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Addition Of Decimals

Addition of Decimals and Whole Numbers

  • To add decimals and whole numbers, we are going to work with the wholes and the parts of the numbers separately.
  • Suppose, we want to add \(12.2\) and \(15\).
  • Since both numbers have different place values, so first make the place values same by adding a zero. Thus, \(15\) is written as \(15.0\)
  • Write one number just below the other such that the bottom decimal point is directly below and lined up with the top decimal point.

\(\begin{array} {c} \hline &\text{Tens}&\text{Ones}&\text{Decimal point}&\text{Tenths}\\ \hline &1&2&\cdot&2\\ +&1&5&\cdot&0\\ \hline &&&\cdot&\\ \hline \end{array}\)

  • Now we add the columns vertically starting from the right most column and moving towards left.
  • Add the digits of tenths column,

\(2+0=2\)

Write \(2\) in the tenths column.

\(\begin{array} {c} &1&2&\cdot&2\\ +&1&5&\cdot&0\\ \hline &&&\cdot&2\\ \hline \end{array}\)

  • Now, add the digits of ones column.

\(2+5=7\)

Write \(7\) in the ones column.

\(\begin{array} {c} &1&2&\cdot&2\\ +&1&5&\cdot&\\ \hline &&7&\cdot&2\\ \hline \end{array}\)

  • At last, add the digits of tens column.

\(1+1=2\)

Write \(2\) in the tens column.

\(\begin{array} {c} \hline &\text{Tens}&\text{Ones}&\text{Decimal point}&\text{Tenths}\\ \hline &1&2&\cdot&2\\ +&1&5&\cdot&\\ \hline &2&7&\cdot&2\\ \hline \end{array}\)

\(\therefore\) The sum of \(12.2\) and \(15\) is \(27.2\)

Illustration Questions

What is the sum of \(5\) and \(2.75\)?

A \(2.80\)

B \(52.75\)

C \(5.77\)

D \(7.75\)

×

Both the numbers should have the same place values.

So, \(5\) can be written as \(5.00\)

Write \(2.75\) just below \(5.00\) so that the bottom decimal point is directly below and lined up with the top decimal point.

\(\begin{array} {c} \hline &\text{Ones}&\text{Decimal point}&\text{Tenths}&\text{Hundredths}\\ \hline &5&\cdot&0&0\\ +&2&\cdot&7&5\\ \hline &&\cdot&&\\ \hline \end{array}\)

Add the digits in the hundredths column.

\(0+5=5\)

\(\begin{array} {c} &5&\cdot&0&0\\ +&2&\cdot&7&5\\ \hline &&\cdot&&5\\ \hline \end{array}\)

Add the digits in the tenths column.

\(0+7=7\)

\(\begin{array} {c} &5&\cdot&0&0\\ +&2&\cdot&7&5\\ \hline &&\cdot&7&5\\ \hline \end{array}\)

Now, add the digits of ones column.

\(5+2=7\)

Write \(7\) in the ones column.

\(\begin{array} {c} \hline &\text{Ones}&\text{Decimal point}&\text{Tenths}&\text{Hundredths}\\ \hline &5&\cdot&0&0\\ +&2&\cdot&7&5\\ \hline &7&\cdot&7&5\\ \hline \end{array}\)

\(\therefore\;7.75\) is the sum of \(5\) and \(2.75\)

Hence, option (D) is correct.

What is the sum of \(5\) and \(2.75\)?

A

\(2.80\)

.

B

\(52.75\)

C

\(5.77\)

D

\(7.75\)

Option D is Correct

Addition of Decimals up to Tenths Place

  • To add decimals and whole numbers, we are going to work with the wholes and the parts of the numbers separately.
  • Suppose, we want to add \(2.4\) and \(1.2\)

i.e., \(2.4+1.2\)

  • Write one number just below the other, so that the bottom decimal point is directly below and lined up with the top decimal point.

\(\begin{array} {c} \hline &\text{Ones}&\text{Decimal point}&\text{Tenths}\\ \hline &2&\cdot&4\\ +&1&\cdot&2\\ \hline &&\cdot&\\ \hline \end{array}\)

  • First, add the digits of tenths column.

\(4+2=6\)

Write \(6\) in the tenths column.

\(\begin{array} {c} &2&\cdot&4\\ +&1&\cdot&2\\ \hline &&\cdot&6\\ \hline \end{array}\)

  • Now, add the digits of ones column.

\(2+1=3\)

Write \(3\) in the ones column.

\(\begin{array} {c} \hline &\text{Ones}&\text{Decimal point}&\text{Tenths}\\ \hline &2&\cdot&4\\ +&1&\cdot&2\\ \hline &3&\cdot&6\\ \hline \end{array}\)

\(\therefore\;3.6\) is the sum of \(2.4\) and \(1.2\)

Illustration Questions

What is the sum of \(6.8\) and \(2.1\)?

A \(8.9\)

B \(62.81\)

C \(8\)

D \(26.18\)

×

Write \(2.1\) just below \(6.8\), such that the bottom decimal point is directly below and lined up with the top decimal point.

\(\begin{array} {c} \hline &\text{Ones}&\text{Decimal point}&\text{Tenths}\\ \hline &6&\cdot&8\\ +&2&\cdot&1\\ \hline &&\cdot&\\ \hline \end{array}\)

First, add the digits of tenths column.

\(8+1=9\)

Write \(9\) in the tenths column.

\(\begin{array} {c} &6&\cdot&8\\ +&2&\cdot&1\\ \hline &&\cdot&9\\ \hline \end{array}\)

Now, add the digits of ones columns.

\(6+2=8\)

Write \(8\) in the ones column.

\(\begin{array} {c} \hline &\text{Ones}&\text{Decimal point}&\text{Tenths}\\ \hline &6&\cdot&8\\ +&2&\cdot&1\\ \hline &8&\cdot&9\\ \hline \end{array}\)

\(\therefore\,8.9\) is the sum of \(6.8\) and \(2.1\)

Hence, option (A) is correct.

What is the sum of \(6.8\) and \(2.1\)?

A

\(8.9\)

.

B

\(62.81\)

C

\(8\)

D

\(26.18\)

Option A is Correct

Addition of Decimals up to Hundredths Place

  • To add decimals and whole numbers, we are going to work with the wholes and the parts of the numbers separately.
  • Suppose, we want to add \(4.96\) and \(1.38\)

i.e., \(4.96+1.38\)

  • Write one number just below the other so that the bottom decimal point is directly below and lined up with the top decimal point.

\(\begin{array} {c} \hline &\text{Ones}&\text{Decimal point}&\text{Tenths}&\text{Hundredths}\\ \hline &4&\cdot&9&6\\ +&1&\cdot&3&8\\ \hline &&\cdot&\\ \hline \end{array}\)

  • First, add the digits of hundredths column.

\(6+8=14\)

Now, \(4\) is to be written in the hundredths column and \(10\) is carried to the next column (tenths).

\(\begin{array} {c} &4&\cdot&^\underline{\color{red}1}9&6\\ +&1&\cdot&\;\;\;3&8\\ \hline &&\cdot&&4\\ \hline \end{array}\)

  • Add the digits of tenths column.

\(1+9+3=13\)

\(3\) is to be written in the tenths column and \(10\) is carried to the next column (ones).

\(\begin{array} {c} &^\underline{\color{red}1}4&\cdot&9&6\\ +&\;\;\;1&\cdot&3&8\\ \hline &&\cdot&3&4\\ \hline \end{array}\)

  • Add the digits of ones column.

\(1+4+1=6\)

\(6\) is to be written in the ones column.

\(\begin{array} {c} \hline &\text{Ones}&\text{Decimal point}&\text{Tenths}&\text{Hundredths}\\ \hline &4&\cdot&9&6\\ +&1&\cdot&3&8\\ \hline &6&\cdot&3&4\\ \hline \end{array}\)

\(\therefore\;6.34\) is the sum of \(4.96\) and \(1.38\)

Illustration Questions

Add \(7.43\) and \(4.58\)

A \(11.101\)

B \(12.01\)

C \(74.4358\)

D \(47.5843\)

×

Write \(4.58\) just below \(7.43\), so that the bottom decimal point is directly below and lined up with the top decimal point.

\(\begin{array} {c} \hline &\text{Ones}&\text{Decimal point}&\text{Tenths}&\text{Hundredths}\\ \hline &7&\cdot&4&3\\ +&4&\cdot&5&8\\ \hline &&\cdot&\\ \hline \end{array}\)

First, add the digits of hundredths column.

\(3+8=11\)

\(1\) is to be written in the hundredths column and \(10\) is carried to the next column (tenths).

\(\begin{array} {c} &7&\cdot&^\underline{\color{red}1}4&3\\ +&4&\cdot&\;\;\;5&8\\ \hline &&\cdot&&1\\ \hline \end{array}\)

Add the digits of tenths column.

\(1+4+5=10\)

\(0\) is to be written in the tenths column and \(10\) is carried to the next column (ones).

\(\begin{array} {c} &^\underline{\color{red}1}7&\cdot&4&3\\ +&\;\;4&\cdot&5&8\\ \hline &&\cdot&0&1\\ \hline \end{array}\)

Now, add the digits of ones column.

\(1+7+4=12\)

\(2\) is to be written in the ones column and \(10\) is carried to the next column (tens).

So, \(1\) is written in the tens column.

\(\begin{array} { r c} \hline \text{Tens}&\text{Ones}&\text{Decimal point}&\text{Tenths}&\text{Hundredths}\\ \hline \text{Carried}\to\underline{\color{red}1}&7&\cdot&4&3\\ +&4&\cdot&5&8\\ \hline 1&2&\cdot&0&1\\ \hline \end{array}\)

\(\therefore\;12.01\) is the sum of \(7.43\) and \(4.58\)

Hence, option (B) is correct.

Add \(7.43\) and \(4.58\)

A

\(11.101\)

.

B

\(12.01\)

C

\(74.4358\)

D

\(47.5843\)

Option B is Correct

Addition of Decimals up to Thousandths Place

  • To add decimals and whole numbers, we are going to work with the wholes and the parts of the numbers separately.
  • Suppose, we want to add \(31.343\) and \(42.456\)
  • Write one number just below the other so that the bottom decimal point is directly below and lined up with the top decimal point.

\(\begin{array} {c} \hline &\text{Tens}&\text{Ones}&\text{Decimal point}&\text{Tenths}&\text{Hundredths}&\text{Thousandths}\\ \hline &3&1&\cdot&3&4&3\\ +&4&2&\cdot&4&5&6\\ \hline &&&\cdot&&&\\ \hline \end{array}\)

  • First, add the digits of thousandths column.

\(3+6=9\)

\('9'\) is to be written in the thousandths column.

\(\begin{array} {c} &3&1&\cdot&3&4&3\\ +&4&2&\cdot&4&5&6\\ \hline &&&\cdot&&&9\\ \hline \end{array}\)

  • Now, add the digits of hundredths column.

\(4+5=9\)

\('9'\) is to be written in the hundredths column.

\(\begin{array} {c} &3&1&\cdot&3&4&3\\ +&4&2&\cdot&4&5&6\\ \hline &&&\cdot&&9&9\\ \hline \end{array}\)

  • Add the digits of tenths column.

\(3+4=7\)

\('7'\) is to be written in the tenths column.

\(\begin{array} {c} &3&1&\cdot&3&4&3\\ +&4&2&\cdot&4&5&6\\ \hline &&&\cdot&7&9&9\\ \hline \end{array}\)

  • Now add the digits of ones column.

\(1+2=3\)

\(3\) is to be written in the ones column.

\(\begin{array} {c} &3&1&\cdot&3&4&3\\ +&4&2&\cdot&4&5&6\\ \hline &&3&\cdot&7&9&9\\ \hline \end{array}\)

  • Add the digits of tens column.

\(3+4=7\)

\(7\) is to be written in the tens column.

\(\begin{array} {c} \hline &\text{Tens}&\text{Ones}&\text{Decimal point}&\text{Tenths}&\text{Hundredths}&\text{Thousandths}\\ \hline &3&1&\cdot&3&4&3\\ +&4&2&\cdot&4&5&6\\ \hline &7&3&\cdot&7&9&9\\ \hline \end{array}\)

\(\therefore\;73.799\) is the sum of \(31.343\) and \(42.456\)

Illustration Questions

Add \(7.175+5.325=?\)

A \(12.5\)

B \(2.500\)

C \(75.500\)

D \(12.55\)

×

Write \(5.325\) just below \(7.175\), so that the bottom decimal point is directly below and lined up with the top decimal point.

\(\begin{array} {c} \hline &\text{Ones}&\text{Decimal point}&\text{Tenths}&\text{Hundredths}&\text{Thousandths}\\ \hline &7&\cdot&1&7&5\\ +&5&\cdot&3&2&5\\ \hline &&\cdot&&&\\ \hline \end{array}\)

First, add the digits of thousandths column.

\(5+5=10\)

\('0'\) is to be written in the thousandths column and \(10\) is carried to the next column (hundredths).

\(\begin{array} {c} &7&\cdot&1&^\underline{\color{red}1}7&5\\ +&5&\cdot&3&\;\;\;2&5\\ \hline &&\cdot&&&0\\ \hline \end{array}\)

Now, add the digits of hundredths column.

\(1+7+2=10\)

\(0\) is to be written in the hundredths column and \(10\) is carried to the next column (tenths).

\(\begin{array} {c} &7&\cdot&^\underline{\color{red}1}1&7&5\\ +&5&\cdot&\;\;\;3&2&5\\ \hline &&\cdot&&0&0\\ \hline \end{array}\)

Add the digits of tenths column.

\(1+1+3=5\)

\(5\) is to be written in the tenths column.

\(\begin{array} {c} &7&\cdot&1&7&5\\ +&5&\cdot&3&2&5\\ \hline &&\cdot&5&0&0\\ \hline \end{array}\)

Now, add the digits of ones column.

\(7+5=12\)

\(2\) is to be written in the ones column and \(10\) is carried to the next column (tens).

\(1\) is to be written in the tens column.

\(\begin{array} { r c} \hline \text{Tens}&\text{Ones}&\text{Decimal point}&\text{Tenths}&\text{Hundredths}&\text{Thousandths}\\ \hline \text{Carried}\to\underline{\color{red}1}&7&\cdot&1&7&5\\ +&5&\cdot&3&2&5\\ \hline 1&2&\cdot&5&0&0\\ \hline \end{array}\)

\(12.500\) can be written as \(12.5\)

\(\therefore\;12.5\) is the sum of \(7.175\) and \(5.325\)

Hence, option (A) is correct.

Add \(7.175+5.325=?\)

A

\(12.5\)

.

B

\(2.500\)

C

\(75.500\)

D

\(12.55\)

Option A is Correct

Addition of Mixed Decimals (Place Values)

  • To add decimals and whole numbers, we are going to work with the wholes and the parts of the numbers separately.
  • Suppose, we want to add \(8.6,\;4.310\) and \(10.6055\)

i.e. \(8.6+4.310+10.6055\)

  • Write each number just below the other, so that the bottom decimal point is directly below and lined up with the middle and the top decimal points.
  • Add zeros, so that all the numbers have the same place values.
  • The given decimal numbers can be written as:

\(\begin{array} {c} \hline &\text{Tens}&\text{Ones}&\text{Decimal point}&\text{Tenths}&\text{Hundredths}&\text{Thousandths}&\text{Ten thousandths}\\ \hline &&8&\cdot&6&0&0&0\\ &&4&\cdot&3&1&0&0\\ +&1&0&\cdot&6&0&5&5\\ \hline &&&\cdot&&&&\\ \hline \end{array}\)

  • First, we add the digits of ten-thousandths.

\(0+0+5=5\)

\(5\) is to be written in the ten-thousandths column.

\(\begin{array} {c} &&8&\cdot&6&0&0&0\\ &&4&\cdot&3&1&0&0\\ +&1&0&\cdot&6&0&5&5\\ \hline &&&\cdot&&&&5\\ \hline \end{array}\)

  • Now, add the digits of thousandths column.

\(0+0+5=5\)

\(5\) is to be written in the thousandths column.

\(\begin{array} {c} &&8&\cdot&6&0&0&0\\ &&4&\cdot&3&1&0&0\\ +&1&0&\cdot&6&0&5&5\\ \hline &&&\cdot&&&5&5\\ \hline \end{array}\)

  • Add the digits of hundredths column.

\(0+1+0=1\)

\(1\) is to be written in the hundredths column.

\(\begin{array} {c} &&8&\cdot&6&0&0&0\\ &&4&\cdot&3&1&0&0\\ +&1&0&\cdot&6&0&5&5\\ \hline &&&\cdot&&1&5&5\\ \hline \end{array}\)

  • Now, add the digits of tenths column.

\(6+3+6=15\)

\(5\) is to be written in the tenths column and \(10\) is carried to the next column (ones).

\(\begin{array} {lrc} \text{Carried}\to&&^\underline{\color{red}1}8&\cdot&6&0&0&0\\ &&4&\cdot&3&1&0&0\\ +&1&0&\cdot&6&0&5&5\\ \hline &&&\cdot&5&1&5&5\\ \hline \end{array}\)

  • Add the digits of ones column.

\(1+8+4+0=13\)

\(3\) is to be written in the ones column and \(10\) is carried to the next column (tens).

\(\begin{array} {lrc} &&8&\cdot&6&0&0&0\\ &\text{Carried}\to\underline{\color{red}1}&4&\cdot&3&1&0&0\\ +&1&0&\cdot&6&0&5&5\\ \hline &&3&\cdot&5&1&5&5\\ \hline \end{array}\)

  • Now, add the digits of tens column.

\(1+1=2\)

\(2\) is to be written in the tens column.

\(\begin{array} {c} \hline &\text{Tens}&\text{Ones}&\text{Decimal point}&\text{Tenths}&\text{Hundredths}&\text{Thousandths}&\text{Ten thousandths}\\ \hline &&8&\cdot&6&0&0&0\\ &&4&\cdot&3&1&0&0\\ +&1&0&\cdot&6&0&5&5\\ \hline &2&3&\cdot&5&1&5&5\\ \hline \end{array}\)

\(\therefore\;23.5155\) is the sum of \(8.6,\;4.310\) and \(10.6055\).

Illustration Questions

Add \(16.375+5.25+11.5\)

A \(16511.375255\)

B \(32.405\)

C \(33.125\)

D \(33.405\)

×

Write \(5.25\) just below \(16.375\) and \(11.5\) just below \(5.25\), so that the decimal points are lined up exactly one above the other. 

Add zeros, so that all the numbers have the same place values.

The given decimal numbers can be written as:

\(\begin{array} {c} \hline &\text{Tens}&\text{Ones}&\text{Decimal point}&\text{Tenths}&\text{Hundredths}&\text{Thousandths}\\ \hline &1&6&\cdot&3&7&5\\ &0&5&\cdot&2&5&0\\ +&1&1&\cdot&5&0&0\\ \hline &&&\cdot&&&\\ \hline \end{array}\)

First, we add the digits of thousandths column.

\(5+0+0=5\)

\(5\) is to be written in the thousandths column.

\(\begin{array} {c} &1&6&\cdot&3&7&5\\ &0&5&\cdot&2&5&0\\ +&1&1&\cdot&5&0&0\\ \hline &&&\cdot&&&5\\ \hline \end{array}\)

Now, add the digits of hundredths column.

\(7+5+0=12\)

\(2\) is to be written in the hundredths column and \(10\) is carried to the next column (tenths).

\(\begin{array} {c} &1&6&\cdot&^\color{red}13&7&5\\ &0&5&\cdot&\;\,2&5&0\\ +&1&1&\cdot&\;\,5&0&0\\ \hline &&&\cdot&&2&5\\ \hline \end{array}\)

Add the digits of tenths column.

\(1+3+2+5=11\)

\(1\) is to be written in the tenths column and \(10\) is carried to the next column (ones).

\(\begin{array} {c} &1&^\color{red}16&\cdot&3&7&5\\ &0&\;\,5&\cdot&2&5&0\\ +&1&\;\,1&\cdot&5&0&0\\ \hline &&&\cdot&1&2&5\\ \hline \end{array}\)

Now, add the digits of ones column.

\(1+6+5+1=13\)

\(3\) is to be written in the ones column and \(10\) is carried to the next column (tens).

\(\begin{array} {c} &^\color{red}11&6&\cdot&3&7&5\\ &\;\,0&5&\cdot&2&5&0\\ +&\;\,1&1&\cdot&5&0&0\\ \hline &&3&\cdot&1&2&5\\ \hline \end{array}\)

Add the digits of tens column.

\(1+1+0+1=3\)

\(3\) is to be written in the tens column.

\(\begin{array} {c} \hline &\text{Tens}&\text{Ones}&\text{Decimal point}&\text{Tenths}&\text{Hundredths}&\text{Thousandths}\\ \hline &1&6&\cdot&3&7&5\\ &0&5&\cdot&2&5&0\\ +&1&1&\cdot&5&0&0\\ \hline &3&3&\cdot&1&2&5\\ \hline \end{array}\)

\(\therefore\;33.125\) is the sum of \(16.375,\;5.25\) and \(11.5\)

Hence, option (C) is correct.

Add \(16.375+5.25+11.5\)

A

\(16511.375255\)

.

B

\(32.405\)

C

\(33.125\)

D

\(33.405\)

Option C is Correct

Adding Decimals Using Hundredths Grid

  • We can add decimals with the help of grid.
  • For example: We want to add \(0.25\) and \(0.35\)

  • In the first grid, \(25\) squares are shaded, so first grid represents \(0.25\)
  • In the second grid, \(35\) squares are shaded, so second grid represents \(0.35\)
  • By combining them we get-

Here, there are \(60\) shaded squares in total. So, now the grid represents \(0.60\)

Thus, \(0.25+0.35=0.60\)

Illustration Questions

What is the sum of shaded squares in the grids below?

A \(0.30\)

B \(0.20\)

C \(0.50\)

D \(0.10\)

×

In the given grids, green shaded squares represent \(0.30\) of grid and red shaded squares show \(0.20\)

To calculate the total, we will combine the shaded squares of both the grids in a single grid.

image

The total shaded square parts represent \(0.50\)

Thus, \(0.30+0.20=0.50\)

Hence, option (C) is correct.

What is the sum of shaded squares in the grids below?

image
A

\(0.30\)

.

B

\(0.20\)

C

\(0.50\)

D

\(0.10\)

Option C is Correct

Practice Now