Grade 4 · 4.NF.C.5 · 4.NF.C.6

Tenths and hundredths

A race is 100 meters long and you stop at the 70-meter mark. What part of the race have you run, and how can that part be written with a point instead of a fraction bar?

You need

Number lines with different scalesGrade 2

Unlocks

Comparing and ordering decimalsGrade 4 Decimals on a number lineGrade 4 Multiplying and dividing by ten, one hundred and one thousandGrade 4 Thousandths: reading, writing and expanded formGrade 5 Adding tenths and hundredths as fractionsGrade 4 Multiplying a decimal by ten, one hundred or one thousandGrade 5 Converting between fractions and decimalsGrade 5 +4 more

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

The school track has a straight that is exactly 100 meters long, with a painted mark every 10 meters. Tevita runs from the start and stops at the 70-meter mark. How much of the straight has he run?

You can already say it with a fraction. The marks cut the straight into 10 equal parts and he has run 7 of them, so he has run \displaystyle \frac{7}{10} of it. But a fraction needs a top, a bar and a bottom, and when the bottom is 10 or 100 there is a shorter way. We write

\frac{7}{10} = 0.7

and read it “seven tenths”. The dot is the decimal point. Digits to its left count wholes, as they always have. Digits to its right count the small pieces you get by cutting a whole into ten, and then into ten again.

This lesson builds the first two of those places, tenths and hundredths. It uses two pictures you will see again and again: a strip or number line cut into ten parts, and a square cut into a hundred small squares. A number line with marks that step by some fixed amount is in number lines with different scales; here the step is one tenth.

Definition

Cut one whole into 10 equal parts. One part is one tenth:

0.1 = \frac{1}{10}

Cut one whole into 100 equal parts. One part is one hundredth:

0.01 = \frac{1}{100}

A decimal writes a number of tenths and hundredths with a point. The first place to the right of the point is the tenths place. The second place is the hundredths place.

Place	One of it, as a decimal	One of it, as a fraction
ones	1	1
tenths	0.1	\displaystyle \frac{1}{10}
hundredths	0.01	\displaystyle \frac{1}{100}

So 0.7 means 7 tenths, \displaystyle \frac{7}{10}. And 0.38 means 3 tenths and 8 hundredths, which is 38 hundredths in all, \displaystyle \frac{38}{100}. A digit to the left of the point counts ones, so 2.63 means 2 ones, 6 tenths and 3 hundredths.

A decimal is a part of a whole, and you must know what the whole is. On the track the whole is the 100-meter straight. In a jar of pennies the whole is one dollar.

Intuition

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

Part 1: tenths on a strip and a number line

Picture the straight as a long strip. The painted marks cut it into ten equal parts, so each part is 10 meters, one tenth of the straight.

Figure 1The 100-meter track cut into ten equal parts. Each part is 10 meters, one tenth of the track. Stopping at 70 meters means 7 tenths have been run, the point 0.7 on the number line.

Count the parts from the start. After one part you are at \displaystyle \frac{1}{10}, written 0.1. After two parts you are at \displaystyle \frac{2}{10}, written 0.2. Tevita stopped after seven parts, at 0.7. After all ten parts you have run the whole straight: \displaystyle \frac{10}{10} = 1.

The number line under the strip shows the same thing as points. The space from 0 to 1 is cut into ten equal steps, and each tick is one more tenth. The decimals 0.1, 0.2, \ldots, 0.9 are the nine ticks between 0 and 1, in order. The 0 in front of the point says there are no whole straights yet, only tenths.

Part 2: hundredths on a grid

A tenth can be cut into ten as well. On the track, each 10-meter part is 10 single meters, and the whole straight is 100 meters. One meter is one hundredth of the straight.

To see hundredths, take a square as the whole and cut it into a 10 by 10 grid. Each small square is \displaystyle \frac{1}{100}, one hundredth, and each full column of ten squares is one tenth.

Figure 2A whole cut into 100 equal squares. Each square is one hundredth. Three full columns and eight more squares make 38 hundredths, 0.38.

Oluchi shades 38 small squares. That is \displaystyle \frac{38}{100} of the square, written 0.38. Look at how the squares sit: three full columns and eight more. Three full columns are 3 tenths and the eight extra squares are 8 hundredths. That is why the digits of 0.38 are a 3 in the tenths place and an 8 in the hundredths place.

Now shade 7 full columns. That is 7 tenths. It is also 70 small squares, which is 70 hundredths. The amount shaded is the same either way, so

\frac{7}{10} = \frac{70}{100} = 0.70 = 0.7

Figure 3Cut each tenth into ten small squares and 7 tenths become 70 hundredths. The shaded part does not change size, so 0.7 = 0.70.

Writing 0.7 as 0.70 adds a 0 in the hundredths place: zero extra hundredths, so no change. That is the tool for writing tenths as hundredths whenever two amounts need the same size of piece.

Part 3: decimals greater than one, words, and money

A decimal can have wholes in front of the point. If Parisa runs the straight twice and then 45 meters more, she has run 2 whole straights and \displaystyle \frac{45}{100} of another, written 2.45. The place value chart gives each digit its column.

Figure 4The number 2.63 in a place value chart: 2 ones, 6 tenths and 3 hundredths.

To read a decimal aloud, say the whole part, say and for the point, then say the digits after the point as one number followed by the name of the last place. So 2.63 is “two and sixty-three hundredths”, and 0.7 is “seven tenths”, with no whole part and no “and”. The word “and” is kept for the point alone.

Why may the digits 63 be read as one number? Because 6 tenths is 60 hundredths, so 6 tenths and 3 hundredths are 60 + 3 = 63 hundredths.

Money uses the same two places. One hundred pennies make one dollar, so one penny, one cent, is one hundredth of a dollar:

1 \text{ cent} = \frac{1}{100} \text{ dollar} = 0.01 \text{ dollar}

A dime is ten cents, which is \displaystyle \frac{10}{100} = \frac{1}{10} of a dollar. So in an amount like 2.63 dollars, the 2 counts dollars, the 6 counts dimes (tenths of a dollar) and the 3 counts pennies (hundredths of a dollar).

Results

The two places.

\begin{aligned} 0.1 &= \frac{1}{10} \\ 0.01 &= \frac{1}{100} \end{aligned}

Trading. Ten hundredths make one tenth, and ten tenths make one whole:

\begin{aligned} 10 \text{ hundredths} &= 1 \text{ tenth} \\ 10 \text{ tenths} &= 1 \\ 100 \text{ hundredths} &= 1 \end{aligned}

Tenths as hundredths. Any number of tenths is ten times as many hundredths:

\frac{7}{10} = \frac{70}{100}, \qquad 0.7 = 0.70

A decimal as a fraction. For digits a (tenths) and b (hundredths),

0.ab = \frac{a}{10} + \frac{b}{100} = \frac{10a + b}{100}

so the two digits after the point, read as one number, count hundredths: \displaystyle 0.38 = \frac{38}{100}.

Word form. Say the whole part, “and” for the point, the digits after the point as one number, then the name of the last place: 2.63 is two and sixty-three hundredths.

Why it is true

Every fact here comes from how the pieces are made. A tenth is a whole cut into ten equal parts, so ten tenths put back together make the whole. A hundredth is a tenth cut into ten equal parts, so ten hundredths make one tenth. Since the whole holds 10 tenths and each tenth holds 10 hundredths, the whole holds 10 \times 10 = 100 hundredths. That is why one hundredth is \displaystyle \frac{1}{100}: on the grid, the ten columns of ten squares are 100 squares.

The same cutting explains \displaystyle \frac{7}{10} = \frac{70}{100}. Each of the 7 shaded columns is cut into 10 small squares, so the shaded part is 7 \times 10 = 70 small squares. Nothing was added or taken away; only the pieces got smaller and more numerous.

The two digits after the point can be read as one number because the tenths digit can be traded down. In 0.38, the 3 tenths are 30 hundredths, so

0.38 = \frac{30}{100} + \frac{8}{100} = \frac{38}{100}

A zero at the far right adds zero pieces of the smaller size, so it changes nothing: 0.7 = 0.70. A zero just after the point is a count of zero tenths. Taking it out would move the next digit one place left, into the tenths place, where it would be worth ten times as much. \blacksquare

Worked examples

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

Example 1

Seventy meters of a hundred

Tevita runs from the start of the 100-meter straight to the 70-meter mark. Write the part he has run as a fraction and as a decimal, and say where it sits on the number line in the first figure.

Solution

The marks every 10 meters cut the straight into 10 equal parts, so each part is one tenth. Seventy meters is 70 \div 10 = 7 of those parts. He has run 7 tenths:

\frac{7}{10} = 0.7

On the number line from 0 to 1, each tick is one more tenth, so 0.7 is the seventh tick after 0. It sits right under the end of the shaded part of the strip, which is the dashed line in the figure.

Example 2

How far, and how much is left

Parisa stops at 0.4 of the straight. How many meters has she run? What part of the straight is still ahead of her, as a decimal and in meters?

Solution

0.4 is 4 tenths. Each tenth of the straight is 10 meters, so she has run 4 \times 10 = 40 meters.

The whole straight is 10 tenths. She has run 4 of them, so 10 - 4 = 6 tenths are still ahead. That is 0.6 of the straight, or 6 \times 10 = 60 meters. Check: 40 + 60 = 100 meters, the whole straight.

Example 3

Thirty-eight hundredths on a grid

Oluchi shades 38 small squares of a 10 by 10 grid, as in the second figure. Write the shaded part as a fraction and a decimal. Then say how many tenths and how many extra hundredths it holds, and how many meters of the straight the same decimal stands for.

Solution

The grid has 100 equal squares, so each is one hundredth. Thirty-eight of them are

\frac{38}{100} = 0.38

Each full column is 10 squares, one tenth. Thirty-eight squares fill 3 columns (30 squares) with 8 left over, so the part is 3 tenths and 8 hundredths. Those counts are the digits of 0.38: a 3 in the tenths place and an 8 in the hundredths place. Check: \displaystyle \frac{3}{10} + \frac{8}{100} = \frac{30}{100} + \frac{8}{100} = \frac{38}{100}.

On the straight, one hundredth is one meter, so 0.38 of the straight is 38 meters.

Example 4

Seven tenths as hundredths

Show that \displaystyle \frac{7}{10}, \displaystyle \frac{70}{100} and 0.70 all name the same amount, and say what each means on the track.

Solution

Use the third figure. Seven of the ten columns are shaded, which is \displaystyle \frac{7}{10}. Each column is ten small squares, so the same shading covers 7 \times 10 = 70 small squares, which is \displaystyle \frac{70}{100}. As a decimal, 70 hundredths is 7 tenths and 0 hundredths, written 0.70, and the 0 in the hundredths place adds nothing, so 0.70 = 0.7.

On the track, \displaystyle \frac{7}{10} counts seven 10-meter parts and \displaystyle \frac{70}{100} counts seventy single meters. Both reach the 70-meter mark, where Tevita stopped.

Example 5

Five hundredths or five tenths

Write \displaystyle \frac{5}{100} and \displaystyle \frac{5}{10} as decimals. Explain why the 0 in one of them cannot be left out.

Solution

\displaystyle \frac{5}{100} is 5 hundredths. On the grid it is 5 small squares, not even one full column, so there are 0 tenths. The tenths place gets a 0 and the 5 goes in the hundredths place: \displaystyle \frac{5}{100} = 0.05.

\displaystyle \frac{5}{10} is 5 tenths, five full columns, so \displaystyle \frac{5}{10} = 0.5, which is also 0.50 or \displaystyle \frac{50}{100}.

The 0 in 0.05 holds the tenths place. Without it the 5 would slide into the tenths place and the number would become 0.5, which is 50 squares instead of 5. On the track, 0.05 is 5 meters and 0.5 is 50 meters, ten times as far.

Example 6

Counting the penny jar

Oluchi empties her penny jar and counts 263 pennies. Write the amount in dollars as a decimal and as a mixed number, and read it aloud.

Solution

One hundred pennies make one dollar. So 263 pennies are 200 pennies, which is 2 dollars, and 63 pennies more, which is \displaystyle \frac{63}{100} of a dollar. The amount is

2\frac{63}{100} \text{ dollars} = 2.63 \text{ dollars}

In the chart (the fourth figure), 2 is in the ones place, and the 63 hundredths are 6 tenths and 3 hundredths. To read it, say the whole part, “and” for the point, then 63 and the last place: two and sixty-three hundredths dollars, which is two dollars and sixty-three cents.

Example 7

From words and meters to decimals

(a) Write six and nine hundredths in standard form.

(b) Write eleven and four tenths in standard form.

(c) Over a week Tevita runs 325 meters on the 100-meter straight. How many straights is that, as a decimal? Read your answer aloud.

Solution

(a) “And” marks the point, so the whole part is 6. The last word, hundredths, says two places follow the point. Nine hundredths puts the 9 in the second place, so the tenths place gets a 0: 6.09.

(b) The whole part is 11. The last word, tenths, says one place follows the point, and four tenths puts a 4 there: 11.4.

(c) Each straight is 100 meters. Three hundred meters is 3 whole straights, and the other 25 meters are \displaystyle \frac{25}{100} of a straight. So he has run

3\frac{25}{100} = 3.25

straights, read three and twenty-five hundredths.

Boundaries

This lesson stops at hundredths. The places keep going to the right, each one tenth of the place before it; the next is Thousandths: reading, writing and expanded form. Nothing in the rule stops at two places.
Zeros at the end, zeros inside. A zero at the far right of the digits after the point changes nothing: 0.7 = 0.70. A zero between the point and a later digit holds a place and cannot be removed: 0.05 is not 0.5.
A decimal is a part of a stated whole. 0.38 of the straight is 38 meters, and 0.38 of a dollar is 38 cents. The same decimal of a different whole is a different amount, so the whole must be named.
Tenths and hundredths only reach some fractions. A fraction whose bottom is 10 or 100, or one equal to such a fraction, can be written with these two places: \displaystyle \frac{1}{2} = \frac{50}{100} = 0.50. Others cannot. One third is more than \displaystyle \frac{33}{100} and less than \displaystyle \frac{34}{100}, so no number of whole hundredths equals it; see repeating decimals.
“And” means the point. In word form the word “and” is used only for the decimal point: “three hundred and five hundredths” is 300.05, while “three hundred five hundredths” has no “and”, so the whole phrase counts hundredths: \displaystyle \frac{305}{100} = 3.05.

Where this goes next

Tenths and hundredths can be put together once they are pieces of the same size, which is exactly what writing \displaystyle \frac{7}{10} as \displaystyle \frac{70}{100} allows: that is adding tenths and hundredths as fractions. Placing a decimal between two whole numbers, such as 2.63 between 2 and 3, is decimals on a number line, and deciding which of two decimals is larger is comparing and ordering decimals.

A third place to the right holds Thousandths: reading, writing and expanded form. Prices with dollars and cents, added and subtracted, are in money written as decimals, and keeping only the places a job needs is rounding decimals. Turning a fraction like \displaystyle \frac{3}{4} into a decimal when its bottom is not 10 or 100 is converting between fractions and decimals. Underneath all of it is a fraction as part of a whole: a decimal is that same idea with the whole cut into tens.

Exercises

Exercises 1 to 8 go with Part 1, 9 to 16 with Part 2, and 17 to 23 with Part 3. The reasoning group draws on all three.

Part 1: tenths on a strip and a number line

Write as a decimal: (a) \displaystyle \frac{3}{10}, (b) \displaystyle \frac{9}{10}.
Write as a fraction: (a) 0.2, (b) 0.1.
A strip is cut into 10 equal parts and 8 are shaded. Write the shaded part as a fraction and as a decimal, and the unshaded part as a decimal.
Tevita stops at the 30-meter mark of the 100-meter straight. What decimal of the straight has he run? How many tenths are left?
A number line from 0 to 1 is cut into ten equal steps. What decimal sits at the second tick after 0? At the ninth?
How many tenths make one whole? Write \displaystyle \frac{10}{10} as a whole number.
Parisa runs 0.1 of the straight. How many meters is that?
Oluchi walks from the start to the 0.8 mark. How many 10-meter parts has she passed, and how many meters has she walked?

Part 2: hundredths on a grid

Write as a decimal: (a) \displaystyle \frac{27}{100}, (b) \displaystyle \frac{64}{100}.
Write as a fraction with bottom 100: (a) 0.53, (b) 0.91.
A 10 by 10 grid has 4 full columns and 6 more squares shaded. Write the shaded part as a fraction and a decimal.
Write as a decimal: (a) \displaystyle \frac{9}{100}, (b) \displaystyle \frac{90}{100}.
Write \displaystyle \frac{3}{10} as hundredths, and as a decimal with two places.
Tevita runs 83 meters of the straight. Write that as a decimal of the straight, and say how many tenths and how many extra hundredths it holds.
Which of these equal 0.2: \displaystyle \frac{2}{100}, \displaystyle \frac{20}{100}, 0.20, 0.02?
Oluchi shades 0.15 of a grid. How many small squares is that, and how many full columns and extra squares?

Part 3: decimals greater than one, words, and money

Write in words: (a) 0.71, (b) 4.6.
Write in words: (a) 12.08, (b) 3.50.
Write in standard form: (a) nine and sixty-two hundredths, (b) twenty and one tenth.
Write five and four hundredths in standard form.
Parisa’s penny jar holds 315 pennies. Write the amount in dollars as a decimal and as a mixed number.
A jar holds 8 pennies. Write the amount as a decimal of a dollar.
Write as decimals: (a) \displaystyle 4\frac{7}{10}, (b) \displaystyle 2\frac{3}{100}.

Reasoning

Tevita says 0.90 is more than 0.9 because 90 is more than 9. Parisa says they are equal. Use a 10 by 10 grid to explain who is right.
Explain why one penny is one hundredth of a dollar, and why 0.03 dollars and 0.3 dollars are different amounts. Say how many cents each one is.

Answers

(a) 0.3, 3 tenths. (b) 0.9, 9 tenths.
(a) \displaystyle \frac{2}{10}. (b) \displaystyle \frac{1}{10}.
Shaded \displaystyle \frac{8}{10} = 0.8. Unshaded 10 - 8 = 2 parts, 0.2.
0.3: thirty meters is 3 parts of 10 meters. 10 - 3 = 7 tenths are left, 0.7.
0.2 at the second tick and 0.9 at the ninth, since each tick is one more tenth.
10 tenths make one whole, so \displaystyle \frac{10}{10} = 1.
10 meters. One tenth of 100 meters is 100 \div 10 = 10 meters.
0.8 is 8 tenths, so she has passed 8 parts and walked 8 \times 10 = 80 meters.
(a) 0.27. (b) 0.64.
(a) \displaystyle \frac{53}{100}. (b) \displaystyle \frac{91}{100}.
\displaystyle \frac{46}{100} = 0.46. Four columns are 40 squares, and 6 more make 46.
(a) 0.09: the tenths place holds a 0. (b) 0.90, the same as 0.9.
\displaystyle \frac{30}{100}, since each tenth is 10 hundredths, so 0.30.
0.83: one meter is one hundredth of the straight. It holds 8 tenths and 3 extra hundredths.
\displaystyle \frac{20}{100} and 0.20. The other two are 2 hundredths, 0.02, ten times smaller.
15 squares, since \displaystyle 0.15 = \frac{15}{100}: 1 full column and 5 extra squares.
(a) seventy-one hundredths. (b) four and six tenths.
(a) twelve and eight hundredths: the digits 08 are the number 8, ending in the hundredths place. (b) three and fifty hundredths, which is the same number as three and five tenths.
(a) 9.62. (b) 20.1.
5.04. Four hundredths puts the 4 in the second place, so the tenths place holds a 0.
3.15 dollars, or \displaystyle 3\frac{15}{100} dollars: 300 pennies are 3 dollars, and 15 more are \displaystyle \frac{15}{100} of a dollar.
0.08 dollars, since 8 pennies are \displaystyle \frac{8}{100} of a dollar.
(a) 4.7. (b) 2.03, with a 0 holding the tenths place.
Parisa is right. 0.9 is 9 full columns of the grid. 0.90 is 90 small squares, and 90 squares fill exactly 9 columns, so the shading is the same. The 90 in 0.90 counts hundredths and the 9 in 0.9 counts tenths, pieces ten times as big, so comparing 90 with 9 compares counts of different pieces.
One hundred pennies make one dollar, so the dollar is cut into 100 equal parts and one penny is one of them, \displaystyle \frac{1}{100} of a dollar. 0.03 dollars is 3 hundredths, 3 cents. 0.3 dollars is 3 tenths, and each tenth of a dollar is 10 cents, so it is 30 cents. The 3 sits in a different place, so it counts a different size of piece.