=\begin{align*}7,380 \div 123 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
15. 102,000÷200=\begin{align*}102,000 \div 200 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
16.  \begin{align*}10,976 \div 98 = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

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### Vocabulary Language: English

TermDefinition
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.
Remainder A remainder is the value left over if the divisor does not divide evenly into the dividend.