Grade 5 · 5.NBT.A.3 · 5.NBT.A.3A

Thousandths: reading, writing and expanded form

A bobsled run is timed to the thousandth of a second. In a time like 52.374 seconds, what is each digit worth, and how do you say the number out loud?

You need

Expanded formGrade 2 Tenths and hundredthsGrade 4

Where you are 4 in Decimal place value

Number9

Place value15

Decimal place value4

Tenths and hundredths Comparing and ordering decimals Decimals on a number line

Thousandths: reading, writing and expanded form

Why this exists

In a bobsled race, a sled of two or four riders slides down an icy track of curves, and a clock times each run. The runs are very close. Two sleds can finish within a few hundredths of a second of each other, and sometimes in the same hundredth.

Suppose Aarav’s sled and Camila’s sled both show 52.37 seconds on a clock that stops at hundredths. Who won? That clock cannot say. A finer clock reads 52.374 seconds for one and 52.378 seconds for the other, and now the race has a winner. The new digit lives in a new place, one step further right than the hundredths: the thousandths place.

You already know tenths and hundredths, the first two places to the right of the point, and expanded form for whole numbers, where a number is written as the sum of what its digits are worth. This lesson adds the third place, shows the rule that makes every place fit with its neighbors, and then writes decimals three ways: in digits, in expanded form, and in words.

Definition

Cut one whole into 1000 equal parts. One of those parts is one thousandth:

0.001 = \frac{1}{1000}

The thousandths place is the third place to the right of the decimal point. A digit there counts thousandths, so the 4 in 52.374 means 4 thousandths, \displaystyle \frac{4}{1000}, or 0.004.

Here are the places this lesson uses, with what one of each is worth.

Place	As a decimal	As a fraction
tens	10	10
ones	1	1
tenths	0.1	\displaystyle \frac{1}{10}
hundredths	0.01	\displaystyle \frac{1}{100}
thousandths	0.001	\displaystyle \frac{1}{1000}

A decimal can be written in three forms.

Standard form is the usual digits with a point: 52.374.
Expanded form is the sum of what each digit is worth: 50 + 2 + 0.3 + 0.07 + 0.004.
Word form is the name of the number: fifty-two and three hundred seventy-four thousandths.

All three name the same number. They are three ways of writing it, not three different amounts.

Intuition

The lesson is taught in four parts, one a day.

Part 1: reading and writing thousandths

A place value chart gives each digit a column. The name of the column says what kind of piece the digit counts, and the value under the name says what one such piece is worth.

Figure 1The time 52.374 seconds in a place value chart. Each digit is worth its own amount times the value at the top of its column.

Read 52.374 in the chart from left to right: 5 tens, 2 ones, then the point, then 3 tenths, 7 hundredths and 4 thousandths. The point does not get a digit of its own. It only marks where the whole part ends and the parts smaller than one begin.

Base-ten pictures show the same pieces as shapes. Take a large square as one whole. Cut it into ten equal strips: each strip is a tenth. Cut one strip into ten equal squares: each small square is a hundredth. Now cut one small square into ten equal slivers. Each sliver is a thousandth.

Figure 2Cut a whole into ten and you get tenths. Cut one tenth into ten and you get hundredths. Cut one hundredth into ten and you get thousandths. Each picture zooms in on the shaded piece of the one before.

A thousandth is tiny, so a picture usually zooms in, as the right-hand drawing does. To write a number from a picture, count the pieces of each kind and put each count in its own column.

Part 2: each place is one tenth of the place to its left

Look at the pictures again. Each new piece came from cutting the piece before it into ten equal parts. So:

one tenth is one tenth of a one,
one hundredth is one tenth of a tenth,
one thousandth is one tenth of a hundredth.

Read the other way, ten of any piece make one of the piece to its left: ten thousandths make one hundredth, ten hundredths make one tenth, ten tenths make one whole. This is the same rule you know for whole numbers, where ten ones make a ten and ten tens make a hundred. The chart simply keeps going past the point, and the rule does not change.

Two facts follow. A digit is worth ten times as much as the same digit one place to its right, and one hundred times as much as the same digit two places to its right. And the number of thousandths in one whole is 10 \times 10 \times 10 = 1000: ten tenths, each made of ten hundredths, each made of ten thousandths.

Part 3: expanded form

Expanded form for a whole number such as 352 is 300 + 50 + 2. A decimal is expanded the same way, one term for each digit, using what the digit is worth in its place.

For 4.086, the chart shows 4 ones, 0 tenths, 8 hundredths and 6 thousandths.

Figure 3The number 4.086. The 0 holds the tenths place open, so the 8 stays in the hundredths place and the 6 in the thousandths place.

Each digit is worth its count times its place value, so

4.086 = 4 + 0.08 + 0.006

The zero tenths add nothing, so that term may be left out. The same expansion can be written with fractions, which shows each place value openly:

\begin{aligned} 4.086 &= 4 \times 1 + 8 \times \frac{1}{100} \\ &\quad + 6 \times \frac{1}{1000} \end{aligned}

A single term can also be a fraction alone: \displaystyle \frac{6}{1000} means six thousandths, the same as \displaystyle 6 \times \frac{1}{1000} and the same as 0.006.

Part 4: word form and zeros as placeholders

To say a decimal aloud, say the whole part, say and for the point, then say the digits after the point as one whole number followed by the name of the last place used. So 52.374 is “fifty-two and three hundred seventy-four thousandths”.

Why can the three digits 374 be read as one number? Because 3 tenths is 300 thousandths and 7 hundredths is 70 thousandths, so 0.374 is 300 + 70 + 4 = 374 thousandths. If there is no whole part, as in 0.374, there is no “and” either: “three hundred seventy-four thousandths”. The word “and” is kept for the point alone, so “three hundred seventy-four” never has an “and” inside it.

Writing from words runs the other way. The last word (tenths, hundredths, thousandths) says how many places follow the point. Then the number is written so its last digit lands in that last place, and any empty places in between are filled with zeros. Those zeros are placeholders: in 4.086, the 0 holds the tenths place, and without it the 8 would slide into the tenths place and the number would become 4.86, a different number.

Results

Place value rule. Each place is worth one tenth of the place to its left, and ten times the place to its right. In particular

\begin{aligned} 0.001 &= \frac{1}{1000} \\ 10 \text{ thousandths} &= 1 \text{ hundredth} \\ 1000 \text{ thousandths} &= 1 \end{aligned}

Expanded form. A decimal equals the sum of its digits’ values, each digit times the value of its place. For digits a, b, c, d in the ones, tenths, hundredths and thousandths places,

a.bcd = a + \frac{b}{10} + \frac{c}{100} + \frac{d}{1000}

Word form. The digits after the point, read as one whole number, count units of the last place. For example \displaystyle 0.374 = \frac{374}{1000}, three hundred seventy-four thousandths.

Zeros. A zero between the point and a later digit holds a place and cannot be removed. A zero at the far right of the digits after the point adds no value: 0.37 = 0.370.

Why it is true

The place value rule comes from how the pieces are made. A tenth is one whole cut into ten equal parts. A hundredth is one tenth cut into ten equal parts, and a thousandth is one hundredth cut into ten equal parts. So each piece is one tenth of the piece before it, and ten of a piece put back together rebuild the piece before it. Since a whole holds 10 tenths, each tenth holds 10 hundredths and each hundredth holds 10 thousandths, the whole holds 10 \times 10 \times 10 = 1000 thousandths. That is why one thousandth is \displaystyle \frac{1}{1000}.

Expanded form is true because a digit in standard form is a count. The 8 in 4.086 says “eight hundredths”, so it stands for \displaystyle 8 \times \frac{1}{100}. Putting every digit’s count back together by adding gives the whole number again.

The word form rule is true because every place can be traded down into the last place. In 0.374, each tenth is 100 thousandths and each hundredth is 10 thousandths, so

0.374 = \frac{300}{1000} + \frac{70}{1000} + \frac{4}{1000} = \frac{374}{1000}

A zero at the far right adds no value because it adds 0 pieces of a smaller size: 0.370 is 0.37 plus zero thousandths. A zero in the middle is different. It is a count of zero in that place, and taking it out moves every digit after it one place to the left, which makes each of them worth ten times as much. \blacksquare

Worked examples

Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, Examples 5 and 6 with Part 3, and Examples 7 and 8 with Part 4.

Example 1

What each digit of a race time is worth

Haruki’s sled finishes in 52.374 seconds. Give the place and the value of every digit, and then say how many whole seconds and how many thousandths of a second the time is made of.

Solution

Put the digits in the chart (the first figure) and read each column.

Digit	Place	Value
5	tens	50
2	ones	2
3	tenths	0.3
7	hundredths	0.07
4	thousandths	0.004

Each value is the digit times the place value at the top of its column: 3 \times 0.1 = 0.3, 7 \times 0.01 = 0.07 and 4 \times 0.001 = 0.004.

The whole part is 52 seconds. The part after the point is 3 tenths, 7 hundredths and 4 thousandths. A tenth is 100 thousandths and a hundredth is 10 thousandths, so that part is 300 + 70 + 4 = 374 thousandths. The time is 52 seconds and 374 thousandths of a second:

52 + \frac{374}{1000} = 52.374

Example 2

Writing a number from a base-ten picture

Camila draws two numbers with base-ten pieces like the ones in the second figure.

(a) 3 whole squares, 1 strip, 5 small squares and 8 slivers.

(b) 6 strips and 9 slivers, and nothing else.

Write each number in standard form.

Solution

(a) A whole square is a one, a strip is a tenth, a small square is a hundredth and a sliver is a thousandth. Put each count in its column: 3 ones, 1 tenth, 5 hundredths, 8 thousandths. The number is 3.158. Check by adding the values: 3 + 0.1 + 0.05 + 0.008 = 3.158.

(b) There are no whole squares, so the ones digit is 0. There are 6 tenths and 9 thousandths, but no hundredths. The hundredths column still needs a digit, or the 9 would sit in the hundredths place. Write 0 there: the number is 0.609. Check: \displaystyle \frac{6}{10} + \frac{9}{1000} = 0.6 + 0.009 = 0.609.

Example 3

Five fives

A practice run takes 55.555 seconds. Find what each 5 is worth, and show that each is ten times the one to its right.

Solution

Every digit is a 5, so the value depends only on the place.

\begin{aligned} \text{tens: } 5 \times 10 &= 50 \\ \text{ones: } 5 \times 1 &= 5 \\ \text{tenths: } 5 \times 0.1 &= 0.5 \\ \text{hundredths: } 5 \times 0.01 &= 0.05 \\ \text{thousandths: } 5 \times 0.001 &= 0.005 \end{aligned}

Each place is one tenth of the place to its left, so each value is one tenth of the value above it in the list, and ten times the value below it. For the last two: ten thousandths make a hundredth, so ten groups of 0.005 make 0.05, and 10 \times 0.005 = 0.05. The same step works all the way up: 10 \times 0.05 = 0.5, 10 \times 0.5 = 5 and 10 \times 5 = 50.

Example 4

One place apart, two places apart

(a) Compare the 8 in 2.08 with the 8 in 2.008.

(b) Compare the 4 in 3.47 with the 4 in 3.704.

Solution

(a) In 2.08 the 8 is in the hundredths place, worth 0.08. In 2.008 it is in the thousandths place, worth 0.008. The hundredths place is one place to the left of the thousandths place, so the first 8 is worth 10 times the second: 10 \times 0.008 = 0.08.

(b) In 3.47 the 4 is in the tenths place, worth 0.4. In 3.704 it is in the thousandths place, worth 0.004. These places are two apart. One step left multiplies by 10, and a second step multiplies by 10 again, so the first 4 is worth 10 \times 10 = 100 times the second: 100 \times 0.004 = 0.4.

(c) One tenth is 10 hundredths and each hundredth is 10 thousandths, so one tenth is 100 thousandths. Four tenths are 4 \times 100 = 400 thousandths: \displaystyle 0.4 = \frac{400}{1000}.

Example 5

Expanded form with a zero inside

Write 4.086 in expanded form, first with decimals and then with fractions.

Solution

The third figure shows the digits in their places: 4 ones, 0 tenths, 8 hundredths, 6 thousandths. Each digit is worth its count times its place value.

4.086 = 4 + 0 + 0.08 + 0.006 = 4 + 0.08 + 0.006

The zero tenths contribute nothing to the sum, so the term may be dropped from the expanded form. It may not be dropped from the standard form, where it holds the tenths place open.

With fractions, each place value is written as a fraction:

\begin{aligned} 4.086 &= 4 \times 1 + 8 \times \frac{1}{100} \\ &\quad + 6 \times \frac{1}{1000} \end{aligned}

Check: \displaystyle 8 \times \frac{1}{100} = 0.08 and \displaystyle 6 \times \frac{1}{1000} = 0.006, and 4 + 0.08 + 0.006 = 4.086.

Example 6

From expanded form back to standard form

Write each in standard form.

(a) \displaystyle 7 + \frac{3}{1000} + \frac{2}{10}

(b) 60 + 0.5 + 0.009

Solution

(a) The terms are not in place order, and adding does not care about order (the commutative property of addition), so sort them by place first. 7 is 7 ones. \displaystyle \frac{2}{10} is 2 tenths. \displaystyle \frac{3}{1000} is 3 thousandths. There are no hundredths, so a 0 goes in the hundredths place:

7 + \frac{2}{10} + \frac{3}{1000} = 7.203

(b) 60 is 6 tens and 0 ones. 0.5 is 5 tenths. 0.009 is 9 thousandths. The ones place and the hundredths place are empty, so each gets a placeholder 0: 60.509. Writing 6.59 or 60.59 would move digits into the wrong places.

Example 7

Saying decimals aloud

Write each number in word form: (a) 52.374, (b) 0.046, (c) 13.005.

Solution

In each one, say the whole part, then “and” for the point, then the digits after the point as one number, then the name of the last place.

(a) Whole part fifty-two. The digits after the point are 374, and the last one is in the thousandths place: fifty-two and three hundred seventy-four thousandths.

(b) The whole part is 0, so it is not said, and there is no “and”. The digits after the point are 046, which is the number 46 (a zero in front of a whole number adds nothing). The last digit is in the thousandths place: forty-six thousandths. Check: \displaystyle 0.046 = \frac{0}{10} + \frac{4}{100} + \frac{6}{1000} = \frac{40 + 6}{1000} = \frac{46}{1000}.

(c) Whole part thirteen. The digits after the point are 005, the number 5, ending in the thousandths place: thirteen and five thousandths.

Example 8

Placeholders from words, and where the “and” goes

Write each in standard form.

(a) fifty-three and nine thousandths

(b) fifty-three and ninety thousandths

(d) four hundred and six thousandths, and then four hundred six thousandths

Solution

In (a) to (c), “and” marks the point, so the whole part is 53. The last word, thousandths, says three places follow the point, and the number of thousandths is written so its last digit lands in the thousandths place.

(a) Nine thousandths: the 9 goes in the third place, and the first two places get placeholder zeros. The number is 53.009.

(b) Ninety thousandths: 90 fills the last two places, so the number is 53.090. The zero at the end adds no value, so this is also 53.09, fifty-three and nine hundredths.

The three answers 53.009, 53.09 and 53.9 are different numbers. Where the 9 sits decides what it is worth.

(d) In “four hundred and six thousandths”, the “and” is the point. The whole part is four hundred, and six thousandths follow: 400.006. In “four hundred six thousandths” there is no “and”, so there is no whole part: the whole phrase counts thousandths, 406 of them, giving 0.406. One small word moves the point, which is exactly why “and” is kept for the point alone.

Boundaries

The places keep going. To the right of the thousandths come the ten-thousandths, the hundred-thousandths and so on, each one tenth of the place before it. This lesson stops at thousandths, but nothing in the rule stops there.
Zeros on the right, zeros inside. A zero at the far right of the digits after the point can be added or removed without changing the value: 0.5 = 0.50 = 0.500. A zero between the point and a later digit, or between two digits, holds a place and cannot be removed: 4.086 is not 4.86.
Word form names the last place written. The same number can be named in more than one way: 0.5 is five tenths and 0.500 is five hundred thousandths. Both names are correct, because both count the same amount in different pieces.
“And” means the point. In word form the word “and” is used only for the decimal point. A stray “and” moves the point: “three hundred and twelve thousandths” names 300.012, while “three hundred twelve thousandths” names 0.312.
Not every number ends. Some numbers never stop needing more places. One third written as a decimal is 0.333\ldots with threes forever, so it has no last place to name in word form; see repeating decimals.
Base ten only. Every place being one tenth of its neighbor is a fact about writing numbers in tens. Numbers written with other groupings, in other bases, have a different rule.

Where this goes next

With three places you can tell numbers apart that tenths and hundredths could not, and the next step is to say which is larger: comparing and ordering decimals reads the places from the left, the way the bobsled clock decides a race. Placing these numbers between whole numbers is decimals on a number line, and keeping only as many places as a job needs is rounding decimals.

The rule that each place is one tenth of its left neighbor is what makes multiplying and dividing by ten so tidy: every digit slides one place, as in multiplying a decimal by ten, one hundred or one thousand and dividing a decimal by ten, one hundred or one thousand, which build on how multiplying by ten shifts place value. The numbers 10, 100 and 1000 themselves, and a short way to write them, are in powers of ten. Prices such as 3.75 dollars use two of these places, in money written as decimals, and adding decimals place by place is adding and subtracting decimals.

Exercises

Exercises 1 to 6 go with Part 1, 7 to 12 with Part 2, 13 to 18 with Part 3, and 19 to 23 with Part 4. The reasoning group draws on all four.

Part 1: reading and writing thousandths

A bobsled run takes 38.615 seconds. Give the value of each digit.
Name the place of the 9 in each number: (a) 0.209, (b) 9.42, (c) 1.596.
Write the number with 4 ones, 7 tenths, 2 hundredths and 6 thousandths.
A base-ten picture shows 1 whole square, no strips, 3 small squares and 5 slivers. Write the number.
Haruki’s picture shows 2 strips and 7 slivers, and nothing else. Write the number.
A run time has 5 tens, 1 one, 9 tenths, 8 hundredths and 2 thousandths. Write the time in seconds.

Part 2: each place is one tenth of the place to its left

How many thousandths make (a) 0.01, (b) 0.1, (c) 1?
The 3 in 0.3 is worth how many times the 3 in 0.03?
Compare the 6 in 4.65 with the 6 in 4.506. How many times as much is the first worth?
Write as a decimal: (a) 47 thousandths, (b) 470 thousandths.
How many hundredths make 0.8? How many thousandths?
Aarav has 1 tenth and 15 thousandths. Write the total as one decimal.

Part 3: expanded form

Write 7.452 in expanded form with decimals.
Write 30.608 in expanded form with decimals.
Write 2.719 in expanded form with fractions.
Write 5 + 0.3 + 0.004 in standard form.
Write \displaystyle 9 + \frac{6}{1000} + \frac{4}{10} in standard form.
Write \displaystyle 50 + 2 + \frac{1}{100} + \frac{7}{1000} in standard form.

Part 4: word form and placeholders

Write 0.638 in word form.
Write in word form: (a) 14.02, (b) 6.205.
Write eighty-one thousandths in standard form.
Write fifty-four and seven thousandths in standard form.
Write in standard form: (a) two hundred and five thousandths, (b) two hundred five thousandths.

Reasoning

Camila says 0.07 and 0.070 are the same number, but 0.07 and 0.007 are not. Explain why she is right, using thousandths.
Use a picture of one hundredth cut into pieces to explain why ten thousandths make one hundredth, and then why a whole is 1000 thousandths.

Answers

3 tens is 30, 8 ones is 8, 6 tenths is 0.6, 1 hundredth is 0.01 and 5 thousandths is 0.005. Check: 30 + 8 + 0.6 + 0.01 + 0.005 = 38.615.
(a) thousandths, the third place after the point. (b) ones, just left of the point. (c) hundredths, the second place after the point.
4.726. Each count goes in its own column: 4, point, 7, 2, 6.
1.035. No strips means 0 tenths, and that 0 holds the tenths place so the 3 stays in the hundredths place.
0.207. No wholes gives 0 ones, 2 tenths, 0 hundredths as a placeholder, 7 thousandths.
51.982 seconds. Tens and ones make 51; then 9, 8, 2 follow the point in order.
(a) 10, since a hundredth is cut into ten thousandths. (b) 100, since a tenth is ten hundredths of ten thousandths each. (c) 1000, since a whole is ten tenths of 100 thousandths each.
10 times. The tenths place is one place left of the hundredths place: 10 \times 0.03 = 0.3.
The first 6 is in the tenths place, worth 0.6; the second is in the thousandths place, worth 0.006. Two places apart, so the first is 100 times the second: 100 \times 0.006 = 0.6.
(a) 0.047: the 7 goes in the thousandths place, the 4 in the hundredths, and a 0 in the tenths. (b) 0.470, which is the same as 0.47.
80 hundredths, because each tenth is 10 hundredths. 800 thousandths, because each tenth is 100 thousandths.
0.115. Fifteen thousandths is 10 thousandths and 5 more, which is 1 hundredth and 5 thousandths. Together with 1 tenth: 0.1 + 0.015 = 0.115.
7.452 = 7 + 0.4 + 0.05 + 0.002.
30.608 = 30 + 0.6 + 0.008. The zero ones and zero hundredths add nothing, so they have no term.
\displaystyle 2.719 = 2 \times 1 + 7 \times \frac{1}{10} + 1 \times \frac{1}{100} + 9 \times \frac{1}{1000}.
5.304. There are no hundredths, so a 0 holds the hundredths place.
9.406. Sorted by place: 9 ones, 4 tenths, no hundredths, 6 thousandths.
52.017. 50 + 2 is 52; no tenths, 1 hundredth, 7 thousandths.
six hundred thirty-eight thousandths. No whole part, so no “and”; the digits 638 end in the thousandths place.
(a) fourteen and two hundredths: the digits 02 are the number 2, ending in the hundredths place. (b) six and two hundred five thousandths.
0.081. Eighty-one fills the last two of three places, and the tenths place gets a 0.
54.007. The “and” is the point; seven thousandths puts the 7 in the third place, with two placeholder zeros.
(a) 200.005: “and” is the point, with two hundred before it and five thousandths after. (b) 0.205: no “and”, so the whole phrase counts thousandths, 205 of them.
0.07 is 7 hundredths, which is 70 thousandths, and 0.070 is 0 tenths, 7 hundredths and 0 thousandths, also 70 thousandths. Same count of the same piece, so same number. But 0.007 is only 7 thousandths. The 7 has moved one place right, so it is worth one tenth as much: 70 thousandths against 7.
Draw one hundredth as a small square and cut it into ten equal slivers. Each sliver is one thousandth, so ten of them put back together fill the square: ten thousandths make one hundredth. In the same way a tenth is ten hundredths and a whole is ten tenths. So a whole is 10 tenths, each of 10 hundredths, each of 10 thousandths, which is 10 \times 10 \times 10 = 1000 thousandths.