Fractions and Decimals

Category : 4th Class

 Fractions and Decimals

 

Fraction

Fraction is used to indicate a part of a whole. Fraction is written, for example, as\[\frac{a}{b}\]. The top number in a fraction is called numerator and the bottom number is called denominator of the fraction. Hence in the given example 'a' is numerator and 'b' is denominator.

Look at the shaded part in the following figures which has been represented by fractions:

 

  •             Example

Tom cuts an apple into 5 equal parts. He eats one part. Give the part of the apple eaten by Tom in fraction.

Solution:

Tom eats 1 out of 5 parts.

So parts of apple eaten by Tom \[=\frac{1}{5}\]

Like Fraction

The fractions having same denominator are called like fractions.

For example, \[\frac{1}{5},\frac{4}{5},\frac{7}{5}\] are like tractions.

 

  •            Example

Choose the like fractions from the following?

\[\frac{45}{41},\frac{25}{13},\frac{13}{25},\frac{57}{61},\frac{6565}{13}\]

Solution:

\[\frac{25}{13}\]and \[\frac{6565}{13}\]are like fractions because they have same denominator.

 

Unlike Fraction

The fractions having different denominators are called unlike fractions.

For example, \[\frac{1}{2},\frac{6}{8},\frac{14}{17},\frac{13}{9}\]are unlike fractions.

 

  •            Example

Which of the following are unlike fractions?

\[\frac{4}{7},\frac{3}{5},\frac{8}{9},\frac{7}{5},\frac{11}{12},\frac{13}{17}\]

Solution:

\[\frac{4}{7},\frac{8}{9},\frac{11}{12},\frac{13}{17}\]are unlike fractions because they have different denominators.

 

Conversion of Unlike Fraction into Like Fraction

Multiply the numerator and denominator of each fraction by a suitable number (one number for one fraction) such that denominator becomes same in all fraction.

 

  •             Example

Convert \[\frac{5}{7}\] and, \[\frac{7}{8}\] into like fractions.

Solution:

\[\frac{5\times 8}{7\times 8}=\frac{40}{56}\]

And, \[\frac{7\times 7}{8\times 7}=\frac{49}{56}\]

\[\frac{40}{56}\] and \[\frac{49}{56}\] are like fractions.

 

Equivalent Fraction

The fractions which have same value are called equivalent fractions.

For example, \[\frac{7}{9}\]and \[\frac{28}{36}\]are equivalent fractions.

 

Finding an Equivalent Fraction of Given Fraction

To find an equivalent fraction of a given fraction, numerator and denominator of the fraction is multiplied or divided by same number.

 

  •            Example

Find an equivalent fraction of \[\frac{15}{17}\].

Solution:

Equivalent fraction of\[\frac{15}{17}=\frac{45}{51}\]

Note: You may find other equivalent fraction of \[\frac{15}{17}\]

 

Unit Fraction

The fractions in the form \[\frac{p}{q}(p=1,q\] is a natural number) are called unit fractions.

For example, \[\frac{1}{4},\frac{1}{5},\frac{1}{6}\] are unit fractions.

Proper Fraction

The fractions having greater denominator are called proper fractions.

For example, \[\frac{21}{25}\] is a proper fraction because in this fraction numerator is smaller than denominator.

 

Improper Fraction

The fractions having smaller denominator are called improper fractions.

For example, \[\frac{21}{17}\] is an improper fraction because in this fraction numerator is greater than denominator

 

Mixed Fraction

Mixed fraction is a sum of a whole number and a proper fraction.

For example, \[1\frac{4}{17}\] is a mixed fraction, because 1 is a whole number and \[\frac{4}{17}\] is a proper fraction.

 

Conversion of Mixed Fraction into Improper Fraction

If \[a\frac{b}{c}\] is a mixed fraction/ then \[\frac{(a\times c)+b}{c}\] is an improper fraction equivalent to given mixed fraction.

 

  •            Example

Convert \[\mathbf{12}\frac{3}{11}\] into the improper fraction.

Solution:

\[\mathbf{12}\frac{3}{11}=\frac{11\times 12+3}{11}=\frac{132+3}{11}=\frac{135}{11}\]

 

Point to note

·         \[\frac{p}{q}\]is a proper fraction, if\[p<q\]

·         \[\frac{p}{q}\]is an improper fraction, if \[p>q\]

·         \[p\frac{q}{r}\]is a mixed fraction where p is whole number and\[\frac{q}{r}\] is a proper fraction

 

Addition of Like Fractions

Like fractions are the fractions with same denominator. So, for the addition of like fractions simply add the numerators and write the sum over the common denominator.

 

  •             Example

Add the following fractions.

\[\frac{5}{7},\frac{1}{7},\frac{6}{7}\]

Solution:

\[\frac{5}{7}+\frac{1}{7}+\frac{6}{7}=\frac{12}{7}\]

 

Subtraction of Like Fractions

In subtraction, like addition, simply subtract the numerators and write the sum over the common denominator.

  •              Example

Subtract \[\frac{7}{16}\]from \[\frac{9}{16}\]

Solution

\[\frac{9}{16}-\frac{7}{16}=\frac{2}{16}\]

 

Decimals

A decimal number is broadly divided into two parts.

(i)     Whole number part

(ii)    Decimal part

The two parts are separated by a dot (.) called the decimal point.

From the decimal point as you move on the left the place value is multiplied by 10 and as you move on the right it is divided by 10.

Decimal fraction

Whole part

Decimal part

5.3

5

3

147.81

147

.81

15.679

15

.679

.8

0

.8

1.004

1

.004

 

Look at the following table:

Thousand

Hundreds

Tens

Ones

4

4

4

4

The number given in the place value chart can be written \[4000+400+40+4.\] We notice/ as we move right in the place value chart, the value of the place gets one-tenths of the previous one.

So far we have studied the place value chart having unit place in extreme right side.

Now we will know about the decimal place value chart, which goes right to the unit place.

 

Decimal Place Value Chart

Thousand

Hundred

Tens

Ones

Tenth

Hundredths

Thousandths

Ten thousandths

1000

100

10

1

1/10

1/100

1/1000

1/10000

 

Expanded form of Decimals

The expanded form of a decimal number is the number written as the sum of its whole number and decimal place values.

 

  •             Example

Write the decimal 54.321 into expanded form.

Solution

\[50+4+\frac{3}{10}+\frac{2}{100}+\frac{1}{1000}\]

  •            Example

Write the required decimal for \[\mathbf{300+4+}\frac{\mathbf{0}}{\mathbf{10}}\mathbf{+}\frac{\mathbf{5}}{\mathbf{100}}\mathbf{+}\frac{\mathbf{7}}{\mathbf{1000}}\]

 

Solution:

304.057

 

Like Decimals

The decimals having equal number of decimal places are called like decimals.

  •            Example

\[4\mathbf{5}.\mathbf{326},\mathbf{3698}.\mathbf{222},\mathbf{0}.\mathbf{003}\]are like decimals because they have equal number of decimal places.

 

Unlike decimals

The decimals having unequal number of decimal place are called unlike decimals.   

 

  •             Example

78.036, 98,000032 are unlike decimals, because they have unequal number of decimal places.

 

Equivalent Decimals

The decimals having same value are called equivalent decimals.

 

  •              Example

78.325 and 78.3250 are equivalent decimals, because they have same value

 

Comparison of decimals

            If there are two decimal numbers we can compare them, that is, whether a number is greater than less than, or equal to other number?

A decimal is just a fractional number. For .ample, compare 0.9 and 0.09 becomes clear if we compare\[\frac{9}{10}\] to \[\frac{9}{100}\]. The fraction\[\frac{9}{10}\], is equivalent to \[\frac{9}{100}\] which is clearly greater than \[\frac{9}{100}\].

Therefore, when decimals are compared, start with tenth place and then go on. If one decimal has a higher number in teeth place, it is large then the decimal with fewer tenth. If tenth are equal, compare the hundreds, then thousands etc., until one decimal is larger, or there are no more place to compare.

 

  •             Example

Find the greater decimal. 

5465.252, 3465.655

Solution:

5465252 > 3465.655

 

  •             Example

Which one is greater

5447.87901, 5447.87910

Solution:

\[5447.87901<5447.87910\]

 

Addition of Decimals

Put whole part under whole part and decimal part under decimal part. Then do simple addition.

  •             Example

Add: \[\mathbf{5411}.\mathbf{322}+\mathbf{8254}.\mathbf{5461}+\mathbf{0}.\mathbf{5422}\]

Solution:

 

  •              Example

\[\mathbf{545}.\mathbf{23}+\mathbf{54232}.\mathbf{544}+\mathbf{0}.\mathbf{00001}+\mathbf{8755}.\mathbf{2}\]

Solution:

 

Subtraction of Decimals

Put whole part under whole part and decimal part under decimal part and do simple subtraction.

  •             Example

\[\mathbf{5446}.\mathbf{20}-\mathbf{455}.\mathbf{0005}\]

Solution:

 

  •              Example

\[\mathbf{74}.\mathbf{154}-\mathbf{54}.\mathbf{23}\]

Solution:

 

Multiplication of Decimals by Power of 10

Shift the decimal point to right as many places as there are zeroes in the power of 10.

 

  •              Example

Multiply: \[\mathbf{394}.\mathbf{5607}\times \mathbf{1000}\]

Solution:

\[394.5607\times 1000=394560.7\]

 

  •             Example

\[\mathbf{A}=\mathbf{2465}.\mathbf{21},\text{ }\mathbf{B}=\mathbf{100}\]. Find the value of \[\mathbf{A}\times \mathbf{B}.\]

Solution:

\[2465.21\times 100=246521\]

 

Multiplication of a Decimal and a Whole Number

Multiply the decimals like normal multiplication and put the decimal point such that the product have same number of decimal places as in the multiplicand.

 

  •             Example

Find the product of 6556.65 and 5.

Solution:

\[6556.65\times 5=32783.25\]

 

Division of a Decimal by the Power of 10

Shift the decimal point, starting from right towards left to the number of places, equal to number of zeroes in denominator

 

  •              Example

Divide 54462. 21 by 1000

Solution:

\[54462.21\div 1000=54.46221\]

 



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