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Grade 4 · 3.NBT.A.3 · 4.NBT.A.1

Multiplying and dividing by ten, one hundred and one thousand

A button factory packs buttons in bags of ten, boxes of one hundred and cartons of one thousand. How can you know at once how many buttons are in 46 boxes, and why do the digits simply slide over?

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Why this exists

At a button factory, Divya, Nikau and Amaru pack buttons. A bag holds 10 buttons. A box holds 100 buttons. A carton holds 1{,}000 buttons. All day long they ask questions like these: how many buttons are in 46 boxes? How many bags can 4{,}700 buttons fill?

You could answer by adding 100 again and again, 46 times. That is slow, and it is easy to lose count. But our way of writing numbers has a pattern built into it that makes these questions quick. Every place is worth ten times the place to its right. Because of that, multiplying by 10, 100 or 1{,}000 does not change the digits at all. It only moves them to new places. Dividing by 10, 100 or 1{,}000 moves them back.

This lesson takes two days. Part 1 looks at the places themselves: why a digit moved one place to the left is worth ten times as much, so that 700 is ten times 70. Part 2 uses that to multiply and divide by 10, 100 and 1{,}000 in your head.

Definition

Each place in a whole number is worth ten times the place just to its right. Ten ones make one ten. Ten tens make one hundred. Ten hundreds make one thousand. Ten thousands make one ten thousand. You met this in Numbers to one million, where the places climb all the way to one million, and in Base ten blocks and trading ten for one, where ten blocks are traded for one block of the next size.

So the same digit is worth ten times as much when it sits one place further to the left. A 7 in the tens place is worth 70. A 7 in the hundreds place is worth 700, and

700 = 10 \times 70.

Going the other way, a digit that sits one place further to the right is worth one tenth as much: 70 is 700 \div 10.

Multiplying a whole number by 10 moves every one of its digits one place to the left. The ones place is left empty, and a 0 is written there to hold it. Multiplying by 100 moves every digit two places to the left, and multiplying by 1{,}000 moves every digit three places.

Dividing by 10, 100 or 1{,}000 moves every digit one, two or three places to the right. For the answer to be a whole number, the number being divided has to end in at least that many zeros, so that no digit is pushed past the ones place.

Intuition

Part 1: ten times the place to the right

Picture base ten blocks. A rod is ten small cubes in a row, so a rod is one ten. A flat is ten rods side by side, so a flat is one hundred.

7 tens = 70 7 hundreds = 700
Figure 1Each flat is ten rods. So seven flats are worth ten times as much as seven rods, and 700 is ten times 70.

Seven rods are 7 tens, which is 70. Seven flats are 7 hundreds, which is 700. Every flat is made of ten rods, so seven flats hold ten times as many cubes as seven rods. That is why 700 is ten times 70. The digit 7 is the same in both numbers. Only its place changed, and moving one place to the left made it worth ten times as much.

The factory works the same way. A box is ten bags, and a carton is ten boxes. So 3 boxes hold ten times as many buttons as 3 bags, and 3 cartons hold ten times as many as 3 boxes.

Two places to the left is ten times, and then ten times again. Since 10 \times 10 = 100, a digit moved two places to the left is worth 100 times as much. A carton holds 100 bags.

Part 2: every digit moves together

Now take a whole number with several digits, such as 46. It is 4 tens and 6 ones. Multiply it by 10. Each of the 4 tens becomes ten tens, which is a hundred, so the 4 tens become 4 hundreds. Each of the 6 ones becomes a ten, so the 6 ones become 6 tens. Every digit has moved one place to the left:

46 \times 10 = 460.

The ones place is now empty, and the 0 in 460 is there to hold it. Without it you would write 46, which is a different number.

ten thousands thousands hundreds tens ones
46
 4 6
46 \times 10
 4 6 0
46 \times 100
 4 6 0 0
46 \times 1{,}000
 4 6 0 0 0
Figure 2Multiplying by 10 moves every digit one place to the left. Multiplying by 100 moves it two places, and by 1{,}000 three places. The shaded zeros hold the empty places.

Read the chart down a column and you see the pattern. Each row is ten times the row above, and each row has the same digits 4 and 6, one place further to the left. In particular

46 \times 100 = 4{,}600.

Dividing runs the chart upward. To find 4{,}700 \div 10, ask which number, times 10, gives 4{,}700. Moving every digit of 4{,}700 one place to the right gives 470, and 470 \times 10 is 4{,}700, so

4{,}700 \div 10 = 470.
thousands hundreds tens ones
4{,}700
 4 7 0 0
4{,}700 \div 10
 4 7 0
4{,}700 \div 100
 4 7
Figure 3Dividing by 10 moves every digit one place to the right, and dividing by 100 moves it two places. The zeros that held the ones and tens places are no longer needed.

The digits never change. A shortcut that says “write a zero on the end” describes what you see for whole numbers, but the reason is the move: every digit goes up one place, and the 0 only fills the place that was left empty.

Results

Ten times the place to the right. In a whole number, each place is worth 10 times the place just to its right. A digit moved one place to the left is worth 10 times as much. A digit moved two places to the left is worth 100 times as much, and three places, 1{,}000 times as much.

Multiplying by 10, 100 and 1{,}000. Multiplying a whole number by 10, 100 or 1{,}000 moves every digit one, two or three places to the left. Zeros fill the empty places at the right.

Dividing by 10, 100 and 1{,}000. Dividing a whole number that ends in at least one, two or three zeros by 10, 100 or 1{,}000 moves every digit one, two or three places to the right. The zeros that held the empty places drop away.

Undoing. Multiplying by 10 and then dividing by 10 gives back the number you started with. The same holds for 100 and for 1{,}000.

Why it is true

A number is the sum of what its digits are worth (Expanded form). Take 46 = 40 + 6. To multiply a sum by 10, you can multiply each part by 10 and add, which is the The distributive property:

46 \times 10 = 40 \times 10 + 6 \times 10 = 400 + 60 = 460.

Each part is a digit times a place. The 4 tens times 10 is 4 hundreds, because ten tens make a hundred. The 6 ones times 10 is 6 tens, because ten ones make a ten. So every digit lands in the next place to the left. Nothing else can happen, since each place is exactly ten times the one to its right.

Multiplying by 100 is the same as multiplying by 10 and then by 10 again, because 100 = 10 \times 10. Grouping the factors either way gives the same product (the associative property of multiplication), so 46 \times 100 = (46 \times 10) \times 10. Each factor of 10 moves the digits one place, so 100 moves them two places. In the same way 1{,}000 is 10 \times 10 \times 10 and moves them three places.

Division undoes multiplication (Multiplication and division undo each other). The question 4{,}700 \div 10 asks for the number that, times 10, makes 4{,}700. Multiplying moves digits left, so the number we want has the same digits one place to the right: 470. A check confirms it, since 470 \times 10 = 4{,}700.

Worked examples

Examples 1 to 3 go with Part 1. Examples 4 to 7 go with Part 2.

Example 1

Two sevens

In the number 770, how many times as much is the left 7 worth as the right 7?

Solution

The left 7 is in the hundreds place, so it is worth 7 hundreds, which is 700. The right 7 is in the tens place, so it is worth 7 tens, which is 70. Each hundred is ten tens, so 7 hundreds is ten times 7 tens:

700 = 10 \times 70.

The left 7 is worth 10 times as much as the right 7. That fits the rule: the left 7 sits one place further to the left. Check: 70 \times 10 = 700.

Example 2

Three eights

In 8{,}808, write the value of each 8. Then say how many times as much the 8 in the thousands place is worth as each of the other two.

Solution

Read the places from the left: thousands, hundreds, tens, ones. The digits are 8, 8, 0, 8. So the 8s are worth 8{,}000, 800 and 8. The 0 in the tens place is worth nothing, but it holds the tens place so that the last 8 stays in the ones place.

The thousands 8 is one place left of the hundreds 8, so it is worth 10 times as much: 8{,}000 = 10 \times 800.

The thousands 8 is three places left of the ones 8. Each place to the left is another factor of 10, and 10 \times 10 \times 10 = 1{,}000, so it is worth 1{,}000 times as much: 8{,}000 = 1{,}000 \times 8.

Example 3

Bags, boxes and cartons

Amaru packs 4 bags, Divya packs 4 boxes and Nikau packs 4 cartons. How many buttons does each pack? How many times as many buttons does Nikau pack as Divya? As Amaru?

Solution

A bag holds 10, so 4 bags hold 4 tens, which is 40 buttons. A box holds 100, so 4 boxes hold 4 hundreds, which is 400 buttons. A carton holds 1{,}000, so 4 cartons hold 4 thousands, which is 4{,}000 buttons.

The digit 4 is the same each time; only its place changes. From 400 to 4{,}000 the 4 moves one place to the left, so Nikau packs 10 times as many as Divya: 4{,}000 = 10 \times 400. From 40 to 4{,}000 the 4 moves two places, so Nikau packs 100 times as many as Amaru: 4{,}000 = 100 \times 40. This kind of “times as many” question is a Times as many.

Example 4

Forty-six boxes

The factory fills 46 bags, 46 boxes and 46 cartons. How many buttons is each?

Solution

The 46 bags hold 46 \times 10 buttons. Multiplying by 10 moves every digit one place to the left: the 4 tens become 4 hundreds and the 6 ones become 6 tens. The ones place is empty, so a 0 holds it: 46 \times 10 = 460.

The 46 boxes hold 46 \times 100 buttons. Multiplying by 100 moves every digit two places to the left, and two zeros fill the tens and ones places: 46 \times 100 = 4{,}600.

The 46 cartons hold 46 \times 1{,}000 buttons. The digits move three places, and three zeros fill the hundreds, tens and ones: 46 \times 1{,}000 = 46{,}000.

The chart in Part 2 shows all three rows. Check the middle one another way: 46 boxes is 40 boxes and 6 boxes, which is 4{,}000 + 600 = 4{,}600 buttons.

Example 5

A zero in the middle

Find 305 \times 100 and 250 \times 1{,}000.

Solution

In 305 the digits are 3 hundreds, 0 tens and 5 ones. Multiplying by 100 moves every digit two places to the left, and that includes the 0. The 3 goes to the ten thousands place, the 0 to the thousands place and the 5 to the hundreds place. Zeros fill the tens and ones:

305 \times 100 = 30{,}500.

The 0 in the middle had to move with the others. It keeps the 3 and the 5 apart, so the 3 stays worth 30{,}000.

In 250 the digits are 2, 5 and 0. Multiplying by 1{,}000 moves all three digits three places to the left. The 0 that was in the ones place moves to the thousands place, and three new zeros fill the hundreds, tens and ones:

250 \times 1{,}000 = 250{,}000.

Check: 250 thousands is 250{,}000, read as “two hundred fifty thousand”.

Example 6

Bags from a pile

Divya has 4{,}700 loose buttons. How many bags of 10 can she fill? How many boxes of 100?

Solution

The number of bags is 4{,}700 \div 10. Dividing by 10 moves every digit one place to the right: the 4 thousands become 4 hundreds, the 7 hundreds become 7 tens, and the 0 that held the tens place moves to the ones place. The last 0 is no longer needed, so

4{,}700 \div 10 = 470.

She can fill 470 bags. Check: 470 \times 10 = 4{,}700.

The number of boxes is 4{,}700 \div 100. Now every digit moves two places to the right: the 4 lands in the tens place and the 7 in the ones place. So 4{,}700 \div 100 = 47, and she can fill 47 boxes. Check: 47 \times 100 = 4{,}700.

Cartons would be 4{,}700 \div 1{,}000. Moving three places would push the 7 past the ones place, so the answer is not a whole number: 4 full cartons, with 700 buttons left over.

Example 7

A day at the factory

In one day the machines make 23{,}000 buttons. Amaru packs them all in cartons. Then Nikau asks: if they had all gone in boxes instead, how many boxes would that be? And how many bags?

Solution

Cartons: 23{,}000 \div 1{,}000. Every digit moves three places to the right, so the 2 ten thousands become 2 tens and the 3 thousands become 3 ones: 23{,}000 \div 1{,}000 = 23 cartons.

Boxes: 23{,}000 \div 100. Every digit moves two places, so 23{,}000 \div 100 = 230 boxes.

Bags: 23{,}000 \div 10. Every digit moves one place, so 23{,}000 \div 10 = 2{,}300 bags.

Notice that each answer is ten times the one before it. A box is one tenth of a carton, so it takes ten times as many boxes as cartons, and ten times as many bags as boxes. Check the boxes by multiplying back: 230 \times 100 = 23{,}000.

Boundaries

Only ten, one hundred and one thousand shift the digits. Multiplying by 20 or by 11 does not simply move the digits. To find 46 \times 20 you can split 20 into 2 \times 10: multiply by 2 first, then move the digits one place. That is the next lesson, Multiplying by tens and hundreds.

Dividing needs enough zeros at the end. In whole numbers, dividing by 10 moves the ones digit past the ones place unless that digit is 0. So 480 \div 10 = 48 is a whole number, but 365 \div 10 is not: 365 is 36 tens and 5 ones. Dividing by 100 needs two zeros at the end, and dividing by 1{,}000 needs three.

The same shift reaches past the ones place. One tenth times 10 is one, so the place to the right of the ones is worth one tenth of a one (Tenths and hundredths). Numbers with digits in those places are moved the same way in Multiplying a decimal by ten, one hundred or one thousand and Dividing a decimal by ten, one hundred or one thousand. There, writing a zero on the end does not multiply by ten, which is one more reason to think of the rule as a move of digits.

Where this goes next

This is the first step of multiplying large numbers in your head. In Multiplying by tens and hundreds a product such as 40 \times 600 becomes a basic fact, 4 \times 6 = 24, followed by a shift of three places. In Partial products a number such as 46 is split into 40 and 6, and each part is multiplied on its own; the tens part is a shift like the ones in this lesson.

The places keep going in both directions. Writing 10, 100 and 1{,}000 as ten multiplied by itself is the idea of Powers of ten. And the rule that each place is ten times the next carries on to the right of the ones place, into tenths and hundredths, which is how the shift works for numbers that are not whole.

Exercises

Part 1: ten times the place to the right

  1. In the number 660, how many times as much is the left 6 worth as the right 6?
  2. Write the value of each 9 in 9{,}990. How many times as much is the first 9 worth as the third 9?
  3. Fill in the blank: 900 is \_\_\_ times 90.
  4. Fill in the blank: 5{,}000 is \_\_\_ times 50.
  5. 30{,}000 is ten times what number?
  6. A digit 2 moves from the tens place to the thousands place. How many times as much is it worth now?
  7. Write a three-digit number in which one digit 5 is worth ten times as much as another digit 5 in the same number. Explain how you know.
  8. Amaru packs 6 bags and Nikau packs 6 boxes. How many buttons does each pack, and how many times as many does Nikau pack?
  9. Divya packs 2 cartons and Amaru packs 2 bags. How many times as many buttons does Divya pack?
  10. How many bags hold as many buttons as one carton? How many boxes?

Part 2: multiplying and dividing by 10, 100 and 1,000

  1. Find 52 \times 10, 52 \times 100 and 52 \times 1{,}000.
  2. Find 9 \times 1{,}000.
  3. Find 407 \times 10.
  4. Find 160 \times 100.
  5. Find 2{,}050 \times 100.
  6. Find 630 \div 10.
  7. Find 9{,}100 \div 100.
  8. Find 72{,}000 \div 1{,}000.
  9. Find 50{,}000 \div 10.
  10. Fill in the blanks: 86 \times \_\_\_ = 8{,}600, and \_\_\_ \times 10 = 3{,}400.
  11. Nikau fills 34 cartons. How many buttons is that?
  12. Divya has 6{,}500 buttons. How many boxes of 100 can she fill? How many bags of 10?

Reasoning

  1. Amaru says, “To multiply by 10, just write a 0 on the end.” Using place value, explain why the 0 appears in 83 \times 10 = 830.
  2. Explain why 3{,}600 \div 100 is a whole number but 3{,}600 \div 1{,}000 is not.
  3. Explain why multiplying a whole number by 1{,}000 moves every digit three places to the left.
Answers
  1. 10 times. The left 6 is in the hundreds place and is worth 600; the right 6 is in the tens place and is worth 60; 600 = 10 \times 60.
  2. The 9s are worth 9{,}000, 900 and 90. The first is 100 times the third, since it sits two places to the left: 9{,}000 = 100 \times 90.
  3. 10. The 9 moves one place to the left, from tens to hundreds: 900 = 10 \times 90.
  4. 100. The 5 moves two places to the left, from tens to thousands: 5{,}000 = 100 \times 50.
  5. 3{,}000. The number one place to the right has the 3 in the thousands place: 10 \times 3{,}000 = 30{,}000.
  6. 100 times. Tens to thousands is two places to the left, 10 \times 10 = 100: 20 becomes 2{,}000, and 2{,}000 = 100 \times 20.
  7. For example 550: the left 5 is worth 500 and the right 5 is worth 50, and 500 = 10 \times 50. Any answer works where the two 5s sit side by side, such as 155 or 553, because neighboring places differ by ten times.
  8. Amaru packs 60 buttons and Nikau packs 600. Nikau packs 10 times as many: 600 = 10 \times 60.
  9. 100 times. Divya packs 2{,}000 and Amaru packs 20, and 2{,}000 = 100 \times 20.
  10. 100 bags, since 1{,}000 \div 10 = 100. And 10 boxes, since 1{,}000 \div 100 = 10.
  11. 52 \times 10 = 520, 52 \times 100 = 5{,}200 and 52 \times 1{,}000 = 52{,}000. The digits move one, two and three places to the left, and zeros fill the empty places.
  12. 9{,}000. The 9 moves three places to the left, from ones to thousands.
  13. 4{,}070. Every digit moves one place to the left, the 0 included, so the 4 lands in the thousands place and the 7 in the tens place.
  14. 16{,}000. The digits 1, 6, 0 move two places to the left, and two more zeros fill the tens and ones.
  15. 205{,}000. Every digit moves two places: the 2 goes from thousands to hundred thousands.
  16. 63. Every digit moves one place to the right; the 0 in the ones place drops away. Check: 63 \times 10 = 630.
  17. 91. Every digit moves two places to the right. Check: 91 \times 100 = 9{,}100.
  18. 72. Every digit moves three places to the right. Check: 72 \times 1{,}000 = 72{,}000.
  19. 5{,}000. The 5 moves one place to the right, from ten thousands to thousands.
  20. 86 \times 100 = 8{,}600, since the digits moved two places. And 340 \times 10 = 3{,}400, since 3{,}400 \div 10 = 340.
  21. 34{,}000 buttons: 34 \times 1{,}000 = 34{,}000.
  22. 65 boxes, since 6{,}500 \div 100 = 65. And 650 bags, since 6{,}500 \div 10 = 650.
  23. 83 is 8 tens and 3 ones. Times 10, each ten becomes a hundred and each one becomes a ten, so the product is 8 hundreds and 3 tens. The digits moved one place to the left. Nothing is left in the ones place, so a 0 is written there to hold it. The 0 is the result of the move, not a separate rule.
  24. Dividing by 100 moves every digit two places to the right, and 3{,}600 ends in two zeros, so only those zeros drop away: 3{,}600 \div 100 = 36. Dividing by 1{,}000 moves every digit three places, which would push the 6 past the ones place. So 3{,}600 is 3 thousands and 600 more, and 3{,}600 \div 1{,}000 is 3 with 600 left over, not a whole number.
  25. 1{,}000 = 10 \times 10 \times 10, so multiplying by 1{,}000 is multiplying by 10 three times. Each time, every digit moves one place to the left, because each place is worth ten times the place to its right. Three factors of 10 make three moves.
kg1128 · updated 2026-10-20 · CC-BY-SA-4.0