Informative line

# Introduction to Decimals

• Decimals are numbers with one visible point somewhere in the number. This point is called decimal point.
• The decimal point is denoted by a dot (.).
• One way of representing a part of a whole is through decimal numbers.

Example: $$0.33,\;2.05,\;3.71$$

Decimal numbers consist of wholes, a decimal point and parts of whole.

For example: $$2.33$$

To the left of the decimal point are the wholes and to the right of the decimal point are the parts of whole.

• To the left of the decimal point are the wholes and to the right of the decimal point are the parts of whole.

#### Which one of the following is a decimal number?

A $$2$$

B $$0$$

C $$1.5$$

D $$100$$

×

Option (A):

$$2$$ does not have any decimal point, so $$2$$ is a whole number.

Hence, option (A) is incorrect.

Option (B):

$$0$$ (zero) does not have any decimal point, so $$0$$ (zero) is a whole number.

Hence, option (B) is incorrect.

Option (C):

$$1.5$$ has a decimal point, so $$1.5$$ is a decimal number.

Hence, option (C) is correct.

Option (D):

$$100$$ does not have any decimal point, so $$100$$ is a whole number.

Hence, option (D) is incorrect.

### Which one of the following is a decimal number?

A

$$2$$

.

B

$$0$$

C

$$1.5$$

D

$$100$$

Option C is Correct

# Place Value of Decimals

• Place value refers to the location of any digit in a number.
• In a number, the same digit can have different place values.
• Tabular representation of the location of each digit in a number is called the place value chart.
• Take any number and represent it in place value chart.

For example: $$59022.4056$$

In the place value chart, location of wholes:

(Starting from left of the decimal point)

$$2$$ at ones place

$$2$$ at tens place

$$0$$ at hundreds place

$$9$$ at thousands place

$$5$$ at ten thousands place

Location of parts:

(Starting from right of the decimal point)

$$4$$ at tenths place

$$0$$ at hundredths place

$$5$$ at thousandths place

$$6$$ at ten thousandths place

#### $$341.7029$$ From the given decimal number, find the place value of $$7.$$

A Ones

B Tens

C Hundredths

D Tenths

×

To find the place value of $$7$$ in $$341.7029$$, prepare a place value chart.

Through chart, we can observe that the location of digit $$7$$ in $$341.7029$$ is at tenths place.

Hence, option (D) is correct.

### $$341.7029$$ From the given decimal number, find the place value of $$7.$$

A

Ones

.

B

Tens

C

Hundredths

D

Tenths

Option D is Correct

# Representation of Decimals Using Square Grid

• Grids help us to understand that decimals are parts of the wholes.
• Consider the following situations-

(1) In a grid of $$10$$ squares, if all the squares are shaded, then the grid represents the whole number $$1$$.

(2) In the same grid of $$10$$ squares, if only $$1$$ square is shaded, then the shaded square represents one tenth of the grid.

(3) One fourth shaded part of the whole grid represents twenty five hundredths.

(4) Now, in a grid of $$100$$ squares, if all the squares are shaded, then the grid represents the whole number $$1$$.

(5) In the grid of $$100$$ squares, if a column of $$10$$ squares is shaded, then the shaded column of squares represents one tenth of the grid.

(6) In the same grid of $$100$$ squares, if only $$1$$ square is shaded, then the shaded square represents one hundredth of the grid.

For example: If we want to represent $$0.26$$ using area model, then we will shade $$26$$ squares out of the grid of $$100$$ squares.

#### The model shown represents a decimal number. Which decimal number does this model represent?

A $$67$$

B $$33$$

C $$0.33$$

D $$0.67$$

×

In the given model, $$67$$ squares are shaded out of $$100$$.

This means that the shaded region represents $$0.67$$ of the grid.

Hence, option (D) is correct.

### The model shown represents a decimal number. Which decimal number does this model represent?

A

$$67$$

.

B

$$33$$

C

$$0.33$$

D

$$0.67$$

Option D is Correct

# Decimals in Expanded Form

• Expanded form of a number is such where the number is stretched out.
• In an expanded form, we write the place values of each digit.

## Whole numbers in expanded form

• Suppose, we want to represent the whole number $$240$$, in the expanded form.

For this, we first represent the number in the place value chart.

Hundreds Tens Ones    .      Tenths Hundredths Thousandths
2 4 0

In the place value chart, we put $$2$$ at hundreds place, $$4$$ at tens place, and $$0$$ at ones place.

Now, we write it in words-

$$2$$ Hundreds $$+\;4$$ Tens $$+\;0$$ Ones

$$=2$$ times Hundreds $$+\;4$$ times Tens $$+\;0$$ times Ones

Now, the number of digits $$=200+40+0$$

This is the required form.

### Decimal numbers in expanded form

Example: $$0.24$$

We represent this number in the place value chart.

Hundreds Tens Ones          Tenths Hundredths Thousandths
. 2 4

In the place value chart, we put $$2$$ at tenths place and $$4$$ at hundredths place.

Now, we write it in words-

$$2$$ Tenths $$+\,4$$ Hundredths

Now, in numbers

$$0.2+0.04$$

This is the required form.

Note: Zeros are placed to make sure that the digit has the correct value because if we write $$0.4$$, it will show tenths place and not of hundredths.

#### Represent the given number in expanded form: $$0.548$$

A $$0.5+0.4+0.8$$

B $$0.5+0.04+0.008$$

C $$5+4+8$$

D $$500+40+8$$

×

Firstly represent $$0.548$$ in the place value chart.

Hundreds Tens Ones          Tenths Hundredths Thousandths
. 5 4 8

Expand the number in words-

$$5$$ Tenths $$+\,4$$ Hundredths $$+\,8$$ Thousandths

$$=5$$ times tenths $$(0.1)$$ $$+\,4$$ times hundredths $$(0.01)$$ $$+\,8$$ times thousandths $$(0.001)$$

Write the words in numbers-

$$0.5+0.04+0.008$$

This is the required solution.

Hence, option (B) is correct.

### Represent the given number in expanded form: $$0.548$$

A

$$0.5+0.4+0.8$$

.

B

$$0.5+0.04+0.008$$

C

$$5+4+8$$

D

$$500+40+8$$

Option B is Correct

# Number line Representation of Decimals up to Tenths Place

• Decimals can be represented on a number line.
• To represent them, we move in the right direction from zero on the number line.
• First, find the wholes and then the parts of the whole on the number line.

For example: We want to represent $$3.7$$ on the number line.

• To find the decimal number $$3.7$$ on the number line, find the whole $$3.$$
• To find $$3$$ on the number line, move in the right direction from zero and mark the whole at $$3.$$

• Now, we have to represent the part $$0.7$$ on the number line.
• To represent $$0.7$$, divide the interval between $$3$$ and $$4$$ into $$10$$ equal sections, each having a scale of $$0.1$$

• Now, start from $$3$$ and count  $$7$$ slashes by moving in the right direction. Mark the 7th slash.

We have reached the whole $$3$$ and the parts $$0.7(7×0.1)$$ on the number line.

So, by combining these, we get the decimal number $$3.7$$ on the number line.

#### Which number represents the location of point $$E$$ on the number line?

A $$1.4$$

B $$1.5$$

C $$1.6$$

D $$0.8$$

×

In the given number line, point $$E$$ lies between $$1$$ and $$2,$$ so the whole is $$1.$$

Start from zero and move in the forward direction till we reach at $$1.$$

Now, we need to find the parts.

Here, the interval between $$1$$ and $$2$$ is divided in $$10$$ equal sections, each having a scale of $$0.1$$

To determine the location of point $$E,$$ start from $$1$$ and move in right direction, one slash at a time.

Now, keep on making such moves till we reach at $$E.$$

We made total $$5$$ moves from $$1$$ to reach part $$0.5(5×0.1)$$.

So, by combining them, we get the decimal number $$(1.5)$$ at point  $$E$$ on the number line.

Hence, option (B) is correct.

### Which number represents the location of point $$E$$ on the number line?

A

$$1.4$$

.

B

$$1.5$$

C

$$1.6$$

D

$$0.8$$

Option B is Correct

# Representation of Decimals up to Hundredths Place on Number Line

• To represent decimal numbers on a number line, we first find the wholes and then the parts of the whole by moving in the right direction from zero.
• For example: We want to represent $$1.45$$ on the number line.
• Here, we first find the whole $$1.$$ To find $$1$$ on the number line, start from $$0$$ and move in the forward direction and mark the whole $$1.$$

• So, we have reached the whole $$1.$$
• Now, we need to find the tenth part, i.e. $$0.4$$
• To find $$0.4$$ on the number line, divide the interval between $$1$$ and $$2$$ into $$10$$ equal sections having a scale of  $$0.10$$ each.

• Start from $$1,$$ and count $$4$$ slashes by moving in the right direction and mark the 4th slash.

• We have reached up to the tenth part, i.e. $$1.4$$
• Now, we need to find the hundredth part, i.e. $$0.45$$
• To find $$0.45$$ on the number line, divide the interval between $$0.40$$ and $$0.50$$ into $$2$$ equal sections of scale $$0.5$$ each.

• Start from $$0.40$$, and count $$1$$ slash by moving forward and mark the 1st slash.

• We have reached up to $$1.45$$ on the number line.
• So, we have successfully learned to represent a decimal $$(1.45)$$ up to the hundredths place on a number line.

#### Points A, B, C and D are shown on the given number line. Which point is located at $$1.75$$ on the number line?

A Point A

B Point C

C Point D

D Point B

×

To find the decimal number $$1.75$$ on the given number line, start from zero and move in the forward direction and mark the whole $$1.$$

Now, we need to find the part $$0.75$$ on the given number line which will lie in the interval between $$1$$ and $$2.$$ There are four equal intervals between $$1$$ and $$2.$$ This means that each interval has a scale of $$0.25$$

Now, start moving from $$1$$ in the forward direction and mark at $$1.75$$.

We can observe that point D represents the decimal number $$1.75$$ on the number line.

Hence, option (C) is correct.

### Points A, B, C and D are shown on the given number line. Which point is located at $$1.75$$ on the number line?

A

Point A

.

B

Point C

C

Point D

D

Point B

Option C is Correct

# Decimal in Words

• We can read or write decimals in words with the help of a place value chart.
• Ignore the decimal point and read the number as a whole number.
• Then add the place value of the last digit.
• For example: Write $$0.4512$$ in words.
• First, ignore the decimal point and write the number as a whole number, i.e.

Four thousand five hundred twelve

• Write the decimal number in the place value chart.
Decimal point Tenths Hundredths Thousandths Ten-thousandths
. 4 5 1 2
• Now, add the place value of the last digit.

Four thousand five hundred twelve ten-thousandths

• Example: (1) $$0.5=$$ Five tenths

(2) $$0.045=$$ Forty-five thousandths

(3) $$0.05=$$ Five hundredths

#### Write $$0.008$$ in words.

A Eight tenths

B Eight hundredths

C Eight thousandths

D Eight

×

First, ignore the decimal point and write the number as a whole number, i.e.

Eight

Write $$0.008$$ in the place value chart.

Wholes Decimal point Tenths Hundredths Thousandths
. 0 0 8

Now, add the place value of the last digit.

So, our decimal number is Eight thousandths.

Hence, option (C) is correct.

### Write $$0.008$$ in words.

A

Eight tenths

.

B

Eight hundredths

C

Eight thousandths

D

Eight

Option C is Correct