Category : 5th Class
FRACTION AND DECIMALS
FUNDAMENTALS
Types of Fraction
Example: \[\frac{1}{3},\frac{2}{3},\frac{4}{5}\]
Example:\[\frac{5}{3},\frac{6}{5},\frac{7}{4},\frac{7}{7}\] etc.
Example:\[1+\frac{2}{3}\] is written as\[1\frac{2}{3}\],\[2+\frac{1}{5}\] is written as\[2\frac{1}{5}\]
Example: \[\frac{1}{8},\frac{2}{8},\frac{5}{8}\] etc.
In all the above fraction denominators are equal, so they are like fraction.
Example: \[\frac{1}{3},\frac{1}{5},\frac{5}{8},\frac{3}{7}\] etc.
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.
Example:\[\frac{3}{10},\frac{1}{100},\frac{1}{1000}\]
Example: \[\frac{1}{2}\]of\[\frac{3}{8}\],\[\frac{1}{3}\] of \[\frac{4}{7}\] etc.
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
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.
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 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.
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.
\[=\frac{\text{ Numerator of fractionwhole number }\!\!~\!\!\text{ }}{\text{ }\!\!~\!\!\text{ }\!\!~\!\!\text{ }\!\!~\!\!\text{ Denominator of the fraction}}\]
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
\[\frac{\text{Product of their numerators}}{\text{Product of their denominators}}\]
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
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
Example: Divide\[\frac{7}{8}\div 14\] \[\therefore \]\[\frac{7}{8}\times \frac{1}{14}=\frac{1}{16}\]
Decimals
Example: 0.7, 1.68, 9.357
Example: In 468.23, whole part is 468 and decimal part is 0. 23. It is read as Four Sixty eight point two three.
Example: 1. 23 have two decimal places and 1. 417 have three decimal places
Example: 5.321, 6.932, 5.834 are like decimals because of each having 3 places of 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
Add 7.35 and 5.26
We have,
Subtraction of Decimal Numbers
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
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
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
\[\therefore \]\[12.8\times 1.2=15.36\]
(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
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
\[\therefore \,\,337.5\div 15=22.5\]
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
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
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
Example: \[0.13=\frac{13}{100},\,\,0.17=\frac{17}{100}\]etc.
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