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.
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.
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 + π² + \(\frac{2}{10}\)+ \(\frac{9}{π²}\)
(ii) 35.265= 30 + 5 + \(\frac{π²}{10}\) + \(\frac{6}{100}\) + \(\frac{5}{π²}\)
(iii) 5672.053= 5000 + 600 + π² + π² + \(\frac{5}{π²}\) + \(\frac{3}{π²}\)
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 + \(\frac{5}{\mathbf{{\color{Red}100}}}\) + \(\frac{3}{\mathbf{{\color{Red}1000}}}\)
β Decimal.
Expanded form of Decimal Fractions.
Changing Unlike to Like Decimal Fractions.
Comparison of Decimal Fractions.
Conversion of a Decimal Fraction into a Fractional Number.
Conversion of Fractions to Decimals Numbers.
Addition of Decimal Fractions.
Problems on Addition of Decimal Fractions
Subtraction of Decimal Fractions.
Problems on Subtraction of Decimal Fractions
Multiplication of a Decimal Numbers.
Multiplication of a Decimal by a Decimal.
Properties of Multiplication of Decimal Numbers.
Problems on Multiplication of Decimal Fractions
Division of a Decimal by a Whole Number.
Division of Decimal Fractions by Multiples.
Division of a Decimal by a Decimal.
Division of a whole number by a Decimal.
Properties of Division of Decimal Numbers
Problems on Division of Decimal Fractions
Conversion of fraction to Decimal Fraction.
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