5. Divide Decimals
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Chapter 3
5.
Divide Decimals
This lesson focuses on teaching the methods for dividing decimals, a crucial skill in mathematics. It introduces the concept of compatible numbers, which are numbers that make mental calculations easier. For instance, if you're trying to divide 612 by 9, rounding 612 to 600 makes the math simpler. The lesson also delves into the traditional method of long division for more complex calculations. These techniques are particularly useful in everyday scenarios where quick mental math is needed, such as calculating discounts or splitting bills. Whether you're a student looking to improve your math skills or an adult wanting to make everyday calculations easier, this lesson offers valuable insights.
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| | 9 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Divide Decimals
Slide of 9
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's Piggy Bank
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?
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c How many dimes does she need to make $3?
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d How many pennies does she need to make $0.3?
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Discussion
Compatible Numbers
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.
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.
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.
| 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
Estimating the Number of Dimes in Piggy Bank
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.
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Hint
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.
Solution
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
Dividing Decimals
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.
In this case, the divisor must be multiplied by 101 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.
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
expand_more 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.
2
Divide in the Same Way as Dividing Whole Numbers
expand_more Start by dividing the whole number part of the dividend by the divisor.
3
Place the Decimal Point and Continue the Division
expand_more 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
How Many Hours Should LaShay Work?
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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Hint
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.
Solution
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 102. Make sure the divided is multiplied by the same number as the divisor.
Example
How Far Does the Drone Go?
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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Hint
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.
Solution
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. 
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 103. Make sure the divided is multiplied by the same number as the divisor.
Pop Quiz
Finding the Quotient of Two Decimal Numbers
Divide the decimal numbers. Make sure that the decimal point is placed correctly. Round the answer to two decimal places, if necessary.
Closure
Money in the Piggy Bank
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?
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b How many ten dollar bills does she need to make $300?
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c How many dimes does she need to make $3?
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d How many pennies does she need to make $0.3?
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Hint
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.
Solution
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.
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. 
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 by 1030÷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.
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 by 10030÷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.
Tiffaniqua spends most of her time on the weekends helping at a farm. Tiffaniqua asks the head farmer a question.
If one gallon of milk weighs about 8.6 pounds, estimate the number of gallons of milk a cow produces each day.
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We know that the average cow produces about 47 pounds of milk per day. We also know that one gallon of milk weighs about 8.5 pounds. Let's divide the weight of milk produced by one cow by the number of pounds in one gallon. 47 ÷ 8.6 This quotient is equal to the number of gallons that one cow produces in one day. Let's round the divisor and the dividend so that they are compatible numbers. Start by rounding 8.6 to the closest whole number. 8.6 ≈ 9 Now, we are looking for a number that is close to 47 and is divisible by 9. 47 ≈ 45 Since 45 is close to 47 and it is a multiple of 9, we round 47 to 45. Now, we can estimate the quotient.
We found that one cow produces about 5 gallons of milk in one day.
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Dylan has 4.6 gigabytes of storage available on his phone. The average size of a song is 0.005 gigabyte.
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We need to divide the available space on Dylan's phone by the average size of a song to find the number of songs the phone can store. 4.6 ÷ 0.005 Dividing decimal numbers is easier if we multiply the divisor and the dividend by the same power of 10 so that the divisor is a whole number. In this case, we will multiply by 10^3, or 1000. 4.6 * 1000 & = 4600 0.005 * 1000& = 5 This means that 4.6 ÷ 0.005 is the same as 4600÷ 5. Let's now calculate it. We will place the first digit of the quotient in the hundreds place because 46 contains nine groups of 5.
This means that Dylan's phone can store 920 songs.
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Zosia rewrites 137.28 divided by 0.64 as follows.
{"type":"choice","form":{"alts":["<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7109375em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">A<\/span><span class=\"mord textbf\">.\u00a0<\/span><\/span><\/span><\/span><\/span> No, the dividend must be <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">3<\/span><span class=\"mord\">.<\/span><span class=\"mord\">7<\/span><span class=\"mord\">2<\/span><span class=\"mord\">8<\/span><span class=\"mord\">.<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7109375em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">B<\/span><span class=\"mord textbf\">.\u00a0<\/span><\/span><\/span><\/span><\/span> No, the dividend must be <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">3<\/span><span class=\"mord\">7<\/span><span class=\"mord\">2<\/span><span class=\"mord\">8<\/span><span class=\"mord\">.<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.73046875em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">C<\/span><span class=\"mord textbf\">.\u00a0<\/span><\/span><\/span><\/span><\/span> No, the divisor must be <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">6<\/span><span class=\"mord\">4<\/span><span class=\"mord\">0<\/span><span class=\"mord\">0<\/span><span class=\"mord\">.<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7109375em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">D<\/span><span class=\"mord textbf\">.\u00a0<\/span><\/span><\/span><\/span><\/span> Yes, Zosia is correct."],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
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We are told that Zosia wants to find the quotient of 137.28 and 0.64 with the given procedure.
We want to find if she is correct. Dividing decimal numbers is easier if we multiply the divisor and the dividend by the same power of 10 so that the divisor is a whole number. In this case, we need to multiply by 100 because the divisor 0.64 has two decimal places. 137.28* 100 &= 13728 0.64* 100 &=64 This means that 137.28÷ 0.64 is the same as 13728÷ 64. We obtained a different answer than our friend. Our friend is not correct because she multiplied the divisor by 100 but divided the dividend by 100. Remember that we must multiply both the divisor and the dividend by the same number when using this method. The answer is then B.
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The diagram shows how Ignacio divides two decimal numbers.
{"type":"choice","form":{"alts":["<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7109375em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">A<\/span><span class=\"mord textbf\">.\u00a0<\/span><\/span><\/span><\/span><\/span> No, the quotient must be <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mord\">.<\/span><span class=\"mord\">0<\/span><span class=\"mord\">9<\/span><span class=\"mord\">4<\/span><span class=\"mord\">.<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7109375em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">B<\/span><span class=\"mord textbf\">.\u00a0<\/span><\/span><\/span><\/span><\/span> No, the dividend must be multiplied by <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8141079999999999em;vertical-align:0em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\"><span class=\"mord\">0<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> first.","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.73046875em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">C<\/span><span class=\"mord textbf\">.\u00a0<\/span><\/span><\/span><\/span><\/span> Yes, Ignacio is correct.","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7109375em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">D<\/span><span class=\"mord textbf\">.\u00a0<\/span><\/span><\/span><\/span><\/span> No, the quotient must be <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mord\">.<\/span><span class=\"mord\">9<\/span><span class=\"mord\">0<\/span><span class=\"mord\">4<\/span><span class=\"mord\">.<\/span><\/span><\/span><\/span>"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}
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Let's examine the steps Ignacio followed.
The operations seem correct at first glance. Let's find the quotient ourselves to find out if it really is. In this case, we divide a decimal number by a whole number. This is why we divide as we would with whole numbers. We also place the decimal point in the quotient directly above its place in the dividend. Let's do it!
We put a zero in the tenths place because there is no group of 7 in 6. Then, we continue as usual. As a result, we obtained a different answer than our friend. The answer is A.
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Divide Decimals
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