Dividing decimals
One morning’s gas for the carpool costs 8.37 dollars, and three families share it equally. How do you split a number with a decimal point into equal parts, where does the point go in the answer, and what do you do when the division will not come out even?
Where you are 8 in Decimal arithmetic
When you reach for it
Haeun, Rajan and Pelumi live on the same street, and their families run a carpool to school. One morning the gas costs 8.37 dollars. The three families split it equally. How much does each family pay?
That is a sharing question, so it is division: 8.37 \div 3. You already know how to share whole numbers with long division (see The standard long division algorithm). The new part is the decimal point. Where does it go in the answer? And what if the sharing does not come out even?
You reach for this lesson whenever an amount with a decimal point is split into a whole number of equal parts. A bill shared by friends. Miles driven in a week, turned into miles per day. Gallons of gas used on several trips, turned into gallons per trip. In every one, the amount being shared has a point, and the number of parts is a whole number.
This lesson takes three days. Part 1 shares a decimal with base-ten pieces and trades a one for ten tenths when the sharing gets stuck. Part 2 writes the same work as long division, with the point placed straight above. Part 3 rounds an answer that will not end, and shares among two-digit numbers of parts.
The procedure
Part 1: sharing a decimal, one place at a time
Think of 8.37 as base-ten pieces: 8 ones, 3 tenths and 7 hundredths (see Tenths and hundredths). A one is a large square. A tenth is a thin strip, one tenth of the square. A hundredth is a tiny square, one tenth of a strip. Now share the pieces among the three families, starting with the biggest pieces.
Ones. There are 8 ones. Each family gets 2 ones, which uses 3 \times 2 = 6 of them. That leaves 2 ones that cannot be shared out whole.
Trade the ones for tenths. One whole is ten tenths, so the 2 leftover ones are 20 tenths. Add the 3 tenths that were already there:
Tenths. Share 23 tenths three ways. Each family gets 7 tenths, which uses 21. That leaves 2 tenths.
Trade the tenths for hundredths. One tenth is ten hundredths, so 2 tenths are 20 hundredths. With the 7 hundredths already there, that is 27 hundredths.
Hundredths. Share 27 hundredths three ways. Each family gets 9 hundredths, and nothing is left.
Each family has 2 ones, 7 tenths and 9 hundredths. Written as a decimal, that is 2.79, so each family pays 2.79 dollars:
Notice what the trades did. A leftover piece was never thrown away. It was broken into ten smaller pieces and shared at the next place. That is the whole idea of this lesson, and it works because ten of each piece make one of the next piece up.
A second view: count in hundredths. The amount 8.37 dollars is 837 cents, and a cent is one hundredth of a dollar (see Money written as decimals). So the question is 837 hundredths shared by 3. That is whole-number division:
Each family gets 279 hundredths, which is 2.79. Same answer.
Check by multiplying. Division undoes multiplication. If each of 3 families pays 2.79 dollars, together they pay 3 \times 2.79, and that must be 8.37 (see Multiplying decimals). Counting hundredths, 3 \times 279 = 837, so 3 \times 2.79 = 8.37. The answer is right.
Part 2: long division with the point straight above
Sharing pieces is clear, but it is slow. Long division does the same steps in a short form. Set up 8.37 \div 3 just as you would for a whole number, and put a point in the answer line straight above the point in 8.37. Then divide as usual.
Read each step against Part 1. The 2 over the 8 is the 2 ones each family gets. Subtracting 6 leaves the 2 leftover ones. Bringing down the 3 makes 23: that is the trade, 2 ones becoming 20 tenths, plus the 3 tenths. The 7 is the tenths each family gets, and the last step shares 27 hundredths.
Why straight above? Each digit of the answer is written over the digit it came from. Sharing ones gives ones. Sharing tenths gives tenths. So the digit above the tenths digit of 8.37 is a tenths digit, and the point between ones and tenths must sit in the same spot in both rows. Once the point is placed, the division itself is exactly the whole-number algorithm. The point just rides along.
When the amount is smaller than the number of parts. Suppose 1.35 gallons of gas is split evenly over 3 trips. There is only 1 one, and 3 trips cannot each get a whole one. So each trip gets 0 ones, and you write 0 in the ones place of the answer. Trade the 1 one for 10 tenths: with the 3 tenths, that is 13 tenths. Each trip gets 4 tenths, using 12, with 1 tenth left. Trade it for 10 hundredths, add the 5, and share 15 hundredths: 5 each.
The answer starts with 0. because it is less than one. Each trip used less than one gallon. The 0 in the ones place holds that place, so the point has a digit in front of it.
The same holding job falls to a zero right after the point. In 0.27 \div 9, there are no ones and only 2 tenths, too few for 9 parts. So each part gets 0 tenths, and you write a 0 in the tenths place. Then 27 hundredths shared by 9 is 3 hundredths, and 0.27 \div 9 = 0.03. Leaving out that zero would give 0.3, ten times too big.
Writing zeros to keep dividing. In one week the carpool drives a total of 14.7 miles over 4 school days, the same distance each day. How far is that per day? Divide 14.7 by 4.
The first digit of 14.7 is 1, too small for 4 parts, so start with 14 ones: 3 each, 2 left. Trade them for 20 tenths, add the 7 tenths, and share 27 tenths: 6 each, 3 left.
Now the digits of 14.7 have run out, but 3 tenths are still left over. They can be shared too. Trade them for 30 hundredths. To show that in the layout, write a 0 after 14.7, in the hundredths place. A zero at the end, after the point, adds nothing, so
This is the same move as writing 5 as 5.000 to find \displaystyle \frac{5}{8} as a decimal (see Converting between fractions and decimals). Share 30 hundredths: 7 each, 2 left. Write one more zero, share 20 thousandths: 5 each, nothing left.
The carpool drives 3.675 miles each day. Check: 4 \times 3.675 is 4 \times 3675 = 14{,}700 thousandths, which is 14.700 = 14.7.
Part 3: rounding to a sensible place, and two-digit divisors
Some divisions never end. Share 10 dollars among 3 families: 10.000 \div 3 gives 3.333\ldots, and each new zero leaves 1 again, so the 3s go on forever (see Terminating and repeating decimals). Even a division that does end can give more digits than make sense: nobody pays 3.675 dollars, because there is no coin for a thousandth of a dollar.
So you choose a sensible place and round to it (see Rounding decimals). The place comes from the story, not from the arithmetic:
- Money is rounded to the cent, the hundredths place.
- A distance driven is usually rounded to the tenth of a mile, as a car’s trip meter shows it.
- A measurement is rounded to the place the measuring tool can read.
To round to a place, you need the digit one place further to the right. So divide one place past the place you want, then round. For money, that means dividing to the thousandths. For 10 \div 3:
The thousandths digit is 3, less than 5, so the cents stay at 3.33. The sign \approx says “is about”: the rounded answer is close, not exact.
A rounded share does not always add back to the bill. Three shares of 3.33 dollars are 3 \times 3.33 = 9.99 dollars, one cent short of 10. In real life one family pays 3.34 dollars. The arithmetic is not wrong: the exact share is between two cents, and someone has to cover the gap.
Two-digit divisors. When the number of parts has two digits, the long division is the one from Dividing by a two-digit number: estimate each digit of the answer, multiply, subtract, bring down. The point is placed straight above, exactly as before. An estimate guards it. For example, 45.60 \div 12: since 45.60 is close to 48, and 48 \div 12 = 4, the answer is close to 4. The long division gives the digits 380, and only 3.80 is close to 4. So 45.60 \div 12 = 3.80, and 12 \times 3.80 = 45.60 checks it.
Worked examples
Examples 1 and 2 go with Part 1, Examples 3 to 5 with Part 2, and Examples 6 and 7 with Part 3.
Parking at the school fair
Parking at the school fair costs 6.45 dollars. The three families split it equally. How much does each pay?
SolutionShare 6.45, which is 6 ones, 4 tenths and 5 hundredths, three ways, biggest pieces first.
The 6 ones share evenly: 2 ones each, none left.
The 4 tenths give 1 tenth each, using 3, with 1 tenth left. Trade it for 10 hundredths; with the 5 hundredths that is 15 hundredths.
The 15 hundredths give 5 hundredths each, with none left.
Each family gets 2 ones, 1 tenth and 5 hundredths, so
Each family pays 2.15 dollars.
Check: in cents, 645 \div 3 = 215, and 3 \times 2.15 = 6.45.
Juice boxes for four riders
Pelumi’s dad buys juice boxes for the 4 kids in the car. They cost 5.16 dollars in all. What is the cost per kid?
SolutionShare 5 ones, 1 tenth and 6 hundredths among 4.
The 5 ones give 1 one each, using 4, with 1 one left. Trade that one for 10 tenths. With the 1 tenth already there, that is 11 tenths.
The 11 tenths give 2 tenths each, using 8, with 3 tenths left. Trade them for 30 hundredths. With the 6 hundredths, that is 36 hundredths.
The 36 hundredths give 9 hundredths each, with none left. So
Each kid’s juice costs 1.29 dollars.
Check: multiply back. 4 \times 1.29 is 4 \times 129 = 516 hundredths, which is 5.16.
This example needed two trades, one at the ones and one at the tenths. Each trade turns a leftover piece into ten pieces of the next size down.
Miles per trip
Rajan’s mom drove 19.56 miles in 6 equal trips. How long was each trip?
SolutionSet up the long division 19.56 \div 6 and place the point in the answer straight above the point in 19.56.
The 1 is too small for 6 parts, so start with 19 ones: 6 \times 3 = 18, write 3 over the 9, and 19 - 18 = 1.
Bring down the 5 to make 15 tenths: 6 \times 2 = 12, write 2 in the tenths place, and 15 - 12 = 3.
Bring down the 6 to make 36 hundredths: 6 \times 6 = 36, write 6, and nothing is left.
Each trip was 3.26 miles.
Check: 19.56 is close to 18, and 18 \div 6 = 3. The answer 3.26 is close to 3, so the point is in the right place.
Less than one gallon a trip
The carpool used 2.45 gallons of gas on 7 school runs, the same amount each run. How much gas did one run use?
SolutionDivide 2.45 by 7, with the point straight above.
There are only 2 ones, and 7 runs cannot each get a whole one. Write 0 in the ones place of the answer, then the point.
Trade the 2 ones for 20 tenths. With the 4 tenths, that is 24 tenths. Now 7 \times 3 = 21, so write 3 in the tenths place, and 24 - 21 = 3.
Bring down the 5: 35 hundredths. 7 \times 5 = 35, so write 5, and nothing is left.
Each run used 0.35 gallon.
Check: 7 \times 0.35 = 7 \times 35 hundredths = 245 hundredths = 2.45. And an answer less than one makes sense, since 2.45 gallons is less than 7 gallons.
A zero to keep going
Over 6 school days the carpool drove 21.3 miles, the same each day. How far is that per day?
SolutionDivide 21.3 by 6, point straight above.
Start with 21 ones: 6 \times 3 = 18, write 3, and 21 - 18 = 3.
Bring down the 3 to make 33 tenths: 6 \times 5 = 30, write 5, and 33 - 30 = 3.
The digits of 21.3 have run out, but 3 tenths are left. Write a zero after 21.3, since 21.3 = 21.30, and bring it down: 30 hundredths. 6 \times 5 = 30, write 5, nothing left.
The carpool drove 3.55 miles a day.
Check: 6 \times 3.55 is 6 \times 355 = 2130 hundredths, which is 21.30 = 21.3.
Sharing a gas bill to the cent
Filling the tank costs 23.50 dollars, and the three families split it. How much should each pay, to the nearest cent? Do the three shares cover the bill?
SolutionMoney is rounded to the cent, the hundredths place, so divide to the thousandths and then round. Write 23.50 as 23.500.
Start with 23 ones: 3 \times 7 = 21, write 7, and 23 - 21 = 2.
Bring down the 5: 25 tenths. 3 \times 8 = 24, write 8, and 25 - 24 = 1.
Bring down the 0: 10 hundredths. 3 \times 3 = 9, write 3, and 10 - 9 = 1.
Bring down the next 0: 10 thousandths. Again 3, with 1 left. Every new step will leave 1 again, so the 3s never stop:
To the cent, look at the thousandths digit. It is 3, less than 5, so round down:
Each family’s share is about 7.83 dollars. Three shares are 3 \times 7.83 = 23.49 dollars, one cent short of 23.50. So two families pay 7.83 dollars and one pays 7.84 dollars, and 7.83 + 7.83 + 7.84 = 23.50.
Gas per school day, then per family
A month of carpool gas cost 61.20 dollars over 18 school days. What did the gas cost per day? Then, with three families sharing each day’s cost, what did one family pay per day, to the nearest cent?
SolutionFirst divide 61.20 by 18, a two-digit divisor. Estimate: 61.20 is close to 60, and 18 is close to 20, so the answer is close to 60 \div 20 = 3.
Start with 61 ones. Try 3: 18 \times 3 = 54, and 61 - 54 = 7, which is less than 18, so 3 is right. Write 3 over the 1.
Bring down the 2: 72 tenths. Try 4: 18 \times 4 = 72, nothing left. Write 4 in the tenths place.
Bring down the 0: 0 hundredths, so write 0.
The gas cost 3.40 dollars per day, close to the estimate of 3. Check: 18 \times 3.40 is 18 \times 340 = 6120 hundredths, which is 61.20.
Now share 3.40 dollars among 3 families. Divide to the thousandths: 3 ones give 1 each, nothing left; 4 tenths give 1 each with 1 left; 10 hundredths give 3 each with 1 left; 10 thousandths give 3 each with 1 left. So 3.40 \div 3 = 1.133\ldots, and the thousandths digit 3 means round down:
Each family paid about 1.13 dollars a day.
Boundaries
- The divisor here is a whole number, and never 0. Sharing among 0 families has no answer, so division by 0 is not defined. Dividing by a decimal, such as 2.45 \div 0.7, is the next step, in Dividing multi-digit decimals with the standard algorithm.
- The point goes straight above only in this setup. It works because sharing ones gives ones and sharing tenths gives tenths. That stays true for any whole-number divisor, and it is why the divisor has to be made whole before long division can start when it is a decimal.
- Zeros may be written only after the last digit, and only after the point. 14.7 = 14.700, because the new zeros add no hundredths or thousandths. A zero written at the end of a whole number, or squeezed between digits, changes the number: 147 is not 1470, and 14.7 is not 14.07.
- A zero in the answer may be needed to hold a place. 1.35 \div 3 = 0.45 needs its 0 in front of the point, and 0.27 \div 9 = 0.03 needs its 0 in the tenths place.
- Not every division ends. With a whole-number divisor, the answer either stops or settles into a repeating pattern, as 3.333\ldots does. A rounded answer is close, not exact, so it is written with \approx, and rounded shares may not add back to the total.
- Dividing by 10, 100 or 1000 has a shortcut. Every digit moves one, two or three places to the right; that is taught in Dividing a decimal by ten, one hundred or one thousand.
Where this goes next
The next step lets the divisor carry a point too, as in “how many 0.35-gallon runs are in 2.45 gallons?” That is Dividing multi-digit decimals with the standard algorithm: scale both numbers by the same power of ten until the divisor is whole, then use exactly the long division of this lesson.
Sharing a total equally is also how an average is found. Add up the miles driven on five days and divide by 5, and you have the Mean, which is very often a decimal. Divisions that never end, like 10 \div 3, are studied in Terminating and repeating decimals. And stories that mix all four operations with decimals, a price per item, a cost per person, a speed in miles per hour, come back in Decimal problems in words.
Exercises
Exercises 1 to 8 go with Part 1, 9 to 16 with Part 2, and 17 to 22 with Part 3. The reasoning group draws on all three.
Part 1: sharing a decimal, one place at a time
- Share the base-ten pieces to find each quotient: (a) 6.9 \div 3, (b) 4.8 \div 2, (c) 0.63 \div 3.
- Find 7.5 \div 3. Say what trade you make, and why.
- How many hundredths are in 4.56? Use that to find 4.56 \div 4.
- Find each quotient: (a) 9.42 \div 3, (b) 5.25 \div 5, (c) 7.68 \div 2.
- A car wash costs 4.26 dollars, and two families split it. How much does each pay?
- Haeun, Rajan and Pelumi split 7.35 dollars for a bag of snacks. How much is each share? Check by multiplying.
- Find 6.92 \div 4, then check by multiplying.
- Rajan shares 5.4 dollars among 3 friends and says each gets 1.8 dollars. Show he is right two ways: by counting tenths, and by multiplying.
Part 2: long division with the point straight above
- Use long division: (a) 17.36 \div 4, (b) 25.83 \div 7.
- Each answer starts with 0.: (a) 3.92 \div 7, (b) 1.44 \div 8.
- Each answer needs a zero to hold a place: (a) 0.35 \div 5, (b) 6.18 \div 6.
- Write zeros after the last digit to keep dividing: (a) 5.4 \div 4, (b) 3.3 \div 6, (c) 2.1 \div 8.
- The carpool drove 26.4 miles in 5 days, the same each day. How many miles a day is that?
- Pelumi divides 28.91 by 7 and gets the digits 413. Use an estimate to place the point.
- Pelumi’s mom shares a 1.5-liter bottle of water equally among 4 kids. How much does each kid get?
- Haeun’s dad drove 11.1 miles in 6 equal trips. How long was each trip?
Part 3: rounding to a sensible place, and two-digit divisors
- Three families split a 14.00-dollar car wash. Find each share to the nearest cent. Do the three shares cover the bill?
- The carpool drives 17.5 miles in 6 days, the same each day. Find the miles per day to the nearest tenth of a mile.
- Use long division: 45.36 \div 12.
- Use long division: 87.75 \div 15.
- A box of 16 granola bars costs 7.25 dollars. Find the cost of one bar, to the nearest cent.
- Gas for 22 school days costs 52.80 dollars. What is the cost per day? If three families share each day’s cost, what does one family pay per day?
Reasoning
- Explain why the point in the answer of 8.37 \div 3 goes straight above the point in 8.37. Use the words ones, tenths and hundredths.
- Haeun divides 3.6 by 9 and writes 4. Explain what happened and give the right answer. How could an estimate have caught it?
- To round 2 \div 7 to the nearest cent, Pelumi stops at 0.28 and writes 0.28 dollars. Divide one more place and explain why the answer to the cent is 0.29, not 0.28.
Answers
- (a) 2.3: 6 ones give 2 each, 9 tenths give 3 each. (b) 2.4: 4 ones give 2 each, 8 tenths give 4 each. (c) 0.21: 0 ones, 6 tenths give 2 each, 3 hundredths give 1 each.
- 2.5. The 7 ones give 2 each with 1 one left. It cannot be shared whole, so trade it for 10 tenths; with the 5 tenths that is 15 tenths, and 15 \div 3 = 5 tenths each.
- 456 hundredths. 456 \div 4 = 114, and 114 hundredths is 1.14.
- (a) 3.14. (b) 1.05: 5 ones give 1 each, 2 tenths give 0 each (write 0), 25 hundredths give 5 each. (c) 3.84.
- 2.13 dollars. 426 \div 2 = 213 cents.
- 2.45 dollars. 7 ones give 2 each with 1 left, 13 tenths give 4 each with 1 left, 15 hundredths give 5 each. Check: 3 \times 2.45 = 7.35.
- 1.73. 6 ones give 1 each with 2 left, 29 tenths give 7 each with 1 left, 12 hundredths give 3 each. Check: 4 \times 173 = 692 hundredths, so 4 \times 1.73 = 6.92.
- Tenths: 5.4 is 54 tenths, and 54 \div 3 = 18 tenths, which is 1.8. Multiplying: 3 \times 1.8 = 5.4. Both show each friend gets 1.8 dollars.
- (a) 4.34: 17 - 16 = 1; 13 - 12 = 1; 16 - 16 = 0. (b) 3.69: 25 - 21 = 4; 48 - 42 = 6; 63 - 63 = 0.
- (a) 0.56: 3 is less than 7, so 0 ones; 39 - 35 = 4; 42 - 42 = 0. (b) 0.18: 0 ones; 14 - 8 = 6; 64 - 64 = 0.
- (a) 0.07: 0 ones, and 3 tenths are too few for 5 parts, so 0 tenths; 35 hundredths give 7. (b) 1.03: 6 ones give 1; 1 tenth gives 0; 18 hundredths give 3.
- (a) 1.35: 5.4 = 5.40; 5 - 4 = 1, 14 - 12 = 2, 20 - 20 = 0. (b) 0.55: 3.3 = 3.30; 33 - 30 = 3, 30 - 30 = 0. (c) 0.2625: 2.1 = 2.1000; 21 - 16 = 5, 50 - 48 = 2, 20 - 16 = 4, 40 - 40 = 0.
- 5.28 miles. 26 - 25 = 1; 14 - 10 = 4; write a zero, 40 - 40 = 0.
- 4.13. 28.91 is close to 28, and 28 \div 7 = 4; of 0.413, 4.13 and 41.3, only 4.13 is close to 4.
- 0.375 liter. 1.5 = 1.500; 0 ones, 15 - 12 = 3, 30 - 28 = 2, 20 - 20 = 0.
- 1.85 miles. 11 - 6 = 5; 51 - 48 = 3; write a zero, 30 - 30 = 0.
- About 4.67 dollars. 14.000 \div 3 = 4.666\ldots, and the thousandths digit 6 rounds the cents up. Three shares of 4.67 are 14.01 dollars, one cent more than the bill, so two families pay 4.67 and one pays 4.66.
- About 2.9 miles. Divide to the hundredths: 17.50 \div 6 = 2.91\ldots, and the hundredths digit 1 rounds down.
- 3.78. 45 - 36 = 9; 93 - 84 = 9; 96 - 96 = 0. Estimate: 48 \div 12 = 4.
- 5.85. 87 - 75 = 12; 127 - 120 = 7; 75 - 75 = 0. Estimate: 90 \div 15 = 6.
- About 0.45 dollars. 7.250 \div 16 = 0.453\ldots: 72 - 64 = 8, 85 - 80 = 5, 50 - 48 = 2. The thousandths digit 3 rounds down.
- 2.40 dollars a day, since 52.80 \div 22 = 2.40 (52 - 44 = 8, 88 - 88 = 0). Then 2.40 \div 3 = 0.80, so each family pays 0.80 dollars a day.
- Long division shares the biggest pieces first. Sharing 8 ones gives 2 ones each, so the 2 is written over the ones digit. The trade brings the leftover into tenths, and sharing tenths gives the tenths digit 7, written over the tenths digit. Then hundredths give the 9. Each answer digit sits over the place it came from, so the point between ones and tenths is in the same spot in both rows.
- 3.6 \div 9 = 0.4. There are only 3 ones, too few for 9 parts, so the ones digit of the answer is 0; 36 tenths shared by 9 is 4 tenths. Writing 4 treats the tenths as ones. An estimate: 3.6 is less than 9, so each of 9 parts is less than 1, and 4 is too big. Check: 9 \times 0.4 = 3.6.
- 2.000 \div 7 = 0.285\ldots (20 - 14 = 6, 60 - 56 = 4, 40 - 35 = 5). The thousandths digit is 5, so the cents round up to 0.29. Stopping at 0.28 cuts off the digit that decides the rounding, and 2 \div 7 is in fact closer to 0.29 than to 0.28.