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Dividing decimals is similar to dividing whole numbers. The difference is that a decimal point is in play. It is crucial to understand where to place it within the quotient. The methods for dividing decimals will be developed in this lesson.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Challenge

LaShay is fascinated by numbers and how she can buy her favorite things. She is currently saving money to buy a used $4K$ drone.

Help LaShay answer the following questions to support her math skills. Doing so will help her improve her saving plan.

a How many one-hundred dollar bills does she need to make $$3000?$

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b How many ten dollar bills does she need to make $$300?$

c How many dimes does she need to make $$3?$

d How many pennies does she need to make $$0.3?$

Discussion

Compatible numbers are numbers that help make doing mental math more manageable. They are typically used to estimate quotients. Consider finding the estimation of the following quotient.

Keep in mind that the estimates hold value as they are close to the exact values of the quotients.

$612÷9 $

It might seem difficult to divide these numbers using mental math. However, both numbers can be rounded to compatible numbers — $612$ to $600$ and $9$ to $10.$ An estimate for $612÷9$ is then found by mentally dividing $600$ by $10,$ which is equal to $60.$ That means the given quotient is estimated to be about $60.$
The compatible numbers previously chosen are not the only options. Any numbers easily divisible and close to the original numbers are worth exploring. For example, rather than rounding $612$ to $600,$ try $630.$ Notice that $63$ is a multiple of $9.$ This makes $630$ evenly divisible by $9.$ The estimate for the quotient is then $630÷9=70.$

Compatible numbers are not used to get precise answers. They are used to get quick and rough approximations. The pairs of numbers in the multiplication table are also compatible numbers. This is why it is a good habit to choose a pair from the multiplication table when calculating a quotient. The following table shows some compatible numbers for different quotients.

Quotients | Exact Value | Use Compatible Numbers | Estimate |
---|---|---|---|

$615÷8$ | $76.875$ | $640÷8$ | $80$ |

$36÷11.25$ | $3.2$ | $36÷12$ | $3$ |

$52.82÷4.75$ | $11.12$ | $50÷5$ | $10$ |

Example

LaShay decides to use one of the all-time classic methods of saving money — a piggy bank! She then weighs it. All of the coins in the bank add up to $137.79$ grams.

There is something interesting about LaShay's piggy bank. LaShay is nit-picky about what coins she puts in the bank — only dimes! The weight of one dime is $2.27$ grams. Use compatible numbers to estimate the number of dimes in the piggy bank.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"About","formTextAfter":null,"answer":{"text":["70"]}}

Divide the total weight of all the money in the piggy bank by the weight of a dime. Then, round the weight of a dime to a whole number.

The total weight of all the money in the piggy bank should be divided by the weight of one dime to find the *exact* number of the dimes.

$137.79÷2.27 $

This quotient can be estimated using compatible numbers. First, round the divisor to a whole number. The digit in the tenths place is $2,$ which is less than $5.$ This is why the divisor is rounded to $2.$
$137.79 ÷ 2.27↓2 $

Recall that compatible numbers are numbers that make doing mental math more straightforward. One way to know that the numbers will be compatible is to round the dividend to a number that is a multiple of $2.$ In this case, it can be $140$ because it is close to $137.79$ and it is a multiple of $2.$ $137.79↓140 ÷÷ 2.27↓2 $

The quotient of $140$ and $2$ is $70.$ LaShay's now knows that her piggy bank contains about $70.$
Discussion

The process of dividing decimals is similar to the process of dividing whole numbers using long division. What differentiates these processes is how the decimal point is placed when dividing decimals. Consider dividing the following decimal numbers.
*expand_more*
*expand_more*
*expand_more*

$58.46÷3.7 $

The following steps can be followed to divide the decimals.
1

Multiply the Divisor and the Dividend by the Same Power of $10$

When dividing decimals, the divisor is changed to a whole number. This requires multiplying the divisor and the dividend by the same power of $10.$ First, write the division using long division notation. The dividend is $58.46$ and the divisor is $3.7.$

In this case, the divisor must be multiplied by $10_{1}$ so that the decimal point moves to the right and the divisor becomes a whole number. The dividend is also multiplied by the same power of $10.$2

Divide in the Same Way as Dividing Whole Numbers

Start by dividing the whole number part of the dividend by the divisor.

3

Place the Decimal Point and Continue the Division

Now place the decimal point in the quotient above the decimal point in the dividend. Then bring down the digit in the tenths place and continue dividing as usual.

The quotient of $58.46$ and $3.7$ is $15.8.$

Example

LaShay needs $$279.99$ to buy the drone. She has already saved $$125.75$ up to this point. She then decided to mow her neighbor's lawn to make the rest of money she needs. Her neighbor agrees to pay her $$9.25$ per hour.

a Estimate the number of hours LaShay needs to work to earn enough to buy the drone.

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b Find the exact number of hours. Round the answer to one decimal place.

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a Subtract the amount that LaShay saved from the price of the drone. Use compatible numbers to estimate the quotient of the difference and $9.25.$

b Start by converting the divisor into a whole number.
Use the standard division algorithm to divide the decimals.

a Start by finding the amount LaShay needs. It is the difference between the price of the drone, $$279.99,$ and the amount LaShay has saved, $$125.75.$

$− 279.99125.75154.24 $

The number of hours LaShay needs to work to earn this difference can be found by dividing it by the hourly wage, $9.25.$
$154.24÷9.25 $

Compatible numbers will be used to estimate the quotient. First, round the divisor to a whole number. For $9.25,$ the digit in the tenths place is $2,$ which is less than $5.$ This is why $9.25$ is rounded to $9.$
$154.24 ÷ 9.25↓9 $

Find a number that is compatible with $9.$ This means that the number should be a multiple of $9$ and close to $154.24.$ It can be either $90$ or $180.$ Since $180$ is closer, the dividend can be replaced with $180.$
$154.24↓180 ÷÷ 9.25↓9 =20 $

The quotient of $180$ and $9$ is $20.$ Therefore, LaShay needs to work about $20$ hours to earn $$154.24.$ Note that the goal here is to find an answer quickly, which leads to finding imprecise answers. The exact value will probably be different from this estimation.
b Recall the quotient from Part A that represents the number of hours LaShay needs to work.

$154.24÷9.25⇓9.25)154.24 $

This is a division of two decimal numbers. The first step in dividing decimals is to convert the divisor into a whole number. Since the divisor has $two$ decimal places, it will be multiplied by $10_{2}.$ Make sure the divided is multiplied by the same number as the divisor.
Now it is a division of two whole numbers. Use long division to calculate the quotient.

The quotient is $16.674$ with a remainder of $550.$ This means that LaShay has to work about $16.7$ hours to save enough to buy the drone.

Example

a LaShay managed to save enough money to buy the drone.

External credits: macrovector

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b LaShay notices a bee nest on a tree branch while flying the drone in the backyard of the house.

Her grandfather, who used to be a beekeeper, tells LaShay the following.

So you want to approximate the number of bees in that hive huh? Well, count the number of bees that leave the hive in one minute. Then multiply it by $3$ and divide by $0.014.$ |

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a Decide if the divisor is a whole number. Then, use long division to divide the decimals.

b What can be done when the divisor is not a whole number? Multiply it by a power of $10.$

a Divide $115.45$ by $5$ to find how far the drone can travel.

$115.45÷5⇓5)115.45 $

The steps for dividing a decimal by a whole number is the same as the steps for dividing whole numbers. However, keep in mind that the decimal point in the quotient is placed above the decimal point of the dividend.
The result of the division is $23.09.$ Therefore, the drone can go $23.09$ meters in one minute.

b Now, it is time to find the number of bees in the nest. The number of bees that leave the hive in one minute is multiplied by $3,$ and then divided by $0.014.$

$(Number of Bees×3)÷0.014 $

LaShay counts $28$ bees that leave the nest in one minute. Multiply this number by $3.$ $× 228384 $

Now, divide this number by $0.014$ using long division.
$84÷0.014⇓0.014)84 $

In this case, the divisor is a decimal. The first step in dividing decimals is to convert the divisor into a whole number. Since the divisor has $three$ decimal places, it will be multiplied by $10_{3}.$ Make sure the divided is multiplied by the same number as the divisor.
Now that this is a division of two whole numbers, they can be divided as usual.

This means that there are about $6000$ bees in the nest.

Pop Quiz

Divide the decimal numbers. Make sure that the decimal point is placed correctly. Round the answer to two decimal places, if necessary.

Closure

When dividing decimals, the divisor is converted into a whole number by multiplying it by a power of $10.$ However, the dividend must also be multiplied by the same power of $10$ to keep the value of the quotient the same. Now take a look at LaShay's piggy bank problem.

a How many one-hundred dollar bills does she need to make $$3000?$

b How many ten dollar bills does she need to make $$300?$

c How many dimes does she need to make $$3?$

d How many pennies does she need to make $$0.3?$

a Use long division to divide $3000$ by $100.$

b Use long division to divide $300$ by $10.$

c A dime is worth $$0.1.$ Divide the decimal $3$ by $0.1.$ Rewrite it so that the divisor is $1.$

d A penny is worth $$0.01.$ Divide the decimal $0.3$ by $0.01.$

a The number of one-hundred dollar bills is found by dividing $3000$ by $100.$

$3000÷100 $

Long division can be used to find this quotient.
The quotient is $30.$ LaShay needs $30$ one-hundred dollar bills to make $$3000.$

b Use the same reasoning as in Part A. Divide $300$ by $10$ to find the number of ten dollar bills.

$300÷10 $

This is also equal to $30.$
LaShay can make $$300$ with $30$ ten dollar bills.

c Recall that a dime is worth $$0.1.$ To find the number of dimes needed to make $$3,$ divide $3$ by $0.1.$

$3÷0.1 $

In this case, the divisor is a decimal number. The quotient must be rewritten so that the divisor is a whole number. Multiplying the divisor by $10$ will give the desired divisor. Keep in mind that the dividend must also be multiplied by the same number.
$Quotient3÷0.1⇓Multiplied by1030÷1 $

When a number is divided by $1,$ the result will be the number itself. Check using the division algorithm to see if it really is.
The result is indeed $30.$ Therefore, $30$ dimes are need to make $$3.$

d A penny is worth $$0.01$ and LaShay wants to make $$0.3$ with pennies. Then, $0.3$ must be divided by $0.01$ to find the number of pennies she needs.

$0.3÷0.01 $

Like in the previous part, the divisor is a decimal number. Multiply the divisor and the dividend by $100$ to make the divisor a whole number.
$Quotient0.3÷0.01⇓Multiplied by10030÷1 $

After multiplication, the quotient becomes the same quotient as in the previous part. Since $30$ divided by $1$ is $30,$ the quotient, $0.3÷0.01,$ is also $30.$
$0.3÷0.01=30 $

This means that LaShay can make $$0.3$ using $30$ pennies. On a final note, examine the table created using the quotients mentioned in these examples. Dividend | Divisor | Quotient |
---|---|---|

$3000$ | $100$ | $30$ |

$300$ | $10$ | $30$ |

$30$ | $1$ | $30$ |

$3$ | $0.1$ | $30$ |

$0.3$ | $0.01$ | $30$ |

The numbers in the first columns get smaller downwards by a factor of $10.$ The same is true for the numbers in the second column. However, the quotient always stays the same. Therefore, when the dividend and divisor **both** increase by the same factor of $10,$ the quotient remains the same.

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