Skip to main content
\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
International Baccalaureate · PYP 5

Decimal Numbers

⌚ ~258 min ▢ 30 blocks ✓ 109 exercises ➣ Prerequisites : Whole Numbers to Millions, Fractions: equivalence and comparison
I What Are Decimals?
I.1 Decimals to the Hundredth: Recap
Recappreviously studied — click to reread

In our number system, the position of a digit tells us its value. Consider the number 235.
\(235=\) Figure
It is made of three parts:
  • The 5 is in the Ones place, meaning 5 ones (\(5\)).
  • The 3 is in the Tens place, meaning 3 tens (\(30\)).
  • The 2 is in the Hundreds place, meaning 2 hundreds (\(200\)).
Our number system follows a pattern based on 10.
  • When we move left \(\leftarrow\), each place is 10 times bigger. (10 ones make a ten, 10 tens make a hundred).
  • When we move right \(\rightarrow\), each place is 10 times smaller. (A hundred divided by 10 is a ten, a ten divided by 10 is a one).
What happens if we keep moving right? What is a one divided by 10?
We get a piece called a tenth. A tenth is a fraction, written as \(\dfrac{1}{10}\).
To write numbers that include these smaller pieces, we use a decimal point to separate the whole part from the fractional part.
\(35.4\) = Figure
This number means 3 tens, 5 ones, and 4 tenths.

Definition — Decimal Number
A decimal number is a number that can be written with a whole part and a fractional part separated by a decimal point, using a finite number of digits after the point. We can represent a decimal number in different ways:
  • Standard Form:
    \(35.4\)
  • Expanded Form $$\begin{aligned} 3\text{ tens} &+& 5\text{ ones} &+& 4\text{ tenths} \\ 30 &+& 5 &+& \frac{4}{10} \end{aligned}$$
  • Place Value Table:
    Tens Ones . Tenths
    3 5 . 4
  • Base Ten Cubes:
    Figure
Definition — The Hundredth
The pattern of our number system does not stop at the tenths: each place is still 10 times smaller than the place on its left.
  • A one divided by 10 is a tenth: \(\dfrac{1}{10} = 0.1\).
  • A tenth divided by 10 is a new, smaller piece called a hundredth. A hundredth is written \(\dfrac{1}{100}\) as a fraction, or \(0.01\) as a decimal.
A hundredth is one tenth of one tenth. Going back up:
  • 10 hundredths make 1 tenth: \(\dfrac{10}{100} = \dfrac{1}{10}\).
  • 100 hundredths make 1 one: \(\dfrac{100}{100} = 1\).
The place-value table gains one more column on the right. For the number \(2.64\) (2 ones, 6 tenths, 4 hundredths):
Ones . Tenths Hundredths
2 . 6 4
\(2.64\) = Figure
Skills to practice
  • Decimals to the Hundredth: Recap
I.2 The Thousandth
Definition — Decimal Number
A decimal number is a number that can be written with a whole part and a fractional part separated by a decimal point, with a finite number of digits after the decimal point. The place-value pattern continues to the right of the hundredths: one hundredth divided by 10 gives one thousandth, written \(\dfrac{1}{1000}\). We can represent a decimal number in different ways:
  • Decimal Form:
    \(35.487\)
  • Expanded Form$$\begin{aligned} 3\text{ tens} &+& 5\text{ ones} &+& 4\text{ tenths} &+& 8\text{ hundredths} &+& 7\text{ thousandths} \\ 30 &+& 5 &+& \frac{4}{10} &+& \frac{8}{100} &+& \frac{7}{1000}\\ 3\times 10 &+& 5\times 1 &+& 4\times \frac{1}{10} &+& 8\times \frac{1}{100} &+& 7\times \frac{1}{1000}\\ \end{aligned}$$
  • Place Value Table:
    Tens Ones . Tenths Hundredths Thousandths
    3 5 . 4 8 7
  • Decimal Fraction:
    \(35.487=\dfrac{35\,487}{1000}\)
    The smallest place used is the thousandth, so the number counts \(35\,487\) thousandths.
Definition — The Thousandth
The pattern continues past the hundredths: each place is still 10 times smaller than the place on its left.
  • A hundredth divided by 10 is a new, smaller piece called a thousandth. A thousandth is written \(\dfrac{1}{1\,000}\) as a fraction, or \(0.001\) as a decimal.
A thousandth is one tenth of one hundredth. Going back up:
  • 10 thousandths make 1 hundredth: \(\dfrac{10}{1\,000} = \dfrac{1}{100}\).
  • \(1\,000\) thousandths make 1 one: \(\dfrac{1\,000}{1\,000} = 1\).
The place-value table gains one more column on the right. For the number \(2.647\) (2 ones, 6 tenths, 4 hundredths, 7 thousandths):
Ones . Tenths Hundredths Thousandths
\(2\) . \(6\) \(4\) \(7\)
Example
Consider the number \(53.276\).
  1. Write \(53.276\) in expanded form.
  2. Write \(53.276\) as a single decimal fraction.
  3. Decompose \(53.276\) as a whole number plus a fraction smaller than \(1\).

The place value table of \(53.276\) is:
Tens Ones . Tenths Hundredths Thousandths
5 3 . 2 7 6
  1. Expanded form: $$53.276=5\times 10+3\times 1+2\times \frac{1}{10}+7\times \frac{1}{100}+6\times \frac{1}{1000}$$
  2. As a single decimal fraction: $$\begin{aligned} 53.276&=\frac{53\,276}{1000} &&\text{(the last digit, 6, is in the thousandths place)} \end{aligned}$$
  3. As a whole number plus a fraction smaller than \(1\): $$\begin{aligned} 53.276&=53+0.276 &&\text{(whole part and decimal part)}\\ &=53+\frac{276}{1000} &&\text{(276 thousandths)} \end{aligned}$$

Skills to practice
  • Identifying Thousandths
  • Writing Decimals with Thousandths
  • Converting with Thousandths Both Ways
  • Writing Decimals from Word Names
II Decimals on a Number Line
II.1 The Number Line: Recap
Recappreviously studied — click to reread

Every decimal number has an exact position on the number line.
  • Finding Tenths
    If we take the space between two whole numbers, like 0 and 1, and divide it into 10 equal parts, each part is one tenth (0.1).
    Figure
  • Finding Hundredths
    If we magnify a small section, such as the one between 0 and 0.1, and divide it into 10 equal parts, each new part is one hundredth (0.01).
    Figure

Method — Placing a Decimal on a Number Line
To place a decimal like 2.27 on a number line:
  1. Find the start and end points. The number 2.27 is between \(2.2 = 2.20\) and \(2.3 = 2.30\).
  2. Draw the line and divide it into 10 equal parts. Each part will represent a hundredth.
  3. Count up from the start. The marks will be 2.21, 2.22, 2.23, and so on. Find the 7th mark to place 2.27.
Figure
Skills to practice
  • The Number Line: Recap
II.2 Placing and Locating to the Thousandth
Example
On the number line below, the large graduations are one hundredth apart, and the segment from \(4.36\) to \(4.37\) is divided into \(10\) equal parts. What number does point \(A\) mark?
Figure

Between \(4.36\) and \(4.37\) there is one hundredth, cut into \(10\) equal parts: each small part is one thousandth. Point \(A\) sits \(7\) small parts after \(4.36\), so it marks the number $$4.36 + \dfrac{7}{1\,000} = 4.367.$$

Skills to practice
  • Reading a Decimal on a Fine Graduation
  • Marking a Decimal to the Hundredth or Thousandth on a Number Line
  • Reading Graduations and Counting On
III Comparing and Ordering Decimals
III.1 Comparing and Ordering: Recap
Recappreviously studied — click to reread
Method — Comparing Decimals Step-by-Step
To find out which decimal number is bigger, follow these steps:
  1. Line them up: Place the numbers one above the other, making sure the decimal points are perfectly aligned.
  2. Make them the same length: Add zeros to the end of the shorter number until both have the same number of decimal places. This doesn't change their value.
  3. Compare from left to right: Starting from the left, compare the digits in each place value column. The first number with a larger digit is the larger number. If one whole part has more digits than the other, that number is the larger one.
Example
Compare 6.22 and 6.3.

  1. Line them up:
    \(6.22\)
    \(6.3\)
  2. Make them the same length: We add a zero to 6.3 to make it 6.30.
    \(6.22\)
    \(6.30\)
  3. Compare:
    • Ones place: Both have a 6.
    • Tenths place: The top number has a 2, the bottom has a 3.
    Since 3 is greater than 2, the number 6.30 is greater than 6.22. So, \(6.22 < 6.3\).
Figure\(\lt\)Figure

Method — Ordering Several Decimals
To order a list of decimal numbers, place them step by step:
  1. Write all the numbers with the same number of decimal places by adding zeros — this changes nothing and makes them easy to compare.
  2. Find the smallest (or the largest) and set it first.
  3. Repeat with the numbers that remain, until the list is used up.
The order is ascending when it goes from smallest to largest, descending the other way.
Example
Order the numbers \(3.7\), \(3.09\), \(3.71\) and \(3.2\) in ascending order.

$$\begin{aligned}[t] 3.70,\; 3.09,\; 3.71,\; 3.20 && \text{(same number of decimal places)} \\ 3.09 && \text{(smallest: its tenths digit is } 0\text{)} \\ 3.09 < 3.20 < 3.70 < 3.71 && \text{(then } 2 \text{ tenths, then } 70 < 71 \text{ hundredths)} \end{aligned}$$

Method — Comparing and Ordering Decimal Fractions
To compare two decimal fractions, write them with the same denominator, then compare the numerators: the larger numerator gives the larger fraction. Since one tenth is ten hundredths and one hundredth is ten thousandths, \(\dfrac{7}{10}=\dfrac{70}{100}\) and \(\dfrac{7}{100}=\dfrac{70}{1\,000}\). $$\dfrac{7}{10}=\dfrac{70}{100}\quad\text{and}\quad 70>65,\quad\text{so}\quad \dfrac{7}{10}>\dfrac{65}{100}.$$ One can also write them with a decimal point and compare the decimals: \(0.70>0.65\). To order several decimal fractions, write them all with the largest denominator, then order the numerators.
Skills to practice
  • Comparing and Ordering: Recap
III.2 Comparing and Ordering to the Thousandth
Example
Order the decimal fractions \(\dfrac{5}{10}\), \(\dfrac{48}{100}\) and \(\dfrac{503}{1\,000}\) in ascending order.

$$\begin{aligned}[t] &\dfrac{5}{10}=\dfrac{500}{1\,000},\; \dfrac{48}{100}=\dfrac{480}{1\,000},\; \dfrac{503}{1\,000} && \text{(all in thousandths)} \\ &480<500<503 && \text{(order the numerators)} \\ &\dfrac{48}{100}<\dfrac{5}{10}<\dfrac{503}{1\,000} \end{aligned}$$ Check with the decimal writing: \(0.48<0.5<0.503\).

Skills to practice
  • Comparing and Ordering Decimals and Decimal Fractions to the Thousandth
IV Bounding and Inserting Decimals
IV.1 Bounding and Inserting: Recap
Recappreviously studied — click to reread
Example — Bounding and Inserting Decimal Fractions
  1. Bound \(\dfrac{347}{100}\) between two consecutive whole numbers, then between two consecutive tenths.
  2. Give a decimal fraction between \(\dfrac{3}{10}\) and \(\dfrac{4}{10}\).

  1. \(\dfrac{347}{100}=3.47\): its whole part is \(3\), so \(3<\dfrac{347}{100}<4\). In hundredths, the tenths \(\dfrac{34}{10}\) and \(\dfrac{35}{10}\) are \(\dfrac{340}{100}\) and \(\dfrac{350}{100}\), and \(340<347<350\): $$\dfrac{34}{10}<\dfrac{347}{100}<\dfrac{35}{10}.$$
  2. In hundredths, \(\dfrac{3}{10}=\dfrac{30}{100}\) and \(\dfrac{4}{10}=\dfrac{40}{100}\): between them lie \(\dfrac{31}{100}\), \(\dfrac{32}{100}\), …, \(\dfrac{39}{100}\). For example \(\dfrac{3}{10}<\dfrac{35}{100}<\dfrac{4}{10}\): \(\dfrac{35}{100}\) is exactly halfway.

Skills to practice
  • Bounding and Inserting: Recap
IV.2 Bounding and Inserting to the Thousandth
Method — Framing and Inserting Decimals
  1. To frame a decimal between two consecutive whole numbers: the whole part gives the lower bound, and the next whole number gives the upper bound. For \(6.815\): $$6\leq 6.815<7$$
  2. To frame a decimal between two consecutive tenths: keep the digits up to the tenths place for the lower bound, then add one tenth for the upper bound. For \(6.815\): $$6.8\leq 6.815<6.9$$
  3. To frame a decimal between two consecutive hundredths: keep the digits up to the hundredths place for the lower bound, then add one hundredth for the upper bound. For \(6.815\): $$6.81\leq 6.815<6.82$$ The sign \(\leq\) reads “is less than or equal to”: the number can be equal to its lower bound (\(6\leq 6<7\) and \(6.1\leq 6.1<6.2\)), never to its upper bound.
  4. To insert a decimal between two given numbers: first write both numbers with the same number of decimal places, then with one more; the new graduations appear between them. Between \(5.7=5.70\) and \(5.8=5.80\), we can insert \(5.71\), \(5.72\), …, \(5.79\). Between two different decimal numbers, there is always another decimal number.
Example
  1. Frame \(7.062\) between two consecutive whole numbers.
  2. Frame \(7.062\) between two consecutive tenths.
  3. Frame \(7.062\) between two consecutive hundredths.
  4. Find three decimal numbers between \(9.61\) and \(9.62\).

  1. The whole part of \(7.062\) is \(7\), so: $$7\leq 7.062<8$$
  2. The digits up to the tenths place give \(7.0\); adding one tenth gives \(7.1\), so: $$7.0\leq 7.062<7.1$$
  3. The digits up to the hundredths place give \(7.06\); adding one hundredth gives \(7.07\), so: $$7.06\leq 7.062<7.07$$
  4. We write both numbers with one more decimal place: \(9.61=9.610\) and \(9.62=9.620\). The new graduations \(9.611\), \(9.612\), …, \(9.619\) all lie between them. For example: $$9.61<9.612<9.615<9.618<9.62$$
    Figure

Skills to practice
  • Bounding a Decimal to the Thousandth Between Two Consecutive Whole Numbers
  • Bounding and Inserting Decimals to the Thousandth
V Rounding Decimals
V.1 Rounding Decimals
Definition — The Whole Part of a Decimal Number
The whole part of a decimal number is the whole number written before the decimal point: the whole part of \(8.4\) is \(8\), the whole part of \(0.36\) is \(0\), the whole part of \(120.05\) is \(120\). For a whole number, the whole part is the number itself: the whole part of \(8\) is \(8\).
  • A decimal number that is not a whole number lies between its whole part and the next whole number: \(6 < 6.72 < 7\).
  • The whole part is not the rounded value. \(6.72\) has whole part \(6\), but rounded to the nearest whole number it gives \(7\), because it is closer to \(7\). The whole part never goes up; the rounded value goes up when the tenths digit is \(5\) or more.
Method — The Four-Step Rounding Rule
  1. Underline the target digit: Find and underline the digit in the place you are rounding to.
  2. Look at the next digit: Look at the digit immediately to the right of your underlined digit.
  3. Decide to round up or down:
    • If the next digit is 5 or more, add one to your underlined digit (round up). If the underlined digit is \(9\), it becomes \(10\): write \(0\) and add \(1\) to the digit on its left (so \(0.\underline{9}5\) rounds to \(1.0\)); if that digit is a \(9\) too, it becomes \(0\) in turn and the carry moves on to the left (so \(9.\underline{9}5\) rounds to \(10.0\)).
    • If the next digit is 4 or less, your underlined digit stays the same.
  4. Drop the rest: the digits to the right of your underlined digit are dropped when they are after the decimal point; in the whole part, they are replaced by zeros (\(365 \approx 400\) to the nearest hundred).
Example
Round the number \(6.72\) to the nearest whole number.

Let's follow the steps:
  1. Underline the digit (the ones place): \(\underline{6}.72\)
  2. Look at the next digit: the digit to the right (the tenths) is 7.
  3. Decide: since 7 is “5 or more”, round up: the 6 becomes 7.
  4. Drop the rest: the 7 and the 2 are dropped.
The rounded number is 7. On the number line, \(6.72\) lies between \(6\) and \(7\), closer to \(7\).

Example
Round the number \(12.346\) to the nearest tenth.

Let's follow the steps:
  1. Underline the digit (the tenths place): \(12.\underline{3}46\)
  2. Look at the next digit: the digit to the right is \(4\).
  3. Decide: since \(4\) is “4 or less”, the underlined digit \(3\) stays the same.
  4. Drop the rest: the \(4\) and the \(6\) are dropped.
The rounded number is \(\mathbf{12.3}\). Why the rule gives the nearest tenth: \(12.346\) lies between two consecutive tenths, $$12.3\leq 12.346<12.4.$$ It is \(0.046\) above \(12.3\) and \(0.054\) below \(12.4\): the nearer bound is \(12.3\). Rounding to the tenth is choosing the nearer of the two bounds.
Figure

Skills to practice
  • Giving the Whole Part of a Decimal
  • Rounding to the Nearest Whole Number
  • Rounding to the Nearest Tenth
  • Rounding to the Nearest Hundredth
VI Multiplying by 10, 100 and 1 000
VI.1 Multiplying by 10, 100 and 1 000
When we multiply a number by 10, it gets 10 times larger. This means every digit moves one spot to the left in the place value chart. A simple shortcut is to move the decimal point to the right.
Report an error
Method — Moving the Decimal Point to Multiply
To multiply by 10, 100 or \(1\,000\), move the decimal point to the right by the number of zeros.
  • Multiply by 10 (1 zero) \(\rightarrow\) move 1 place right.
  • Multiply by 100 (2 zeros) \(\rightarrow\) move 2 places right.
  • Multiply by 1000 (3 zeros) \(\rightarrow\) move 3 places right.
If you run out of digits, add zeros as placeholders.
Example
Calculate \(10\times 5.24 \)

Figure

Example
Compute: \(100 \times 7.3\) and \(2.456 \times 100\).

$$\begin{aligned}[t] 100 \times 7.3 &= 730 && \text{(2 zeros: move the point 2 places right, append a zero)} \\ 2.456 \times 100 &= 245.6 && \text{(2 places right)} \end{aligned}$$

Method — Multiplying by 5 and by 50
  1. To multiply by 5: multiply by \(10\), then divide by \(2\), because \(5=10\div 2\). $$\begin{aligned} 2.6\times 5&=2.6\times 10\div 2 &&\text{(since } 5=10\div 2\text{)}\\ &=26\div 2\\ &=13 \end{aligned}$$
  2. To multiply by 50: multiply by \(100\), then divide by \(2\), because \(50=100\div 2\). $$\begin{aligned} 1.4\times 50&=1.4\times 100\div 2 &&\text{(since } 50=100\div 2\text{)}\\ &=140\div 2\\ &=70 \end{aligned}$$
Skills to practice
  • Multiplying by 10
  • Multiplying by 100
  • Multiplying by 1 000
  • Multiplying by 5 and by 50
VII Dividing by 10, 100 and 1 000
VII.1 Dividing by 10, 100 and 1 000
When we divide a number by 10, it gets 10 times smaller. This means every digit moves one spot to the right. The shortcut is to move the decimal point to the left.
Report an error
Method — Moving the Decimal Point to Divide
To divide by 10, 100 or \(1\,000\), move the decimal point to the left by the number of zeros.
  • Divide by 10 (1 zero) \(\rightarrow\) move 1 place left.
  • Divide by 100 (2 zeros) \(\rightarrow\) move 2 places left.
  • Divide by 1000 (3 zeros) \(\rightarrow\) move 3 places left.
If you run out of digits, add zeros as placeholders.
Example
Calculate \(23.2\div 10\)

Figure

Example
Compute: \(56 \div 100\) and \(340.8 \div 100\).

$$\begin{aligned}[t] 56 \div 100 &= 0.56 && \text{(2 zeros: move the point 2 places left)} \\ 340.8 \div 100 &= 3.408 && \text{(2 places left)} \end{aligned}$$

Skills to practice
  • Dividing by 10
  • Dividing by 100
  • Dividing by 1 000
  • Converting Decimal Fractions to Decimals
  • Converting Decimals to Decimal Fractions
VIII Going further
Skills to practice
  • Analysing Comparison Errors
  • Justifying Insertion Strategies
  • Placing Numbers and Choosing a Precision
  • Quiz
Jump to section