**Category : **5th Class

**FRACTION AND DECIMALS**

**FUNDAMENTALS**

- A fraction is a number representing a part of a whole
- The fractions One-third, Three-fifths, Two-sevenths are written as: \[\frac{1}{3},\frac{3}{5},\frac{2}{7}\]respectively.
- The lower part of a fraction, which indicates the number of equal parts into which the upper part is divided, is called denominator. The upper number, denotes the number of parts considered of the whole, is called numerator.

**Types of Fraction**

- Proper Fraction: A fraction whose numerator is less than the denominator is called a proper fraction.

Example: \[\frac{1}{3},\frac{2}{3},\frac{4}{5}\]

- In the above fractions, the numerators 1, 2, 3, are less than denominators 3, 3, 5 respectively.
- Improper fraction: A fraction whose numerator is more than or equal to denominator is called improper fraction.

Example:\[\frac{5}{3},\frac{6}{5},\frac{7}{4},\frac{7}{7}\] etc.

- Mixed Fraction: A combination of a whole number and a paper fraction is called a mixed fraction

Example:\[1+\frac{2}{3}\] is written as\[1\frac{2}{3}\],\[2+\frac{1}{5}\] is written as\[2\frac{1}{5}\]

- Like Fraction: Fraction with same denominators are called like fractions.

Example: \[\frac{1}{8},\frac{2}{8},\frac{5}{8}\] etc.

In all the above fraction denominators are equal, so they are like fraction.

- Unlike Fractions: Fractions with different denominators are called unlike Fractions.

Example: \[\frac{1}{3},\frac{1}{5},\frac{5}{8},\frac{3}{7}\] etc.

- Equivalent Fractions: Fractions having the same value are called equivalent fractions.

Example: \[\frac{1}{5},\frac{2}{10},\frac{3}{15}\] etc.

In above fractions value of each fraction is equal so they are equipment fractions.

- Decimal Fractions: A fraction whose denominator is powers of 10 is called a decimal fraction.

Example:\[\frac{3}{10},\frac{1}{100},\frac{1}{1000}\]

- Combined Fraction: A fraction of a fraction is called a combined fraction.

Example: \[\frac{1}{2}\]of\[\frac{3}{8}\],\[\frac{1}{3}\] of \[\frac{4}{7}\] etc.

- Continued Fractions: Fractions which contain an additional fraction in the numerator or the denominator are called continued fractions.

Example: \[4+\frac{1}{1+\frac{1}{1+\frac{2}{3}}},\,\,2+\frac{1}{1-\frac{1}{1-\frac{1}{3}}}\]etc.

**Additional of fractions**

- Additional of like fractions

Sum of like fractions\[=\frac{\text{sum}\,\,\text{of}\,\,\text{numerators}}{\text{sum}\,\,\text{of}\,\,\text{denominators}}\]

Example:\[\frac{3}{7}+\frac{4}{7}=\frac{3+4}{7}=\frac{7}{7}=1,\]

\[\frac{3}{8}+\frac{7}{8}=\frac{10}{8}=\frac{5}{4}=1\frac{1}{4}\]etc.

- Addition of unlike fractions: Get the LCM of denominator of unlike fraction. Then change unlike fractions into like fraction with their LCM as common denominator. Finally, add the like fractions.

Sum of\[\frac{2}{5}\]and \[\frac{1}{3}\]

LCM of 5 and 3= 15

Now,\[\frac{2}{5}\times \frac{5}{3}=\frac{6}{15}\]and \[\frac{1\times 5}{3\times 5}=\frac{5}{15}\]

Then,\[\frac{6}{15}+\frac{5}{15}=\frac{11}{15}\]

- Subtraction of fractions

**Subtraction of lie fractions **

\[=\frac{\text{Difference between the numerators}}{\text{common denominator}}\]

Examples: \[\frac{6}{5}-\frac{2}{5}=\frac{6-2}{5}=\frac{6-2}{5}=\frac{4}{5}\]

\[\frac{8}{3}-\frac{1}{3}=\frac{7}{3}=2\frac{1}{3}\] etc.

- Subtraction of unlike fractions: Get LCM of Denominators of unlike fractions with their LCM as common denominator. Finally get difference between the like fractions so obtained.

Difference of \[\frac{3}{5}\]and\[\frac{1}{2}\]

LCM of 5 and 2=10

\[\frac{3}{5}=\frac{3\times 2}{5\times 2}=\frac{6}{10}\]

\[\frac{1}{2}=\frac{1\times 5}{2\times 5}=\frac{5}{10}\]

Now, \[\frac{6}{10}-\frac{5}{10}=\frac{6-5}{10}=\frac{1}{10}\]

** **

**Multiplication of a fraction by a whole number.**

- Faction\[\times \]whole number

\[=\frac{\text{ Numerator of fractionwhole number }\!\!~\!\!\text{ }}{\text{ }\!\!~\!\!\text{ }\!\!~\!\!\text{ }\!\!~\!\!\text{ Denominator of the fraction}}\]

Multiply \[\frac{2}{5}\] by 4

We have, \[\frac{2}{5}\times 4=\frac{2\times 4}{5}=\frac{8}{5}=1\frac{3}{5}\]

**Multiplication of a fraction by a fraction**

- 1st fraction \[\times \] 2nd fraction

\[\frac{\text{Product of their numerators}}{\text{Product of their denominators}}\]

Multiply \[\frac{3}{10}\] by \[\frac{7}{8}\]

We have, \[\frac{3}{10}\times \frac{7}{8}=\frac{21}{80}\]

**Division of a fraction by a fraction**

- To divide a fraction by a fraction, multiply the fraction by reciprocal of the divisor.

Example: Divide \[\frac{3}{5}\div \frac{7}{8}=\frac{3}{5}=\frac{8}{7}=\frac{24}{35}\]

**Division of a fraction by a fraction**

- Fraction\[\times \]whole number= fraction\[\times \]reciprocal of the whole number

Example: Divide\[\frac{7}{8}\div 14\] \[\therefore \]\[\frac{7}{8}\times \frac{1}{14}=\frac{1}{16}\]

**Decimals**

- The numbers which are written in decimal from are called decimal numbers or decimals.

Example: 0.7, 1.68, 9.357

- A decimal number has two parts, separated by a decimal point. Left part of the decimal point is whole number and right of the decimal point is decimal part.

Example: In 468.23, whole part is 468 and decimal part is 0. 23. It is read as Four Sixty eight point two three.

- Decimals Places: The number of figure which follow the decimal point is called the number of decimal places.

Example: 1. 23 have two decimal places and 1. 417 have three decimal places

- Like Decimals: The decimal numbers having the same number of decimal places are called like decimals.

Example: 5.321, 6.932, 5.834 are like decimals because of each having 3 places of decimals.

- Unlike Decimals: The Decimal numbers having the different number of decimal places are known as unlike decimals.

Example: 5.41, 6.232, 9.2314 are unlike decimals because of each having different number of decimals.

**Equivalent Decimals**

Let there be two decimal numbers having different numbers of decimal after decimal point to the number having less number of decimal places, we add appropriate number of zeros at the extreme right so that the two numbers have same number of decimal places. Then two numbers called equivalent decimals.

Example: Let 9.6 and 8.324 ne two number.

Now, 9.6 can be written as 9.600 so that 9.600 and 8.324 have both 3 decimal places.

Hence 9.600 and 8.324 are equivalent decimals

Similarly, 10.32 = 10.320

7.3 = 7.300

9.142 = 9.142

All the above decimals are equivalent decimals.

**Additional of Decimal Numbers**

- In order to add two or more decimal numbers, write the decimal numbers in column with the decimals points directly below each other and then add them as whole numbers Finally place the decimal point in the answer directly below the other decimal points.

Add 7.35 and 5.26

We have,

**Subtraction of Decimal Numbers**

- To final the difference of two decimal numbers proceed as:

Step I: If the given decimal numbers are unlike decimals, write them into like decimals.

Step II: Write the smaller decimal number under the larger decimal number.

Step III: Subtract as usual ignoring the decimal points.

Step IV: Finally, put the decimal point in the difference under the decimal points of the given number.

Example: Subtract 13.74 from 80.4

On converting the given numbers into like decimals we get, 13.74 and 80.40. Writing the decimals in column and on subtracting, we get,

\[\therefore \]\[80.40-13.74=66.66\]

**Multiplication of Decimal Numbers by a whole number**

- Multiplication of whole numbers, ignoring the decimals.

Step I: Multiply like whole numbers, ignoring the decimals.

Step II: Count the number of decimal places in the decimal numbers

Step III: Show the same number of decimal places in the product.

Example: Multiply 6.238 by 6

First we multiply 6.238 by 6

Since given decimal number has 3 decimal places. So, the product will have 3 decimals places.

So,\[6.238\times 6=37.428\]

**Multiplication of two Decimal Numbers**

- Step I: Multiply two decimal numbers ignoring the decimal.
- Step II: In the product, put the decimal point so that the decimal places in the product is equal to the sum of the decimal places in the given decimal numbers.

Example: Multiply 12.8 by 1.2

Sum of decimal places in the given decimals\[=1+1=2\]

So, place the decimal point in the product so as to have 2 decimal points

\[\therefore \]\[12.8\times 1.2=15.36\]

- Multiplying by 10, 100, 1000 etc.

(a) On multiplying a decimal number by 10, the decimal point moves one place to the right \[10\times 0.212=2012,\,\,10\times 3.163=31.63.\]

(b) On multiplying a decimal number by 100, the decimal point moves two place to the right\[100\times 0.4321=43.21,\,\,100\times 7.832=783.2\].

(c) On multiplying a decimal number by\[1000\], the decimal point moves three places to the right.

\[1000\times 0.2312=231.2\]

\[1000\times 0.12=120\]

\[1000\times 7.3=7300\]

**Decimal of Decimal Numbers**

- Division of decimals by a whole number.

Consider the dividend as a whole number and perform the division, when the division of whole number part of the decimal is complete. Place the decimal point in the question and continue with the division as in the case of whole numbers.

Example: Divide 337.5 by 15

\[\therefore \,\,337.5\div 15=22.5\]

- Division of a decimal by 10, 10, 1000 etc.: When a decimal number is divided by 10, 100, 1000, the decimal point moves to the left, by one place, two place and three places respectively.

Example: \[\frac{13}{10}=1.3,\,\,\frac{151}{100}=1.51,\,\,\frac{1321}{1000}=1.321\]

**Division of a decimal number by another decimal number**

- To divide a decimal number by another decimal number, first of all convert the divisor into a whole number by multiplying the divided and the divisor by 10, 100 and 1000 etc. depending upon the decimal places in the divisor. Finally divide the new dividend by the whole number as usual.

Example: Divided 21.97 by 1.3

\[21.97\div 13=\frac{21.97\times 10}{1.3\times 10}=\frac{219.7}{13}=16.9\]

**Division of a whole numbers by a decimal number**

- To divide a number by a decimal number, change the divisor into a whole number, by multiplying the dividend as well as divisor by 10.100.1000 etc. depending upon the decimal places in the divisor.

Example: Divided 68 by 0.17

We have

\[\frac{68}{0.17}=\frac{68\times 100}{0.17\times 100}=\frac{6800}{17}=400\]

**Converting a Decimal into a vulgar fraction**

- Write down the given number without the decimal point. Then write 1 followed by as many zeroes as there are decimal places in the given decimal as the denominator of the fraction. Simplify the fraction if possible.

Example: \[0.13=\frac{13}{100},\,\,0.17=\frac{17}{100}\]etc.

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