Place Value and How Decimals Work

Each place-value column is worth ten times the one on its right, and those columns only ever build whole numbers. But a length between $14$ and $15$ has no whole-number name. So keep the columns running to the right, each one worth a tenth of the one before it.

Problem

5.83

the digit

8

sits in the first place past the decimal point, the tenths place. What is the value of the

8

alone? Write your answer as a fraction

\frac{a}{b}

Show a hint

The value of a digit is the digit times the worth of its place. The 8 is sitting in the tenths place, and one single tenth is worth $1 0^{- 1} = \frac{1}{10}$ . So you have 8 of those tenths.
Take the digit $8$ and multiply it by the place value $\frac{1}{10}$ . That is $8 \times \frac{1}{10}$ , which stacks 8 copies of one tenth on top of each other. Write the result as a single fraction.

Show the full solution

The

8

sits one step past the point, in the tenths place, and one unit of that place is worth

1 0^{- 1} = \frac{1}{10}

. Eight of those is

8 \times \frac{1}{10} = \frac{8}{10}

. A digit's value is always the digit times the worth of its place, so the

5

and the

3

do not affect this one.

Problem

5.83

the digit

3

sits one step right of the tenths place, in the hundredths place. What is the

3

worth all by itself? Write your answer as a fraction

a / b

Show a hint

The 3 is one step to the right of the tenths place. Stepping right means the place value gets divided by 10 again, so this new place is worth $\frac{1}{10}$ of $\frac{1}{10}$ .
One tenth of one tenth is $\frac{1}{10} \times \frac{1}{10} = \frac{1}{100}$ , which is $1 0^{- 2}$ . The digit there is a $3$ , so its value is $3$ of those, meaning $3 \times \frac{1}{100}$ .

Show the full solution

Step one place right from the tenths and the place value divides by

10

again, so the

3

sits in the hundredths place, worth

\frac{1}{10} \times \frac{1}{10} = \frac{1}{100}

. Three of those is

3 \times \frac{1}{100} = 3 / 100

. That is

1 0^{- 2}

, and it should come out smaller than the

8

tenths next door, since places shrink as you move right.

Reading from the left, each column is worth ten times the one to its right, so the values run

100, 10, 1

and keep falling straight through the point to

\frac{1}{10}, \frac{1}{100}, \frac{1}{1000}

. That dividing-by-ten staircase never pauses at the decimal point. The negative powers

1 0^{- 1}, 1 0^{- 2}, 1 0^{- 3}

are simply the next steps down from

1 0^{0} = 1

. In

5.83

the

5

lands in the ones column as

5

ones, the

8

in tenths as

8

tenths, and the

3

in hundredths as

3

hundredths, while the thousandths cell stays empty. Lining a number up on the point is exactly what drops each digit into its true column.

Problem

Write the decimal with 5 in the ones place, 7 in tenths, nothing in hundredths, and 3 in thousandths as a single decimal.

Show a hint

Set up the columns left to right just as they are named. One column for the ones, then the decimal point, then tenths, then hundredths, then thousandths. Drop each given digit into its own column and ask what belongs in the hundredths column when the part is $\frac{0}{100}$ .
The digits in order are 5, then a point, then 7, then 0, then 3. That zero is not optional. It is the placeholder that keeps the 3 sitting in the thousandths column instead of letting it drift left. Read the columns straight across to get the decimal.

Show the full solution

Drop each digit into its own column, left to right. Ones

5

, then the point, then tenths

7

, hundredths

0

, thousandths

3

. Reading straight across gives

5.703

. That middle zero is doing real work. Without it you would write

5.73

, which puts the

3

in the hundredths place, ten times bigger than intended.

Problem

Write

0.27

as a single fraction over a power of ten in the form

\frac{a}{b}

. Use a common denominator of

100

Show a hint

You have two pieces, $\frac{2}{10}$ and $\frac{7}{100}$ . To add them you need the same denominator. Which power of ten do both pieces fit into, $10$ or $100$ ?
Rewrite $\frac{2}{10}$ over $100$ by multiplying top and bottom by $10$ , which gives $\frac{20}{100}$ . Now add the hundredths together, $\frac{20}{100} + \frac{7}{100}$ , and keep the denominator the same.

Show the full solution

The

2

is in tenths and the

7

is in hundredths, so

0.27 = \frac{2}{10} + \frac{7}{100}

. Rewrite the first piece over

100

\frac{20}{100}

and add,

\frac{20}{100} + \frac{7}{100} = \frac{27}{100}

. Two decimal places always give a denominator of

1 0^{2} = 100

, with the digits after the point read as the numerator.

Problem

A jeweller records a mass as

\frac{9}{10}

of a gram. Write it as a decimal.

Show a hint

A denominator of $10$ means tenths, and the tenths place is the first slot to the right of the decimal point. So the only digit you need to write past the point is the $9$ .
Put the $9$ in the tenths place, just after the point, and write a $0$ in the ones place in front so the answer reads $0.9$ instead of a lonely $. 9$ .

Show the full solution

A denominator of

10

means tenths, and tenths is the first place past the point, so the

9

goes there. There are no whole grams, so a

0

fills the ones place, giving

0.9

. That leading zero is only there so the answer does not arrive as a bare

. 9

that is easy to misread.

Problem

Are

0.3

and

0.30

equal as numbers? Answer yes or no.

Show a hint

A trailing zero lives in a brand new place to the right. Ask yourself what that place is actually holding. The hundredths place in $0.30$ holds a $0$ , so it contributes $\frac{0}{100}$ , and adding nothing to a number leaves it alone.
Turn each reading into a fraction and see if they match. $0.3$ means $\frac{3}{10}$ . For $0.30$ , write $\frac{30}{100}$ and simplify by dividing top and bottom by $10$ to get $\frac{3}{10}$ . Same fraction means same number.

Show the full solution

Read each one as a fraction.

0.3

\frac{3}{10}

. In

0.30

the

3

is still in tenths and the new

0

sits in hundredths, adding

\frac{0}{100} = 0

, so

0.30 = \frac{3}{10}

as well. Same fraction, same number, so

yes

. Tacking on a trailing zero is the same move as multiplying top and bottom by

10

, which never changes a value.

A whole number has an exact spot on the number line, but $7.4$ falls between $7$ and $8$ . Zoom in on that gap. Chop it into ten tenths, then chop each tenth into ten hundredths. The same divide-by-ten step that built the columns lets you land on any decimal.

Problem

The unit from

0

1

is split into

10

equal steps.

0.7

means seven tenths. Which numbered mark does

0.7

land on?

Show a hint

The marks are $0.1, 0.2, 0.3$ and so on. Walk along them one tenth at a time and count how many steps it takes to arrive at $0.7$ .
The number $0.7$ means $7$ tenths, that is $7 \times \frac{1}{10}$ , so it sits $7$ one-tenth steps from $0$ .

Show the full solution

0.7

means seven tenths, that is

7 \times \frac{1}{10}

, so it sits seven one-tenth steps from

0

and lands on mark

7

. The digit in the tenths place gives the count directly, so you never have to walk the marks.

The top line splits one whole into ten tenths. Pick the slice between

0.6

and

0.7

, stretch it wide, and ten new ticks appear, the hundredths. One tenth holds exactly ten hundredths, so the gold dot on

0.62

sits two small steps into the slice that the green

0.7

caps off. Every zoom is the same move you made building the columns, dividing by ten, the negative powers

1 0^{- 1}, 1 0^{- 2}, 1 0^{- 3}

drawn right on the line. Keep zooming and you can pin any decimal as tightly as you like.

Problem

Stack

0.7

and

0.68

with decimal points aligned. Read from left to right one place at a time. Which is larger?

Show a hint

Line up the decimal points and look only at the tenths place to start. One number has $7$ tenths there, the other has $6$ tenths. The hundredths and anything smaller come later and only matter if the tenths tie.
The tenths place already disagrees, so that is where it ends. $7$ tenths beats $6$ tenths, and even the largest possible hundredths digit in $0.68$ cannot add up to a full extra tenth. So the number with $7$ in the tenths place wins.

Show the full solution

Pad

0.7

0.70

so both numbers show the same places, then read from the left. The tenths place has a

7

against a

6

, and that first disagreement settles it, so

0.7

is larger. The most a hundredths digit can be worth is

9

hundredths, still less than one whole tenth, so nothing to the right can close a gap opened in the tenths.

Problem

Which number is larger,

0.5

0.4999

? Stack them with points aligned and compare place by place from the left.

Show a hint

Write them one above the other with the decimal points aligned, $0.5000$ over $0.4999$ . Now scan left to right and find the first place where the two digits are different. That place, and nothing after it, settles the question.
The first differing column is the tenths. There $0.5$ shows a $5$ and $0.4999$ shows a $4$ . Five tenths already beats four tenths, so the larger number is decided before you ever reach those $9$ s. All three $9$ s live in the hundredths, thousandths, and ten-thousandths, places far too tiny to add up to a whole extra tenth.

Show the full solution

Pad to the same length and compare

0.5000

with

0.4999

from the left. The tenths place has a

5

against a

4

, so

0.5

is larger. Over a common denominator the gap is

\frac{5000}{10000} - \frac{4999}{10000} = \frac{1}{10000}

. Those three

9

s buy finer detail, not more size.

Problem

Order

0.4

0.39

0.401

0.4009

, and

0.41

from smallest to largest by aligning decimal points. Which number is third in the list?

Show a hint

First order them. Line up the decimal points and read the tenths place. Four splits begin with $4$ tenths and one with $3$ tenths, so the $3$ -tenths one is smallest. Then pad the rest with trailing zeros to the same length and compare.
Padded to four places the four tied splits are $0.4000, 0.4010, 0.4009, 0.4100$ , so compare $4000, 4010, 4009, 4100$ . The full order is $0.39, 0.4, 0.4009, 0.401, 0.41$ . Pick the one in the middle.

Show the full solution

Only

0.39

opens with

3

tenths, so it is the smallest. Pad the other four to

0.4000, 0.4010, 0.4009, 0.4100

and compare them like the whole numbers

4000, 4010, 4009, 4100

. The full order is

0.39, 0.4, 0.4009, 0.401, 0.41

, so third is

0.4009

. It shows the most digits yet is not the biggest, because those extra digits sit in the thousandths and ten-thousandths where they are worth almost nothing.

Problem

Multiplying by

10

shifts every digit one place left, so the decimal point appears to hop right. Multiplying by

100

shifts two places. What is

3.7 \times 100

Show a hint

Each factor of $10$ shifts every digit one place toward higher value. You have $100$ , which is two tens, so that is two places. Watch where the $3$ and the $7$ land.
Holding the point still, multiplying by $100$ makes it appear to move two places to the right. Starting from $3.7$ , one place gives $37$ , and a second place gives $370$ .

Show the full solution

Multiplying by

100

shifts every digit two places toward higher value. The

3

goes from ones to hundreds, worth

300

, and the

7

goes from tenths to tens, worth

70

. Together that is

300 + 70 = 370

. Watching the point instead of the digits gives the same thing, two places right,

3.7

37

370

Problem

A tag should read

4.50

dollars but printed

45.0

dollars instead, shifting the decimal point one place right. By how many times is the printed price larger than the correct price?

Show a hint

Look at where the digit 4 lives in each price. In 4.50 it is in the ones place, worth 4. In 45.0 it is in the tens place, worth 40. Every digit has jumped up exactly one place.
When each digit moves one place to the left, its value gets ten times bigger. So the whole price is ten times bigger. You can check by asking what you multiply 4.50 by to reach 45.0.

Show the full solution

4.50

the

4

is in the ones place, worth

4

. In

45.0

that same

4

is in the tens place, worth

40

. Every digit moved one place left, so every digit is worth ten times as much and so is the whole price, making the printed tag

10

times larger. Moving the point one place right always multiplies by

10

, and

4.50 \times 10 = 45.0

confirms it.

Practice these ideas

Practice

Four splits in seconds: Runner A

0.205

, B

0.21

, C

0.2

, D

0.2049

. Ordered smallest to largest, which split is the second smallest?

Show the solution

Pad every split to four places,

0.2000, 0.2049, 0.2050, 0.2100

, and read left to right. All four tie at

2

tenths. In the hundredths only

0.2100

has a

1

, so it is the largest. The other three split at the thousandths, where the digits are

0

4

5

. The order is

0.2, 0.2049, 0.205, 0.21

, so the second smallest is

0.2049

Practice

4.36

, swap the tenths and hundredths digits. What new decimal do you get?

Show the solution

The swap touches only the two digits after the point. The

6

moves into tenths and the

3

moves into hundredths, while the

4

stays where it is, giving

4.63

. Swapping changes the number because the tenths place is worth ten times the hundredths place.

Practice

Write

0.013

as a single fraction

a / b

with a power-of-ten denominator.

Show the solution

The last digit

3

sits in the thousandths place, so the denominator is

1, 000

, and the digits after the point read as

13

on top. That gives

13 / 1000

. The leading zero adds nothing to the numerator, but it does push the

3

out to thousandths, which is what sets the denominator.

Practice

Write

\frac{7}{100}

as a decimal.

Show the solution

A denominator of

100 = 1 0^{2}

means hundredths, the second place after the point. Put the

7

there and a

0

in the tenths place to hold it open, giving

0.07

. Drop that inner zero and you get

0.7

, which is ten times too big.

Practice

A cartridge holds

100

drops, each weighing

0.306

grams. What is the total weight in grams? Compute

0.306 \times 100

Show the solution

Multiplying by

100

slides every digit two columns left. The

3

runs from tenths to ones to tens, worth

30

, and the

6

runs from thousandths to hundredths to tenths, worth

0.6

. Adding gives

30.6

grams. Each factor of ten is one slide, so

100

is two.

Practice

A weather station records

8.2

mm of rain, then converts to meters by dividing by

1, 000

. What is the rainfall in meters?

Show the solution

Dividing by

1, 000 = 1 0^{3}

slides every digit three columns right. The

8

runs from ones down to thousandths and the

2

lands in ten-thousandths, with zeros filling the empty tenths and hundredths. So

8.2 \div 1, 000 = 0.0082

. Shrinking by a factor of a thousand should land far below

1

, and it does.

Practice

A point on the number line is at

14.7

. What is the smaller of the two whole numbers it sits between?

Show the solution

14.7

14

whole units plus

7

tenths, so it sits past

14

but short of

15

. The smaller of the two is

14

. The digits left of the point always name the largest whole number the value has fully reached.

Practice

The value

0.73

falls between two neighboring tenths. Which of those two tenths is the larger one?

Show the solution

The tenths digit of

0.73

7

, and the

3

hundredths carries it just past

0.7

. It is

73

hundredths, still short of the

80

hundredths that make

0.8

, so it lies between

0.7

and

0.8

and the larger tenth is

0.8

Practice

Which number is larger,

0.6

0.5999

Show the solution

Pad the short one to

0.6000

and compare with

0.5999

from the left. The tenths place has a

6

against a

5

, so

0.6

is larger. More digits does not mean more value. As fractions,

\frac{6000}{10000}

beats

\frac{5999}{10000}

by a single ten-thousandth.

Practice

Which number is larger,

0.07

0.7

Show the solution

0.7

the

7

is in the tenths place, worth

\frac{7}{10}

. In

0.07

the zero takes the tenths place and pushes the

7

out to hundredths, worth

\frac{7}{100}

. Since

\frac{7}{10} = \frac{70}{100}

, the larger number is

0.7

, ten times as big. An inner zero moves a digit, so it changes the value.

Practice

Four forms:

0.6

0.60

0.600

, and

0.06

. How many of these name the exact same amount as

0.6

Show the solution

0.6

\frac{6}{10}

. Then

0.60 = \frac{60}{100} = \frac{6}{10}

and

0.600 = \frac{600}{1000} = \frac{6}{10}

, so counting

0.6

itself that is

3

forms naming the same amount.

0.06

is the odd one out, since its zero sits before the

6

and drops it to hundredths, ten times smaller.

Practice

A receipt should read

8.30

dollars but printed

83.0

dollars. How many times larger is the printed amount?

Show the solution

8.30

the

8

sits in the ones column, and in

83.0

it has slid into the tens column, so it counts for ten times as much and every other digit does too. The printed amount is

10

times larger. Checking directly,

83.0 \div 8.3 = 10

, which is exactly what moving the point one place right does.

Practice

An app should have multiplied

4.5

grams by

10

but divided by

10

instead, showing

0.45

grams. How many times larger is the correct amount,

45

grams, than

0.45

grams?

Show the solution

Getting from

0.45

up to

45

moves the point two places right, and each place is a factor of

10

, so the correct amount is

10 \times 10 = 100

times larger. Dividing directly agrees,

\frac{45}{0.45} = \frac{4500}{45} = 100

. The gap is two steps because the app divided by

10

where it should have multiplied by

10

Practice

Using tiles

0

3

8

in the three blank slots of

0. \underline{} \underline{} \underline{}

, build the smallest positive decimal greater than zero.

Show the solution

The tenths place counts for the most, so park the

0

there. Of the two digits left, the smaller one goes in the stronger place, so

3

in hundredths and

8

in thousandths, giving

0.038

. Swapping them gives the bigger

0.083

, and any nonzero tile in tenths jumps you to at least

0.3

Place Value and Decimals

Practice these ideas