Informative line

Decimal And Its Representation

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.

Illustration Questions

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

Illustration Questions

\(341.7029\) 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.

image

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.

Illustration Questions

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.

image

Hence, option (D) is correct.

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

image
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.

Illustration Questions

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.

Illustration Questions

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 \(1.\)

image

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 \(0.5(5×0.1)\).

image

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?

image
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.

Illustration Questions

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 \(1.\)

image

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 \(0.25\)

image

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.

image

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?

image
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

Illustration Questions

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

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