Grade 4 · 4.NF.C.7 · 5.NBT.A.3

Comparing and ordering decimals

One gymnast’s cartwheel earns 0.7 of a point and another’s earns 0.68. The second score has more digits. Which score is higher, and how can you tell without guessing?

You need

Comparing and ordering whole numbersGrade 2 Tenths and hundredthsGrade 4

Unlocks

Adding and subtracting decimalsGrade 5 Comparing numbers in scientific notationMath 8

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

When you reach for it

Bhumika, Kahu and Inti train at a gymnastics club, and on weekends they skate at the ice rink. Both sports are scored by judges, and the judges write their scores as decimals. At practice the coach scores each cartwheel from 0 to 1 point. Bhumika’s cartwheel earns 0.7 of a point and Kahu’s earns 0.68. Whose cartwheel scored higher?

You reach for this skill whenever two or more decimals have to be put side by side: which score wins, which time is shorter, which amount of money is more, or how to list a set of numbers from least to greatest. You already know how to do this for whole numbers: line the numbers up by place and look at the biggest place first (Comparing and ordering whole numbers). Decimals work the same way, once you remember from Tenths and hundredths what each place after the point is worth.

Three symbols record the answer. a > b says a is greater than b. a < b says a is less than b. a = b says they are the same number. The open side of > and < always faces the greater number (the symbols for comparing).

The procedure

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

Part 1: compare place by place from the left

Put 0.7 and 0.68 on hundred grids, where the whole is a square cut into 100 small squares. Seven full columns is 7 tenths, which is 70 small squares. Six full columns and eight more squares is 68 small squares.

Figure 10.7 and 0.68 on hundred grids. Seven full columns are 70 small squares, two more than 68, so 0.7 > 0.68.

The left grid has two more shaded squares, so 0.7 is the greater score:

0.7 > 0.68

Grids work, but you cannot draw a grid for every pair. The digits tell the same story. Line the two numbers up so the points sit under each other, then read the places from the left.

	ones	tenths	hundredths
0.7	0	7
0.68	0	6	8

The ones are the same, 0 and 0. In the tenths place, 7 tenths is more than 6 tenths, and that settles it. You do not need to look any further.

Why may you stop as soon as one place differs? Because each place is worth ten of the place to its right. One tenth is 10 hundredths, and the hundredths digit can be at most 9. So whatever the hundredths digits are, they add less than one tenth:

0.09 < 0.1

Kahu has one tenth fewer than Bhumika, and his 8 hundredths cannot make up that one tenth. In the same way, the most that the tenths and hundredths together can add is 0.99, which is less than one whole. So a number with more ones is greater, whatever comes after the point.

This is the same reason the left digit decides for whole numbers. In 500 against 498, the 5 hundreds beat the 4 hundreds, because 98 is less than one more hundred. Decimals just carry the same place value pattern on past the point.

So the move is: line up the points, then go from the left. The first place where the digits differ decides, and the number with the greater digit there is the greater number. If every place matches, the numbers are equal.

Part 2: equivalent decimals, and why the count of digits does not decide

In the table above, 0.7 has an empty box in the hundredths place. An empty place after the last digit means zero of that piece. So 0.7 is 7 tenths and 0 hundredths, and we may write it

0.7 = 0.70

This is the grid fact from Tenths and hundredths: 7 full columns are 70 small squares. Two decimals that name the same amount, such as 0.5 and 0.50, are called equivalent decimals. Five tenths is half the grid, and so is fifty hundredths:

0.5 = 0.50 = \frac{50}{100} = \frac{5}{10}

Equivalent decimals give a second way to compare. Write both numbers with the same number of places, so both count the same size of piece:

\begin{aligned} 0.70 &= 70 \text{ hundredths} \\ 0.68 &= 68 \text{ hundredths} \end{aligned}

Now it is a whole number comparison. Seventy is more than 68, so 0.70 > 0.68, and that is 0.7 > 0.68 again. Both ways always agree, because they read the same places.

Notice what did not matter: how many digits each number has. The number 0.68 is written with two digits after the point and 0.7 with one, yet 0.7 is greater. The reason is place value. A digit’s worth comes from its place, not from how many digits stand beside it. The second digit of 0.68 counts hundredths, pieces ten times smaller than tenths, so it adds only a little. With whole numbers, more digits did mean a bigger number, because every extra digit pushed the others into bigger places: 68 has a tens digit and 7 does not. After the point, an extra digit on the right pushes nothing; it adds a smaller piece. So the length of a decimal never settles a comparison by itself. Sometimes the longer one is greater (0.75 > 0.7) and sometimes the shorter one is (0.7 > 0.68). The places decide.

Zeros may be written or dropped only at the right end, after the point. A zero there counts zero of a small piece and changes nothing. A zero between the point and another digit is different: it holds a place, and 0.05 (five hundredths) is not 0.5 (five tenths).

Part 3: order a list, and decimals greater than one

Kahu’s skating spins get three scores from three judges: 1.3, 1.29 and 1.31 points. To put them in order, compare them all at once, place by place from the left. It helps to write them with the same number of places first, lined up by the point:

\begin{aligned} 1.3 &= 1.30 \\ 1.29 &= 1.29 \\ 1.31 &= 1.31 \end{aligned}

All three have 1 one, so the ones do not decide. Now read the tenths: 3, 2 and 3. The score 1.29 has the fewest tenths, so it is the least. The other two tie on tenths at 3, so go on to the hundredths: 0 against 1. So 1.30 < 1.31. In order from least to greatest:

1.29 < 1.3 < 1.31

The number line shows the same order. Between 1.25 and 1.35 each tick is one hundredth, and 1.3 sits at the tick 1.30, one step right of 1.29 and one step left of 1.31. (Decimals on a number line is about placing decimals on a line like this one.)

Figure 2Between 1.25 and 1.35 each step is one hundredth. The point 1.3 is the tick 1.30, so the three scores sit in the order 1.29, 1.3, 1.31 from left to right.

A list may be put in order from least to greatest or from greatest to least. Either way the comparisons are the same; only the direction of reading changes.

One more thing must be true before any comparison means anything: both decimals must be parts of the same whole. On a hundred grid that is automatic, because both numbers are drawn on the same size of square. In a story it is not. If Inti skates 0.5 of a long lap and Kahu skates 0.6 of a short lap, the decimals compare as 0.6 > 0.5, but that does not say who skated farther. Example 7 works this out.

Worked examples

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

Example 1

Two cartwheels

Bhumika’s cartwheel scores 0.7 of a point and Kahu’s scores 0.68. Compare the scores with a symbol and say whose is higher.

Solution

Line up the points and read from the left. Both have 0 ones. In the tenths place Bhumika has 7 and Kahu has 6. The tenths differ, so they decide: 7 tenths is more than 6 tenths, and Kahu’s 8 hundredths are less than the one tenth he is short. So

0.7 > 0.68

and Bhumika’s cartwheel scored higher. Check with the first figure: 70 shaded squares against 68.

Example 2

When the tenths tie

Compare with >, < or =: (a) 0.46 and 0.43, (b) 3.1 and 2.95.

Solution

(a) Both have 0 ones and 4 tenths, so those places do not decide. Go on to the hundredths: 6 against 3. Since 6 hundredths is more than 3 hundredths,

0.46 > 0.43

The same fact read the other way is 0.43 < 0.46. Both sentences are true; the open side of the symbol faces 0.46 each time.

(b) Start at the ones: 3 against 2. They differ, so they decide. Everything after the point in 2.95 adds 0.95, less than one whole, so it cannot make up the missing one. So 3.1 > 2.95, even though 2.95 has the bigger tenths digit.

Example 3

Five tenths, fifty hundredths

Inti’s handstand scores 0.5 of a point and Bhumika’s scores 0.50. Are the scores the same? Also say whether 2.80 = 2.8.

Solution

Picture a hundred grid. 0.5 is 5 full columns. Each column is 10 small squares, so 5 columns are 5 \times 10 = 50 small squares, which is 50 hundredths, 0.50. Both scores shade exactly half the grid, so

0.5 = 0.50

and the scores are the same. As fractions, \displaystyle \frac{5}{10} = \frac{50}{100}.

For 2.80: line it up with 2.8. Both have 2 ones and 8 tenths. The 0 at the right end of 2.80 is zero hundredths, which adds nothing, so 2.80 = 2.8. The zero is at the right end after the point, which is the one place a zero may be added or dropped.

Example 4

More digits, or fewer

Compare 0.2 with 0.19, and then 0.2 with 0.25. Use the two pairs to explain why the number of digits does not tell which decimal is greater.

Solution

Write each pair with two places. 0.2 = 0.20, which is 20 hundredths, and 0.19 is 19 hundredths. Since 20 > 19,

0.2 > 0.19

For the second pair, 0.20 is 20 hundredths and 0.25 is 25 hundredths, so 0.2 < 0.25.

In both pairs 0.2 is the shorter number. It is greater than one longer number and less than the other. So the count of digits cannot be what decides. Place value says why: the extra digit in 0.19 and in 0.25 sits in the hundredths place, where it adds only hundredths. Whether the total comes out above or below 0.2 depends on the tenths digit first: 1 tenth in 0.19 is less than 2 tenths, while 0.25 ties 0.2 on tenths and then has 5 hundredths more.

Example 5

Three spin scores

Three judges score Kahu’s spin 1.3, 1.29 and 1.31. Write the scores from least to greatest, then from greatest to least.

Solution

Write all three with two places and line up the points: 1.30, 1.29, 1.31. The ones are all 1, so look at the tenths: 3, 2, 3. The score 1.29 has the fewest tenths, so it is the least. The other two tie on 3 tenths, so look at the hundredths: 1.30 has 0 and 1.31 has 1. So 1.30 < 1.31. From least to greatest:

1.29 < 1.3 < 1.31

From greatest to least it is the same list read backward: 1.31 > 1.3 > 1.29. The second figure shows the three scores one hundredth apart on the number line.

Example 6

The scoreboard

Five judges score Bhumika’s floor routine out of 10 points: 9.4, 9.45, 8.95, 9.05 and 9.50. Put the scores in order from greatest to least. Which two judges gave the same score, if any?

Solution

Write every score with two places, points lined up:

score	ones	tenths	hundredths
9.4	9	4	0
9.45	9	4	5
8.95	8	9	5
9.05	9	0	5
9.50	9	5	0

Ones first. Four scores have 9 ones and 8.95 has only 8, so 8.95 is the least, even with its large tenths digit. Among the four with 9 ones, read the tenths: 4, 4, 0, 5. The greatest is 9.50 with 5 tenths, and the least of the four is 9.05 with 0 tenths. The two with 4 tenths are settled by the hundredths: 9.45 has 5 and 9.4 has 0, so 9.45 > 9.4. From greatest to least:

9.50 > 9.45 > 9.4 > 9.05 > 8.95

No two scores are equal: once each is written with two places, no two rows of the table match. The scores 9.4 and 9.50 look alike at first glance, but 9.50 = 9.5, which has 5 tenths, not 4.

Example 7

The same decimal of different laps

The big ice rink has a lap 60 meters long, and the small practice rink has a lap 40 meters long. Inti skates 0.5 of a big rink lap. Kahu skates 0.6 of a practice rink lap. Kahu says he skated farther, since 0.6 > 0.5. Is he right?

Solution

The decimals do compare as 0.6 > 0.5: 6 tenths is more than 5 tenths. But the two tenths are tenths of different wholes, so they are not the same size.

One tenth of the big lap is 60 \div 10 = 6 meters, so Inti skated 5 \times 6 = 30 meters. One tenth of the practice lap is 40 \div 10 = 4 meters, so Kahu skated 6 \times 4 = 24 meters. Since 30 > 24, Inti skated farther, and Kahu is not right.

Figure 30.6 is more than 0.5, but 0.6 of the short lap is less ice than 0.5 of the long lap. A comparison of decimals is a comparison of parts of the same whole.

The comparison 0.6 > 0.5 is still true; it just answers a different question. It says Kahu finished a bigger part of his own lap. To compare distances, the decimals must be parts of the same whole, or be turned into meters first.

Boundaries

The same whole. Comparing two decimals compares two amounts only when both are parts of the same whole: the same grid, the same lap, the same dollar. Of different wholes, the greater decimal can be the smaller amount, as in Example 7.
Zeros only at the right end after the point. Writing 0.4 as 0.40 changes nothing. Writing a zero anywhere else changes the number: 0.04 is 4 hundredths, and putting a zero on the right of a whole number, 4 to 40, makes it ten times as large.
This lesson stops at hundredths. The same left-to-right reading works for any number of places, since every place is still worth ten of the next one; it carries on to Thousandths: reading, writing and expanded form.
Numbers below zero turn the order around. Every decimal here is 0 or more. For numbers below zero the one farther from zero is the smaller, which is comparing and ordering rational numbers.

Where this goes next

Writing two decimals with the same number of places is the same move as giving two fractions the same bottom number: 0.7 and 0.68 become \displaystyle \frac{70}{100} and \displaystyle \frac{68}{100}. For fractions whose bottoms are not 10 or 100, that move is comparing by finding a common denominator, and it rests on Equivalent fractions, the fraction form of 0.5 = 0.50.

Lining up the points is also the first step of adding and subtracting decimals, where only pieces of the same size can be combined. A third place after the point holds Thousandths: reading, writing and expanded form, and the same left-to-right reading compares numbers with three places too.

Exercises

Exercises 1 to 8 go with Part 1, 9 to 16 with Part 2, and 17 to 22 with Part 3. The reasoning group draws on all three.

Part 1: compare place by place from the left

Compare with >, < or =: (a) 0.8 and 0.6, (b) 0.34 and 0.52.
Compare with >, < or =: (a) 0.57 and 0.59, (b) 0.71 and 0.17.
Compare with >, < or =: (a) 4.2 and 3.8, (b) 5.06 and 5.6.
One hundred grid shows 0.4 and another shows 0.39. How many small squares are shaded on each, and which decimal is greater?
Inti’s jump at practice scores 0.75 of a point and Bhumika’s scores 0.8. Whose score is higher? Say which place decides.
True or false: (a) 0.48 > 0.84, (b) 2.7 < 2.9, (c) 0.05 < 0.5.
Which digits can go in the box to make 0.\square 4 < 0.54 true?
Kahu’s skating jump scores 3.65 points and his spin scores 3.56. Which scored higher?

Part 2: equivalent decimals

Write with two places after the point: (a) 0.4, (b) 0.9, (c) 3.1.
True or false: (a) 0.80 = 0.8, (b) 0.08 = 0.8, (c) 1.50 = 1.5, (d) 0.6 = 0.06.
Write 0.6 and 0.58 as numbers of hundredths, then compare them.
Compare 0.49 and 0.5 by writing both in hundredths.
Compare with >, < or =: (a) 1.6 and 1.60, (b) 0.1 and 0.09.
How many hundredths is 0.45, and how many is 0.5? Which decimal is greater?
Inti’s spin scores 2.4 points and Kahu’s scores 2.38. Whose is higher, and by how many hundredths?
Write two decimals equivalent to \displaystyle \frac{3}{10}, one with one place and one with two places.

Part 3: order a list

Order from least to greatest: 0.6, 0.65, 0.56.
Order from least to greatest: 2.1, 1.99, 2.01.
Order from greatest to least: 4.5, 4.05, 4.55, 4.15.
On the vault, Bhumika scores 8.7, Kahu 8.75 and Inti 8.07. Who places first, second and third?
Name a decimal with two places that is greater than 1.4 and less than 1.5.
Bhumika has performed 0.4 of her 90-second floor routine. Kahu has performed 0.5 of his 60-second beam routine. Who has been performing for longer, in seconds?

Reasoning

Explain why, when two decimals have different tenths digits, the hundredths digits cannot change which one is greater.
Explain why writing a zero at the right end of 0.4 does not change its value, but writing a zero right after the point, to make 0.04, does.
To compare 6.1 and 5.99, explain why the ones digits alone settle it.

Answers

(a) 0.8 > 0.6: 8 tenths against 6 tenths. (b) 0.34 < 0.52: 3 tenths against 5 tenths.
(a) 0.57 < 0.59: the tenths tie at 5, and 7 hundredths is less than 9. (b) 0.71 > 0.17: 7 tenths against 1 tenth.
(a) 4.2 > 3.8: the ones decide, 4 against 3. (b) 5.06 < 5.6: the ones tie, then 0 tenths against 6 tenths.
0.4 is 40 squares and 0.39 is 39 squares, so 0.4 > 0.39.
Bhumika’s: 0.8 > 0.75. The ones tie at 0, and the tenths decide, 8 against 7.
(a) False: 4 tenths is less than 8 tenths, so 0.48 < 0.84. (b) True: 7 tenths is less than 9 tenths. (c) True: 0 tenths is less than 5 tenths.
0, 1, 2, 3, 4. The digit is the tenths digit. Any of 0 to 4 tenths is less than 5 tenths. A 5 gives 0.54, which is equal, not less, and 6 to 9 give more.
The jump: 3.65 > 3.56. The ones tie at 3, and 6 tenths beats 5 tenths.
(a) 0.40. (b) 0.90. (c) 3.10. Each extra 0 is zero hundredths.
(a) True. (b) False: 0.08 is 8 hundredths and 0.8 is 80 hundredths. (c) True. (d) False: 0.6 is 60 hundredths and 0.06 is 6 hundredths.
0.6 = 0.60 is 60 hundredths and 0.58 is 58 hundredths. Since 60 > 58, 0.6 > 0.58.
0.49 is 49 hundredths and 0.5 = 0.50 is 50 hundredths, so 0.49 < 0.5.
(a) 1.6 = 1.60: the extra 0 is zero hundredths. (b) 0.1 > 0.09: 10 hundredths against 9 hundredths.
0.45 is 45 hundredths and 0.5 is 50 hundredths, so 0.5 > 0.45.
Inti’s: 2.4 = 2.40, and 2.40 > 2.38. The difference is 40 - 38 = 2 hundredths.
0.3 and 0.30, since \displaystyle \frac{3}{10} = \frac{30}{100}.
0.56 < 0.6 < 0.65. As hundredths the three numbers are 60, 65 and 56.
1.99 < 2.01 < 2.1. The ones put 1.99 first. Then 2.01 has 0 tenths and 2.1 has 1.
4.55 > 4.5 > 4.15 > 4.05. As hundredths after the 4 ones: 55, 50, 15, 5.
First Kahu (8.75), second Bhumika (8.7 = 8.70), third Inti (8.07). The ones tie at 8; tenths 7, 7, 0 put Inti last; hundredths 5 against 0 put Kahu ahead of Bhumika.
Any of 1.41 to 1.49, for example 1.45. It has 1 one and 4 tenths like 1.40, and more hundredths than 0, but fewer than the 5 tenths of 1.50.
Bhumika, 36 seconds against 30. One tenth of 90 seconds is 9 seconds, so 0.4 of it is 4 \times 9 = 36 seconds. One tenth of 60 seconds is 6 seconds, so 0.5 of it is 5 \times 6 = 30 seconds. The decimals compare as 0.5 > 0.4, but they are parts of different wholes.
Say the two decimals differ by at least one tenth in the tenths place. One tenth is 10 hundredths, and a hundredths digit is at most 9, so the hundredths can add at most 9 hundredths, 0.09, which is less than the one tenth between them. So the number with more tenths stays greater whatever its hundredths are.
0.40 is 4 tenths and 0 hundredths: the new zero counts zero of a new, smaller piece, so the amount is the same, 40 squares of a hundred grid either way. In 0.04 the zero takes the tenths place and pushes the 4 into the hundredths place, where it counts hundredths instead of tenths: 4 squares instead of 40, ten times smaller.
6.1 has 6 ones and 5.99 has 5. Everything after the point in 5.99 is 0.99, which is one hundredth short of a whole, so 5.99 is less than 5 + 1 = 6. Any number with 6 ones is at least 6, so 6.1 > 5.99.