Dividing a decimal by ten, one hundred or one thousand
Dividing by ten makes a number ten times smaller. Where do its digits go, and what fills the empty places?
Where you are 15 in Place value
When you reach for it
Ignacio, Natsuki and Tanvi are on a class trip to the planetarium. Out front stands a rocket 56.3 meters tall. In the lobby there is a model of it, made 100 times smaller. How tall is the model? The question is
You reach for this move whenever a number is divided by 10, 100 or 1{,}000. That happens when a group cost is shared among 10 or 100 people, when a length is cut into 10 or 100 equal parts, when a model is built 1{,}000 times smaller, and when you need to undo a multiplication by 10, 100 or 1{,}000. You never need long division for it. Place value does all the work.
The procedure
Day 1: every digit moves to the right
In a place value chart, each place is worth ten times the place to its right. That is what multiplying and dividing by ten rests on. Read it the other way round: each place is worth one tenth of the place to its left. One ten split into ten equal parts is one one. One one split into ten equal parts is one tenth. One tenth split into ten equal parts is one hundredth.
So dividing by 10 takes one tenth of every digit’s value. A 5 in the tens place is worth 50, and 50 \div 10 = 5, which is a 5 in the ones place. A 3 in the tenths place is worth 0.3, and one tenth of that is 0.03, a 3 in the hundredths place. Every digit moves one place to the right. The digits do not change and do not change their order. Only their places change.
Dividing by 100 is dividing by 10 and then by 10 again, because 10 \times 10 = 100. So every digit moves two places to the right. Dividing by 1{,}000 is dividing by 10 three times, so every digit moves three places.
Here is the rocket in a chart. The 5 tens, 6 ones and 3 tenths each move two places.
The model is 0.563 meters tall. The decimal point never moved: the chart’s columns stayed put and the digits slid past them. People often say “move the decimal point left”, which describes the same answer seen from the digits’ side. Both views give the same number.
Zeros hold empty places. Natsuki finds a model of the whole planetarium building, made 1{,}000 times smaller. The real telescope dome on its roof is 9 meters wide, so the model dome is 9 \div 1{,}000 meters wide. The 9 starts in the ones place and moves three places, to the thousandths place.
The ones, tenths and hundredths are now empty. A blank cannot be written in a number, so each empty place gets a 0:
The zeros after the point are needed: without them, the 9 would land in the wrong place. The 0 in the ones place is not needed, but it is written so the point is easy to see. The model dome is 0.009 meters wide, which is 9 millimeters.
Zeros at the right end of a decimal can go. 4{,}500 \div 100 moves the digits to give 45.00, and 45.00 = 45, because 0 tenths and 0 hundredths add nothing.
Day 2: powers of ten, undoing and equal parts
Powers of ten give the divisors a short name: 10^1 = 10, 10^2 = 100 and 10^3 = 1{,}000. The small raised number counts the tens multiplied together, and it is also the number of places every digit moves. Dividing by 10^2 moves every digit 2 places to the right.
Division undoes multiplication. Multiplying by 100 moves every digit two places to the left, and dividing by 100 moves them back. So if you know the answer to a multiplication by 10, 100 or 1{,}000 and want the number you started with, divide.
Splitting into equal parts is division too. Cutting a length into 10 equal parts makes each part one tenth of it, and cutting each part into 10 again makes one hundredth of the whole. The strip in Example 7 shows that picture.
Worked examples
Examples 1 to 4 go with Day 1, and Examples 5 to 7 go with Day 2.
A whole number by ten, one hundred and one thousand
Find 7{,}240 \div 10, 7{,}240 \div 100 and 7{,}240 \div 1{,}000.
Solution7{,}240 has 7 thousands, 2 hundreds, 4 tens and 0 ones. A whole number has its decimal point after the ones, even when it is not written.
Dividing by 10 moves every digit one place right. The 7 becomes 7 hundreds, the 2 becomes 2 tens, the 4 becomes 4 ones and the 0 becomes 0 tenths:
Dividing by 100 moves every digit two places. Now the 4 lands in the tenths place:
Dividing by 1{,}000 moves every digit three places. The 7 is in the ones place:
Each answer is ten times smaller than the one before it, as it should be, since each divisor is ten times bigger. Check: 7.24 \times 1{,}000 = 7{,}240.
Zeros in front
Find 0.4 \div 100 and 3.2 \div 100.
Solution0.4 has a 4 in the tenths place. Moving it two places right puts it in the thousandths place. The tenths and hundredths are then empty, so each gets a zero:
3.2 has 3 ones and 2 tenths. Each moves two places right: the 3 goes to the hundredths and the 2 goes to the thousandths. The tenths place is left empty and gets a zero:
Read it back to be sure: 0.032 is thirty-two thousandths, and 3.2 is thirty-two tenths. One thousandth is one hundredth of one tenth, so the count stayed 32 and each piece got 100 times smaller.
Sharing a group cost
The planetarium show costs 87.50 dollars for a group of 10 students. The bus costs 465 dollars, shared by the 100 students on the trip. How much does each student pay for each?
SolutionAn equal share of a cost is the cost divided by the number of people. For the show,
The 8 tens became 8 ones, the 7 ones became 7 tenths, the 5 tenths became 5 hundredths, and the 0 dropped off the end. Each student pays 8.75 dollars for the show.
For the bus, the 4 hundreds, 6 tens and 5 ones each move two places right:
Each student pays 4.65 dollars for the bus. Check: 100 shares of 4.65 dollars are 465 dollars.
A zero in the middle, and a zero at the end
Find 30.06 \div 10 and 600 \div 1{,}000.
Solution30.06 has 3 tens, 0 ones, 0 tenths and 6 hundredths. All four digits move one place right, zeros included. A zero in the middle of a number is holding a place, and it keeps holding one:
For 600 \div 1{,}000, the 6 hundreds move three places, to the tenths. The two zeros after it move to the hundredths and thousandths:
Those end zeros are worth nothing and can be dropped. The zero in 3.006 cannot: 3.06 is a different number.
Dividing by a power of ten
Find 418 \div 10^2 and 25 \div 10^3.
SolutionThe exponent says how many places each digit moves. 10^2 = 100, so every digit of 418 moves two places right:
10^3 = 1{,}000, so every digit of 25 moves three places right. The 2 lands in the hundredths and the 5 in the thousandths, and the empty tenths place gets a zero:
The same move works for 10^1 = 10: one place.
Undoing a multiplication
Tanvi thinks of a number, multiplies it by 1{,}000 and gets 3{,}750. Then she thinks of another number, multiplies it by 100 and gets 62. What were her numbers?
SolutionMultiplying by 1{,}000 moved every digit three places left. To get back, move them three places right, which is dividing by 1{,}000:
Check: 3.75 \times 1{,}000 = 3{,}750.
The second number times 100 is 62, so the number is 62 \div 100. The 6 tens and 2 ones move two places right:
Check: 0.62 \times 100 = 62. A product can be smaller than 100 when the number being multiplied is less than 1.
A tenth of a tenth
Ignacio has a strip of star stickers 4.8 meters long. He cuts it into 10 equal pieces, then cuts one of those pieces into 10 equal pieces. How long is each small piece? Show that the answer is 4.8 \div 100.
The first cut splits 4.8 meters into 10 equal parts, so each part is
The second cut splits 0.48 meters into 10 equal parts:
Dividing by 10 and then by 10 again is dividing by 10 \times 10 = 100. Each digit of 4.8 moved one place and then one more, two places in all:
The picture says the same thing. The big strip holds 10 middle pieces, and each middle piece holds 10 small ones, so the big strip holds 100 small pieces. Each small piece is 0.048 meters, about 5 centimeters.
Boundaries
The divisor must be 10, 100, 1{,}000 or another power of ten. Dividing by 20 or by 5 does not just move digits, because 20 is not ten tens of anything in the chart. That needs dividing a decimal by a whole number. Dividing by 1, which is 10^0, moves nothing at all.
Dividing by a decimal goes the other way. 10, 100 and 1{,}000 are bigger than 1, so the answer is smaller than the number divided. Dividing by 0.1 or 0.01 gives a bigger answer, and that is a different move, met in dividing decimals.
Digits can move past the thousandths. 3.2 \div 1{,}000 = 0.0032, and the 2 lands in the ten-thousandths place. The rule still works; the chart just needs more columns to the right. Every place there is still one tenth of the place to its left.
An answer in dollars may not be payable. 9 \div 1{,}000 = 0.009 is right as a number, but 0.009 dollars is less than one cent. When a cost does not split into whole cents, the shares have to be rounded, and then they no longer add up to exactly the total.
Only zeros at the far right can be dropped. 0.600 = 0.6, but the zeros in 0.009 and 3.006 hold places and must stay. A zero to the left of the first nonzero digit, after the point, also stays.
Where this goes next
This move is one half of a pair. The other half is multiplying by ten, one hundred and one thousand, where every digit moves left, and each undoes the other. Both rest on powers of ten and on places reaching down to Thousandths: reading, writing and expanded form.
Changing metric units is this move in daily use: 9 millimeters is 9 \div 1{,}000 = 0.009 meters, because a meter is 1{,}000 millimeters (converting metric units). Later, dividing a decimal by any whole number takes the idea of splitting into equal parts beyond 10, 100 and 1{,}000. And when numbers get very large or very small, scientific notation writes them as a number times a power of ten, so moving digits by a power of ten becomes just changing the exponent.
Exercises
Day 1: divide by ten, one hundred and one thousand
- 380 \div 10
- 2{,}615 \div 100
- 47.9 \div 10
- 8{,}302 \div 1{,}000
- 61.5 \div 100
Day 1: zeros in front
- 7 \div 100
- 0.6 \div 10
- 52 \div 1{,}000
- 0.3 \div 100
Day 1: zeros in the middle and at the end
- 9{,}000 \div 1{,}000
- 20.4 \div 10
- 1{,}050 \div 100
Day 1: sharing a cost
- The star dome tour costs 142.50 dollars for a group of 10 students. How much is each student’s share?
- Lunch for the trip costs 735 dollars for 100 students. How much is each student’s share?
Day 2: powers of ten
- 93 \div 10^2
- 4.1 \div 10^1
- 806 \div 10^3
Day 2: undoing a multiplication
- Natsuki multiplied a number by 100 and got 284. What was the number?
- A number times 1{,}000 is 65. What is the number?
- A number times 10 is 3.7. What is the number?
Day 2: equal parts and models
- A rope 8.4 meters long is cut into 100 equal pieces. How long is each piece?
- Jupiter is about 778.5 million kilometers from the Sun. On a floor model at the planetarium, every 100 million kilometers is shown as 1 meter, so the model distance in meters is 778.5 \div 100. How far from the model Sun is model Jupiter?
- The dome show cost 1{,}250 dollars to run for one evening, and 1{,}000 visitors came. If the cost is shared equally, what is each visitor’s share?
Reasoning
- Ignacio wrote 46 \div 100 = 0.046. Explain what went wrong, and give the right answer.
- Tanvi says, “To divide by 100, just take two zeros off the end.” Explain why her rule works for 4{,}500 \div 100, and why it says nothing about 56.3 \div 100. Then describe a way that works for both.
Answers
- 38. The 3 hundreds and 8 tens each move one place right; the 0 becomes 0 tenths and drops off.
- 26.15. Every digit moves two places right: the 6 hundreds become 6 ones.
- 4.79. The 4 tens become 4 ones, the 7 ones become 7 tenths, the 9 tenths become 9 hundredths.
- 8.302. Every digit moves three places right; the 0 stays to hold the hundredths place.
- 0.615. The 6 tens move to the tenths, the 1 to the hundredths, the 5 to the thousandths.
- 0.07. The 7 moves from the ones to the hundredths; the tenths place gets a 0.
- 0.06. The 6 moves from the tenths to the hundredths; the tenths place gets a 0.
- 0.052. The 5 tens move to the hundredths and the 2 ones to the thousandths; the tenths place gets a 0.
- 0.003. The 3 moves from the tenths to the thousandths; the tenths and hundredths get zeros.
- 9. The 9 thousands become 9 ones, and 9.000 = 9.
- 2.04. The zero in the ones place moves to the tenths and keeps holding it.
- 10.5. Every digit moves two places right to give 10.50, and 10.50 = 10.5.
- 14.25 dollars. 142.50 \div 10 = 14.250 = 14.25.
- 7.35 dollars. 735 \div 100 = 7.35.
- 0.93. 10^2 = 100, so every digit moves two places right.
- 0.41. 10^1 = 10, so every digit moves one place right.
- 0.806. 10^3 = 1{,}000, so every digit moves three places right; the 0 holds the hundredths place.
- 2.84. Undo the multiplication by dividing: 284 \div 100 = 2.84. Check: 2.84 \times 100 = 284.
- 0.065. 65 \div 1{,}000 = 0.065. Check: 0.065 \times 1{,}000 = 65.
- 0.37. 3.7 \div 10 = 0.37. Check: 0.37 \times 10 = 3.7.
- 0.084 meters. 8.4 \div 100 = 0.084: the 8 moves from the ones to the hundredths and the 4 from the tenths to the thousandths.
- 7.785 meters. 778.5 \div 100 = 7.785: every digit moves two places right.
- 1.25 dollars. 1{,}250 \div 1{,}000 = 1.250 = 1.25.
- The right answer is 0.46. Ignacio moved the digits three places, which is dividing by 1{,}000. Dividing by 100 moves each digit two places: the 4 tens become 4 tenths and the 6 ones become 6 hundredths. Check: 0.46 \times 100 = 46, while 0.046 \times 100 = 4.6.
- In 4{,}500, moving every digit two places right turns the 4 thousands into 4 tens and the 5 hundreds into 5 ones, and the two zeros land in the tenths and hundredths, where they are worth nothing and can be dropped. So “taking off two zeros” happens to give 45. But 56.3 has no zeros on its end to take off, and the answer 0.563 has new digits in new places. The rule that works for both is the one under Tanvi’s shortcut: every digit moves two places to the right, and zeros fill any empty places between the point and the digits.