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Grade 5 · 5.NBT.A.2

Multiplying a decimal by ten, one hundred or one thousand

A pack of tortillas weighs 0.529 kilograms. How much do 100 packs weigh, and why can you find it without multiplying digit by digit?

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Multiplying and dividing by ten, one hundred and one thousandGrade 4Tenths and hundredthsGrade 4
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Dividing by a decimalGrade 5
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Multiplying a decimal by ten, one hundred or one thousand

When you reach for it

Graciela works at a tortilla factory. Tortillas go out in packs of 10, cases of 100 and pallets of 1{,}000. One pack weighs 0.529 kilograms. What does a stack of 100 packs weigh?

The question is 0.529 \times 100. You could multiply it out digit by digit, but there is a much faster way, and it comes straight from how place value works. Multiplying by 10, 100 or 1{,}000 never changes which digits you write. It only changes the place each digit sits in.

You reach for this move any time one factor is 10, 100, 1{,}000 or another power of ten: counting up packs, cases and pallets, changing kilograms to grams, or turning dollars into cents. It rests on one fact from Multiplying and dividing by ten, one hundred and one thousand: each place is worth ten times the place to its right. The places to the right of the point are tenths, hundredths and thousandths, as in Tenths and hundredths and Thousandths: reading, writing and expanded form.

The procedure

This lesson takes two days. Day 1 multiplies by 10, 100 and 1{,}000 on a place value chart and explains why it is the digits that move. Day 2 writes the factors as powers of ten, fills empty places with zeros, and uses the move in word problems.

Day 1: every digit moves to the left

Start with 0.529 \times 10. The number 0.529 is 5 tenths, 2 hundredths and 9 thousandths, so it is a sum of three parts:

0.529 = 0.5 + 0.02 + 0.009

To multiply a sum by 10, multiply each part by 10 and add. That is the distributive property. Each part is easy on its own. Ten tenths make one whole, so 5 tenths times 10 is 5 ones. Ten hundredths make one tenth, so 2 hundredths times 10 is 2 tenths. Ten thousandths make one hundredth, so 9 thousandths times 10 is 9 hundredths.

\begin{aligned} 0.529 \times 10 &= 0.5 \times 10 + 0.02 \times 10 + 0.009 \times 10 \\ &= 5 + 0.2 + 0.09 \\ &= 5.29 \end{aligned}

Look at what happened. The digits are still 5, 2, 9, in the same order. Each one now sits one place further left, in the place worth ten times as much. Multiplying by 10 moves every digit one place to the left.

Now 100 = 10 \times 10, so multiplying by 100 is multiplying by 10 and then by 10 again. Each step moves every digit one place left, so together they move every digit two places left:

0.529 \times 100 = 5.29 \times 10 = 52.9

That answers Graciela: 100 packs weigh 52.9 kilograms. The chart below shows the move.

tens ones tenths hundredths thousandths 5 2 9 0 5 2 9 0.529 52.9 × 100
Figure 10.529 \times 100 = 52.9. Every digit moves two places to the left. The point stays where it is, between the ones and the tenths.

In the same way 1{,}000 = 10 \times 10 \times 10, so multiplying by 1{,}000 moves every digit three places left: 0.529 \times 1{,}000 = 529.

Why the digits move and not the point. The decimal point is not a digit and has no value of its own. It is a marker. It always sits just to the right of the ones place, to show you which digit is the ones. On the chart it is the heavy line, and it never changes columns. What changes when you multiply by ten is what each digit is worth: the 5 that meant 5 tenths now means 5 ones. A digit’s worth is decided by its place, so a digit worth ten times as much must sit one place to the left. That is the whole move.

You may hear the same result described as “move the point one place to the right.” Written down, the two give the same digits in the same places, because sliding the point right against the digits looks the same as sliding the digits left against the point. The chart shows what really happens: the places stay fixed, the point stays fixed, and the digits move up in value.

Zeros in front. A number less than one is written with a 0 in the ones place, as in 0.529. That 0 only says “no ones.” After the digits move, a real digit may land in the ones place or further left, and the leading 0 is no longer needed. So 0.529 \times 100 is written 52.9, not 052.9.

Day 2: powers of ten, and filling empty places

A power of ten is 10 multiplied by itself some number of times, written with a small raised number called the exponent. Powers of ten covers this notation; here you need only

\begin{aligned} 10^1 &= 10 \\ 10^2 &= 10 \times 10 = 100 \\ 10^3 &= 10 \times 10 \times 10 = 1{,}000 \end{aligned}

The exponent counts the factors of ten. Each factor of ten moves every digit one place left, so the exponent tells you how many places every digit moves. Multiplying by 10^2 moves every digit two places; multiplying by 10^3 moves every digit three places. The same holds for 10^4 = 10{,}000 and beyond.

Now try 3.08 \times 1{,}000, which is 3.08 \times 10^3. The number 3.08 is 3 ones, 0 tenths and 8 hundredths. Each digit moves three places left:

  • 3 ones become 3 thousands,
  • 0 tenths become 0 hundreds,
  • 8 hundredths become 8 tens.
thousands hundreds tens ones tenths hundredths 3 0 8 3 0 8 0 3.08 3,080 × 1,000
Figure 23.08 \times 1{,}000 = 3{,}080. After the move the ones place is empty, so a 0 is written there to hold it.

There is a gap. Nothing has moved into the ones place, because no digit of 3.08 sat three places to the right of the ones. An empty place inside a whole number must be written as 0, or the other digits would slide into the wrong places. Without that zero you would write 308, which is 3 hundreds, 0 tens and 8 ones, a different number. So

3.08 \times 1{,}000 = 3{,}080

The 0 in the hundreds place of 3{,}080 came from the 0 tenths of 3.08. The 0 in the ones place was written to fill a place nothing moved into. Both are needed.

Zeros at the far right of a decimal, after the last nonzero digit to the right of the point, are different. They can be dropped or added freely, because they say “no hundredths” or “no thousandths” about a place that is already past the end of the number: 52.90 = 52.9. Money is the one place where people keep them on purpose, so that every price shows cents, as in 18.50 dollars.

Word problems. In a word problem the units ride along with the number. If one tortilla weighs 0.045 kilograms, then 1{,}000 tortillas weigh 0.045 \times 1{,}000 kilograms. The answer is still in kilograms; only the number of them changed.

Worked examples

Examples 1 to 3 go with Day 1, and 4 to 7 with Day 2.

Example 1

One place to the left

Find 6.3 \times 10.

Solution

In 6.3 the 6 is in the ones place and the 3 is in the tenths place.

tensonestenths
6.363
6.3 \times 1063

Multiplying by 10 makes each digit worth ten times as much. Six ones times ten is six tens. Three tenths times ten is thirty tenths, and ten tenths make one, so thirty tenths are three ones. Every digit moves one place left, and nothing is left to the right of the point:

6.3 \times 10 = 63

Check by size: 6.3 is a little more than 6, and 6 \times 10 = 60, so an answer a little more than 60 makes sense.

Example 2

A ladder of tens

Find 2.517 \times 10, 2.517 \times 100 and 2.517 \times 1{,}000.

Solution

Each factor of ten moves every digit one more place left. Follow the 7, which starts in the thousandths place.

Times 10, one place: the 7 moves to the hundredths. The 2 moves from ones to tens, and the point still sits just right of the ones, which now holds the 5. So 2.517 \times 10 = 25.17.

Times 100, two places: the 7 moves to the tenths. So 2.517 \times 100 = 251.7.

Times 1{,}000, three places: the 7 moves to the ones. Every digit is now left of the point, so there is nothing to write after it, and 2.517 \times 1{,}000 = 2{,}517.

\begin{aligned} 2.517 \times 10 &= 25.17 \\ 2.517 \times 100 &= 251.7 \\ 2.517 \times 1{,}000 &= 2{,}517 \end{aligned}

The same four digits appear in every row. Only their places change.

Example 3

Graciela’s pack and case

One corn tortilla weighs 0.026 kilograms. Graciela packs 10 of them in a pack and 100 of them in a case. How much does a pack weigh, and how much does a case weigh?

Solution

The pack is 0.026 \times 10 kilograms. In 0.026 there are 0 ones, 0 tenths, 2 hundredths and 6 thousandths. Moving each digit one place left puts the 2 in the tenths and the 6 in the hundredths:

0.026 \times 10 = 0.26

The 0 before the point stays, because there are still no ones. A pack weighs 0.26 kilograms.

The case is 0.026 \times 100 kilograms. Moving each digit two places left puts the 2 in the ones and the 6 in the tenths:

0.026 \times 100 = 2.6

The zeros that sat in front of the 2 only held empty places. Now a 2 sits in the ones place, so they are not needed. A case weighs 2.6 kilograms.

Check: a case of 100 tortillas is also 10 packs, and 0.26 \times 10 = 2.6, the same weight.

Example 4

Filling empty places

Find 0.7 \times 10^3.

Solution

The exponent 3 means three factors of ten, since 10^3 = 1{,}000. So the 7 moves three places left: from tenths to ones, to tens, to hundreds. It is now 7 hundreds.

The tens and ones places are empty, because no digit moved into them. They must hold zeros, or the 7 would read as 7 ones:

0.7 \times 10^3 = 700

Check by thinking about tenths: 0.7 is 7 tenths, and 1{,}000 times one tenth is 100. So 1{,}000 times 7 tenths is 7 hundreds, which is 700.

Example 5

A zero inside, and a zero at the end

Find 14.05 \times 10^2 and 14.05 \times 10^3.

Solution

In 14.05 there are 1 ten, 4 ones, 0 tenths and 5 hundredths. The 0 is a real digit of the number: it says there are no tenths, and it moves along with the others.

Times 10^2, every digit moves two places left. The 1 goes to thousands, the 4 to hundreds, the 0 to tens and the 5 to ones:

14.05 \times 10^2 = 1{,}405

Times 10^3, every digit moves three places left. The 1 goes to ten thousands, the 4 to thousands, the 0 to hundreds and the 5 to tens. The ones place is now empty and gets a 0:

14.05 \times 10^3 = 14{,}050

The second answer is ten times the first, as it should be: one more factor of ten, one more place.

Example 6

Jiho’s pallet

Jiho loads a pallet with 1{,}000 flour tortillas. Each tortilla weighs 0.045 kilograms and costs the factory 0.09 dollars to make. How much does the pallet of tortillas weigh, and what did they cost to make?

Solution

The weight is 0.045 \times 1{,}000 kilograms. The 4 hundredths move three places to 4 tens, and the 5 thousandths move three places to 5 ones:

0.045 \times 1{,}000 = 45

The tortillas weigh 45 kilograms.

The cost is 0.09 \times 1{,}000 dollars. The 9 hundredths move three places to 9 tens. The ones place is empty and gets a 0:

0.09 \times 1{,}000 = 90

The tortillas cost 90 dollars to make. Check: 9 cents a tortilla is 9{,}000 cents for 1{,}000 tortillas, and 9{,}000 cents is 90 dollars.

Example 7

Lakshmi finds the factor

Lakshmi’s order sheet is smudged. It shows 0.38 \times 10^{n} = 380 and 6.5 \times 10^{m} = 650, where n and m are exponents she cannot read. Find each one.

Solution

The exponent is the number of places every digit moved, so follow one digit.

In 0.38 the 8 is in the hundredths. In 380 the 8 is in the tens. From hundredths to tenths, to ones, to tens is three places, so n = 3 and the factor is 10^3 = 1{,}000. Check: 0.38 \times 1{,}000 = 380.

In 6.5 the 5 is in the tenths. In 650 it is in the tens. From tenths to ones to tens is two places, so m = 2 and the factor is 10^2 = 100. Check: 6.5 \times 100 = 650.

Counting the zeros on the end of the answer does not find the exponent: 380 and 650 each end in one zero, yet the factors are 1{,}000 and 100. Those zeros fill empty places; how many are needed depends on where the digits started. Following a digit always works.

Boundaries

  • Only powers of ten shift. The move works for 10, 100, 1{,}000 and any 10^{n} where n is a whole number. For 0.529 \times 20 or 0.529 \times 50 the digits themselves change, so you multiply in the usual way. A shift can still help as one step: 0.529 \times 20 is 0.529 \times 2 \times 10.
  • The exponent 0 moves nothing. 10^0 = 1, so 2.517 \times 10^0 = 2.517.
  • It is a base ten fact. Digits move one place per factor of ten because each place is worth ten times the place to its right. In a system that bundles by another number, as in Numbers in other bases, it is that number, not ten, that shifts the digits.
  • Multiplying by 0.1 or 0.01 is the other direction. Those factors make every digit worth less, so the digits move right. That is the same as dividing by 10 or 100, the subject of Dividing a decimal by ten, one hundred or one thousand.
  • Zeros inside a whole number cannot be dropped. 3{,}080 and 308 are different numbers. Zeros at the right end of the part after the point can be dropped: 45.0 = 45.

Where this goes next

The opposite move, dividing by 10, 100 and 1{,}000, sends every digit to the right instead, and it is in Dividing a decimal by ten, one hundred or one thousand. Together the two moves are how the metric system changes units: meters to centimeters is times 100, grams to kilograms is divided by 1{,}000.

When both factors are decimals, as in 0.529 \times 0.3, the shift is one part of the story. Where the point goes in a product counts where the point goes in such a product, and Multiplying multi-digit decimals with the standard algorithm multiplies any two decimals. Later, in Dividing by a decimal, multiplying both numbers by the same power of ten turns a decimal divisor into a whole number, and that rests on exactly this move.

Exercises

Day 1: times ten, one hundred and one thousand

Multiply. Say where each digit moves.

  1. 3.9 \times 10
  2. 0.72 \times 10
  3. 0.64 \times 100
  4. 5.13 \times 100
  5. 0.418 \times 1{,}000
  6. 0.903 \times 10
  7. 0.057 \times 100
  8. 1.234 \times 100
  9. A flour tortilla weighs 0.047 kilograms. How much do a pack of 10 and a case of 100 weigh?
  10. Find 0.268 \times 100. What is the digit 6 worth before and after?
  11. What number goes in the box? 0.35 \times \square = 35
  12. Jiho sells a pack of tortillas for 1.85 dollars. What do 10 packs cost?

Day 2: powers of ten and empty places

  1. 4.6 \times 10^2
  2. 0.8 \times 10^3
  3. 2.75 \times 10^3
  4. 9.04 \times 10^3
  5. 0.06 \times 10^3
  6. 13.7 \times 10^2
  7. 5.067 \times 10^4
  8. Find the exponent n in 0.92 \times 10^{n} = 920.
  9. Lakshmi loads a pallet with 1{,}000 corn tortillas, each weighing 0.028 kilograms. How much do the tortillas weigh?
  10. Each tortilla costs the factory 0.07 dollars to make. What does a pallet of 1{,}000 cost to make?
  11. Which is heavier, and by how much: 1{,}000 tortillas at 0.031 kilograms each, or 100 tortillas at 0.29 kilograms each?

Reasoning

  1. Find 0.4 \times 100. Explain with place value where the 4 goes, and why a 0 must be written in the ones place of the answer.
  2. Use 0.215 to explain why multiplying by 10 and then by 100 gives the same result as multiplying by 1{,}000. Name the property that lets you group the factors.
Answers
  1. 39. The 3 moves from ones to tens and the 9 from tenths to ones.
  2. 7.2. The 7 moves from tenths to ones and the 2 from hundredths to tenths.
  3. 64. Two places left: 6 tenths become 6 tens, 4 hundredths become 4 ones.
  4. 513. Two places left: the 5 to hundreds, the 1 to tens, the 3 to ones.
  5. 418. Three places left: 4 tenths become 4 hundreds, 1 hundredth becomes 1 ten, 8 thousandths become 8 ones.
  6. 9.03. One place left: the 9 to ones, the 0 to tenths, the 3 to hundredths.
  7. 5.7. Two places left: the 5 hundredths become 5 ones, the 7 thousandths become 7 tenths. The leading zeros are no longer needed.
  8. 123.4. Two places left: the 1 to hundreds, the 2 to tens, the 3 to ones, the 4 to tenths.
  9. A pack weighs 0.47 kilograms, since 0.047 \times 10 = 0.47. A case weighs 4.7 kilograms, since 0.047 \times 100 = 4.7.
  10. 26.8. Before, the 6 is in the hundredths and is worth 6 hundredths. After two places, it is in the ones and is worth 6 ones.
  11. 100. The 5 moves from hundredths to ones, which is two places, so the factor is 100.
  12. 18.50 dollars. 1.85 \times 10 = 18.5, written with cents as 18.50.
  13. 460. Two places left puts the 4 in hundreds and the 6 in tens; the ones place gets a 0.
  14. 800. The 8 moves from tenths to hundreds; the tens and ones get zeros.
  15. 2{,}750. The 2 to thousands, the 7 to hundreds, the 5 to tens, and a 0 in the ones.
  16. 9{,}040. The 9 to thousands, the 0 to hundreds, the 4 to tens, and a 0 in the ones.
  17. 60. The 6 moves from hundredths to tens; the ones place gets a 0.
  18. 1{,}370. The 1 to thousands, the 3 to hundreds, the 7 to tens, and a 0 in the ones.
  19. 50{,}670. Four places left: the 5 to ten thousands, the 0 to thousands, the 6 to hundreds, the 7 to tens, and a 0 in the ones.
  20. n = 3. The 9 moves from tenths to hundreds, three places. Check: 0.92 \times 1{,}000 = 920.
  21. 28 kilograms. 0.028 \times 1{,}000 = 28: the 2 moves from hundredths to tens, the 8 from thousandths to ones.
  22. 70 dollars. 0.07 \times 1{,}000 = 70, with a 0 written in the ones place.
  23. The 1{,}000 tortillas, by 2 kilograms. 0.031 \times 1{,}000 = 31 and 0.29 \times 100 = 29, and 31 - 29 = 2.
  24. 0.4 \times 100 = 40. The 4 is 4 tenths. Multiplying by 100 makes it worth a hundred times as much, so it moves two places left, from tenths to ones to tens: 4 tens. No digit moved into the ones place, so it holds a 0. Without that 0 the answer would read 4, which is 4 ones, not 4 tens.
  25. 0.215 \times 10 = 2.15 moves every digit one place, and 2.15 \times 100 = 215 moves them two more, three places in all. 0.215 \times 1{,}000 = 215 moves them three places at once. They agree because (0.215 \times 10) \times 100 = 0.215 \times (10 \times 100) by the associative property, and 10 \times 100 = 1{,}000.
kg1128 · updated 2026-10-19 · CC-BY-SA-4.0