Expanded form of Decimal Fractions

We will discuss here about the expanded form of decimal fractions.

In expanded form of decimal fractions we will learn how to read and write the decimal numbers.

Decimal numbers can be expressed in expanded form using the place-value chart. Let us consider the number 561.129. Let us expand each of the digits using the place-value chart.

Expanded form of Decimal

So, we can write 561.129 in the expanded form as follows.

561.129 = 500 + 60 + 1 + 0.1 + 0.02 + 0.009

             = 5 hundreds + 6 tens + 1 ones + 1 tenths + 2 hundredths + 9 thousandths

             = 500 + 60 + 1 + \(\frac{1}{10}\) + \(\frac{2}{100}\) + \(\frac{9}{1000}\)


Again,

493.2 = 4 hundreds + 9 tens + 3 ones + 2 tenths

         = 400 + 90 + 3 + \(\frac{2}{10}\)


1436.74 = 1 thousands + 4 hundreds + 3 tens + 6 ones + 7 tenths + 4 hundredths

             = 1000 + 400 + 30 + 6 + \(\frac{7}{10}\) + \(\frac{4}{100}\)


Note: When a decimal is missing either in the integral part or decimal part, substitute with 0. 


1. Write the decimal numbers in expanded form:

(i) 3479.105

= 3 thousands + 4 hundreds + 7 tens + 9 ones + 1 tenths + 0 hundredths+ 5 thousandths

= 3000 + 400 + 70 + 9 + \(\frac{1}{10}\) + \(\frac{0}{100}\) + \(\frac{5}{1000}\)


(ii) 7833.45

= 7 thousands + 8 hundreds + 3 tens + 3 ones + 4 tenths + 5 hundredths

= 7000 + 800 + 30 + 3 + \(\frac{4}{10}\) + \(\frac{5}{100}\)


(iii) 21.1097

= 2 tens + 1 ones + 1 tenths + 0 hundredths + 9 thousandths + 7 ten thousandths

= 20 + 1 + \(\frac{1}{10}\) + \(\frac{0}{100}\) + \(\frac{9}{1000}\) + \(\frac{7}{10000}\)


(iv) 524.1

= 5 hundreds + 2 tens + 4 ones + 1 tenths

= 500 + 20 + 4 + \(\frac{1}{10}\)


(v) 143.011

= 1 hundreds + 4 tens + 3 ones + 0 tenths + 1 hundredths + 1 thousandths

= 100 + 40 + 3 + \(\frac{0}{10}\) + \(\frac{1}{100}\) + \(\frac{1}{1000}\)


(vi) 840.006

= 8 hundreds + 4 tens + 0 ones + 0 tenths + 0 hundredths + 6 thousandths

= 800 + 40 + 0 + \(\frac{0}{10}\) + \(\frac{0}{100}\) + \(\frac{6}{1000}\)


(vii) 64.21

= 6 tens + 4 ones + 2 tenths + 1 hundredths

= 60 + 4 + \(\frac{2}{10}\) + \(\frac{1}{100}\)


(viii) 4334.334

= 4 thousands + 3 hundreds + 3 tens + 4 ones + 3 tenths + 3 hundredths + 4 thousandths

= 4000 + 300 + 30 + 4 + \(\frac{3}{10}\) + \(\frac{3}{100}\) + \(\frac{4}{1000}\)


2. Write as decimal fractions:

(i) 8 thousands + 8 ones + 3 tenths + 9 hundredths

= 8008.39


(ii) 4000 + 7 + \(\frac{5}{10}\) + \(\frac{6}{100}\)

= 4007.56


(iii) 6 hundreds + 9 tens + 8 tenths + 4 thousandths

= 690.804


(iv) 3 tens + 7 ones + 6 hundredths + 8 thousandths

= 37.068


(v) 400 + 50 + 1 + \(\frac{9}{100}\)

= 451.09


(vi) 800 + 70 + 2 + \(\frac{8}{10}\) + \(\frac{5}{1000}\)

= 872.805

(vii) 6 tens + 5 tenths + 8 hundredths

= 60.58


(viii) 9 hundreds + 4 tens + 3 tenths + 4 hundredths

= 940.34


3. Write the following in short form.

(i) 100 + 0.5 + 0.06 + 0.008             (ii) 80 + 1 + 0.02 + 0.005


Solution:

(i) 100 + 0.5 + 0.06 + 0.008           

= 100.568            


(ii) 80 + 1 + 0.02 + 0.005

= 81.025


4. Write the place-value of the underlined digits.

(i) 2.47                                (ii) 11.003                           (iii) 5.175


Solution:

(i) 2.47 

Place-value of 7 in 2.47 is 7 hundredths or 0.07.


(ii) 11.003

Place-value of 3 in 11.003 is 3 thousandths or 0.003.


(iii) 5.175

Place-value of 1 in 5.175 is 1 tenths or 0.1.


Expanded form of Decimals:

This is a form in which we add the place value of each digit forming the number.


Practice Problems on Expanded Form of Decimal Fractions:

I. Write each of the following decimals in expanded form:

(i) 38.54

(ii) 83.107

(iii) 627.074

Solution:

(i) 38.54 = 38 + \(\frac{5}{10}\) + \(\frac{4}{100}\) = 30 + 8 + 0.5 + 0.04


(ii) 83.107 = 83 + \(\frac{1}{10}\) + \(\frac{0}{100}\) + \(\frac{7}{1000}\)

                = 80 + 3 + 0.1 + 0 + 0.007

                = 80 + 3 + 0.1 + 0.007


(ii) 627.074 = 627 + \(\frac{0}{10}\) + \(\frac{7}{100}\) + \(\frac{4}{1000}\)

                  = 600 + 20 + 7 + 0 + 0.07 + 0.004

                  = 600 + 20 + 7 + 0.07 + 0.004


II. Write following in short form:

(i) 9 + \(\frac{3}{10}\) + \(\frac{4}{100}\)

(ii) 50 + 7 + \(\frac{6}{10}\) + \(\frac{2}{100}\) + \(\frac{4}{1000}\)

(iii) 100 + 4 + \(\frac{3}{10}\) + \(\frac{6}{1000}\)


Solution:

(i) 9 + \(\frac{3}{10}\) + \(\frac{4}{100}\) = 9.34

(β…±) 50 + 7 + \(\frac{6}{10}\) + \(\frac{2}{100}\) + \(\frac{4}{1000}\) = 57.624

(iii) 100 + 4 + \(\frac{3}{10}\) + \(\frac{6}{1000}\) = 104.306


III. Write the given decimals in expanded form by fractional expansion.

One example has been done for you to get the idea how to do decimals in expanded form by fractional expansion.

1.73 = 1 + \(\frac{7}{10}\) + \(\frac{3}{100}\)

(i) 23.8

(ii) 60.27

(iii) 119.05

(iv) 276.207


Answers:

(i) 20 + 3 + \(\frac{8}{10}\)

(ii) 60 + 0 + \(\frac{2}{10}\) + \(\frac{7}{100}\)

(iii) 100 + 10 + 9 + 0 + \(\frac{5}{100}\)

(iv) 200 + 70 + 6 + \(\frac{2}{10}\) + 0 + \(\frac{7}{100}\)


IV. Write the given decimals in expanded form by decimal expansion.

One example has been done for you to get the idea how to do decimals in expanded form by decimal expansion.

8.461 = 8 + 0.4 + 0.06 + 0.001

(i) 6.08

(ii) 36.505

(iii) 402.613

(iv) 700.037


Answers:

(i) 6 + 0.0 + 0.08

(ii) 30 + 6 + 0.5 + 0.00 + 0.005

(iii) 400 + 0 + 2 + 0.6 + 0.01 + 0.003

(iv) 700 + 0 + 0 + 0.0 + 0.03 + 0.007


V. Write the decimal number for the expansions given below.

(i) 10 + 6 + \(\frac{3}{10}\) + \(\frac{9}{1000}\)

(ii) 600 + 20 + 7 + \(\frac{1}{10}\) + \(\frac{3}{100}\) + \(\frac{7}{1000}\)

(iii) 2000 + 8 + \(\frac{3}{10}\) + \(\frac{9}{100}\)

(iv) 400 + 70 + 1 + 0.5 + 0.07 + 0.002

(v) 5000 + 80 + 0 + 0.2 + 0.002


Answers:

(i) 16.309

(ii) 627.137

(iii) 2008.39

(iv) 471.572

(v) 5080.202


VI. Write the following decimals in expanded form:

(i) 31.5

(ii) 37.53

(iii) 307.85

(iv) 752.34

(Ξ½) 882.146

(vi) 41.005

(vii) 345.083

(viii) 435.202


Answer:

VI. (i) 31.5 = 31 + 05

(ii) 37.53 = 30 + 7 + 0.5 + 0.03

(iii) 307.85 = 300 + 7 + 0.8 + 0.05

(iv) 752.34 = 700 + 50 + 2 + 0.3 + 0.04

(Ξ½) 882.146 = 800 + 80 + 2 + 0.1 + 0.04 + 0.006

(vi) 41.005 = 40 + 1 + 0.005

(vii) 345.083 = 300 + 40 + 5 + 0.08 + 0.003

(viii) 435.202 = 400 + 30 + 5 + 0.2 + 0.002


2. Write each of the following in decimal form:

(i) 9 + 4/10 + 6/100 + 2/1000

(ii) 600 + 40 + 5/1000

(iii) 300 + 3 + 5/10 + 2/1000

(iv) 700 + 40 + 7 + 2/100 + 3/1000


Answer:

2. (i) 9.462

(ii) 640.005

(iii) 303. 502

(iv) 747.023 


3. Fill in the boxes with correct numbers:

(i) 84.29 = 80 + πŸ”² + \(\frac{2}{10}\)+ \(\frac{9}{πŸ”²}\)

(ii) 35.265= 30 + 5 + \(\frac{πŸ”²}{10}\) + \(\frac{6}{100}\) + \(\frac{5}{πŸ”²}\)

(iii) 5672.053= 5000 + 600 + πŸ”² + πŸ”² + \(\frac{5}{πŸ”²}\) + \(\frac{3}{πŸ”²}\)


Answer:

3. (i) 84.29 = 80 + 4 + \(\frac{2}{10}\) + \(\frac{9}{\mathbf{{\color{Red}100}}}\)

(ii) 35.265= 30 + 5 + \(\frac{\mathbf{{\color{Red}2}}}{10}\) + \(\frac{6}{100}\) + \(\frac{5}{\mathbf{{\color{Red}1000}}}\)

(iii) 5672.053= 5000 + 600 + 70 + 2 + \(\frac{5}{\mathbf{{\color{Red}100}}}\) +  \(\frac{3}{\mathbf{{\color{Red}1000}}}\)

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