Addition of numbers:
$\text{Eg. Add the following - } 4.305, 0.7, 19.3973, 101.9$
$\,\,\,\,\,\,\,\,101\,.\,9000$
$+\,\,\,019\,.\,3973$
$+\,\,\,004\,.\,3050$
$\,\,\,\,\,\,\,000\,.\,0700$
---------------
$\,\,\,\,\,\,\,125\,.\,6723$
---------------
Subtraction of numbers:
$\text{Subtract 13.5794 from 31.343:}$
$\,\,\,\,\,\,\,\,31\,.\,3430$
$-\,\,\,13\,.\,5794$
$-------------$
$\,\,\,\,\,\,\,\,17\,.\,7636$
$-------------$
Multiplication of a Decimal number by a non-decimal number:
When multiplying a decimal number by a non-decimal number, in the result the point is placed before as many numbers as it is in the decimal number.
Eg. $\text{Multiply }43\,.\,5037 \text{ from } 93$
$\Rightarrow 435037 \times 93 = 40458441$
$\Rightarrow 43\,.\,5037 \times 93 = 4045\,.\,8441$
Multiplication of a Decimal number by multiples of 10:
When multiplying a decimal number by multiples of 10, the point is moved forward as many times as it is the power of 10 or to as many zeroes . This means that when a decimal number is multiplied by 10, the point is moved one place forward, when by 100 it moves 2 place forward.
(i) $37\,.\,057 \times 10 = 370\,.\,57$
(ii) $11\,.\,241 \times 100 = 1124\,.\,1$
(iii) $13\,.\,4 \times 1000 = 13400\,.\,0$
Multiplication of a Decimal number by a Decimal number:
When multiplying a decimal fraction from another decimal fraction, first multiply both the numbers without the decimal point. In the result, the point is placed before as many numbers (from the right) as there are numbers after the decimal point in both the numbers. For example, multiplying 13.55 by 15.55 the point will be placed 4 placed ahead from the last number on the extreme right.
Eg. $\text{Multiply } 27\,.\,35 \text{ by } 1\,.\,247$
$\Rightarrow 2735 \times 1.247 = 3407810$
$\Rightarrow 27\,.\,35 \times 1\,.\,247 = 34\,.\,07810$
Eg. $\text{Multiply } 1\,.\,01 \text{ by } 0\,.\,001$
$\Rightarrow 101 \times 1 = 101$
$\Rightarrow 101 \times 1 = 0\,.\,00101$
Division of a Decimal number by a non-decimal number:
When dividing a decimal number by a non-decimal number, first divide the number without considering the the decimal point. In the result the point is placed before as many numbers as it is in the decimal number.
Eg. $\text{Divide } 0\,.\,049 \text{ by } 7$
$\Rightarrow 49 \div 7 = 7$
$\Rightarrow 0\,.\,049 \times 7 = 0\,.\,007$
Division of a Decimal number by another Decimal number:
When dividing a decimal number by another decimal number, first convert the decimal numbers by multiples of 10 and then prooceed to divide as normal.
Eg. $\text{Divide } 0.0182 \text{ by } 0.014$
$\Rightarrow \dfrac{0.0182}{0.014}$
$\Rightarrow \dfrac{\dfrac{182}{10000}}{\dfrac{14}{1000}}$
$\Rightarrow \dfrac{182 \times 1000}{14 \times 10000}$
$\Rightarrow \dfrac{182}{14 \times 10} = \dfrac{26}{2 \times 10}$
$\Rightarrow \dfrac{26}{2 \times 10} = \dfrac{13}{10} = 1.3$ Answer
Eg. $\text{Divide } 63.5535 \text{ by } 13.05$
$\Rightarrow \dfrac{63.5535}{13.05}$
$\Rightarrow \dfrac{\dfrac{635535}{10000}}{\dfrac{1305}{100}}$
$\Rightarrow \dfrac{635535 \times 100}{1305 \times 10000}$
$\Rightarrow \dfrac{635535}{1305 \times 100} = \dfrac{487}{100}$ (On dividing 635535 by 1305 we have 487)
$\Rightarrow \dfrac{487}{100} = 4.87$ Answer
Using formulas for solving problems on decimal fractions:
We have following formulas:
(a) $(a + b)^{2} = a^{2} + 2ab + b^{2}$
(b) $(a - b)^{2} = a^{2} - 2ab + b^{2}$
(c) $a^{2} - b^{2} = (a + b) (a - b)$
(d) $(a + b)^{3} = a^{3} + b^{3} + 3ab(a + b)$
(e) $(a - b)^{3} = a^{3} - b^{3} - 3ab(a - b)$
Eg. Solve: $\dfrac{0\,.\,538 \times 0\,.\,538 - 0\,.\,462 \times 0\,.\,462}{1 - 0\,.\,924}$
$\Rightarrow \dfrac{(0\,.\,538)^{2} - (0\,.\,462)^{2}}{0\,.\,076} \qquad (1\,.\,000 - 0\,.\,924 = 0\,.\,076)$
$\Rightarrow \dfrac{(0\,.\,538)^{2} - (0\,.\,462)^{2}}{(0\,.\,538)^{2} - (0\,.\,462)^{2}} \qquad ((0\,.\,538)^{2} - (0\,.\,462)^{2} = 0\,.\,076)$
Now, taking $\text{a } = 0\,.\,538 \text{ and b} = 0\,.\,462 \text{, we have:}$
$\Rightarrow \dfrac{(a^{2} - b^{2})}{a - b} = a + b$
$\therefore a + b = 0\,.\,538 + 0\,.\,462 = 1$ Answer
Eg. Solve: $\dfrac{(2\,.\,3)^{3} - (0\,.\,3)^{3}}{(2\,.\,3)^{2} + 0\,.\,69 + 0\,.\,09}$
Converting $0\,.\,69 = 2\,.\,3 \times 0\,.\,3$
and $0\,.\,09 = (0\,.\,3)^{2}$
We have: $\dfrac{(2\,.\,3)^{3} - (0\,.\,3)^{3}}{(2\,.\,3)^{2} + (2\,.\,3 \times 0\,.\,3) + (0\,.\,3)^{2}}$
Now, taking $\text{a } = 2\,.\,3 \text{ and b} = 0\,.\,3 \text{, we have:}$
$\Rightarrow \dfrac{a^{3} - b^{3}}{a^{2} + ab + b^{2}}$
Now, we know that: $(a + b)^{2} = a^{}2 + ab + b^{2} \text{ or } a^{2} + 2ab + b^{2}$
$\therefore \dfrac{a^{3} - b^{3}}{(a + b)^{2}} = \dfrac{a^{3} - b^{3}}{a^{2} + b^{2}}$
$\Rightarrow \dfrac{a^{3} - b^{3}}{a^{2} + b^{2}} = a - b$
$\Rightarrow a - b = 2\,.\,3 - 0\,.\,3 = 2$ Answer
Eg. Solve: [$(0\,.\,6)^{3} + (0\,.\,4)^{3} + 3 \times 0\,.\,6 \times 0\,.\,4 \times (0\,.\,6 + 0\,.\,4)$]
$\text{Taking a } = 0\,.\,6 \text{ and b} = 0\,.\,4 \text{, we have:}$
$\Rightarrow a^{3} + b^{3} + 3ab(a + b)$
We know that: $(a + b)^{3} = a^{3} + b^{3} + 3ab(a + b)$
$\therefore (a + b)^{3} = (0\,.\,6 + 0\,.\,4)^{3} = (1)^{3} = 1$ Answer
Eg. Solve: $\dfrac{6\,.\,431 \times 6\,.\,431 \times 6\,.\,431 + 0\,.\,569 \times 0\,.\,569 \times 0\,.\,569}{6\,.\,431 \times 6\,.\,431 - 6\,.\,431 \times 0\,.\,569 + 0\,.\,569 \times 0\,.\,569}$
$\text{Taking a } = 6\,.\,431 \text{ and b } = 0\,.\,569 \text{, we have:}$
$\Rightarrow \dfrac{a^{3} - b^{3}}{a^{2} - ab + b^{2}}$
We know that: $(a + b)^{2} = a^{2} + ab + b^{2} \text{ or } a^{2} + 2ab + b^{2}$
$\therefore \dfrac{a^{3} + b^{3}}{(a - b)^{2}} = \dfrac{a^{3} + b^{3}}{a^{2} - b^{2}}$
$\Rightarrow \dfrac{a^{3} + b^{3}}{a^{2} - b^{2}} = a + b$
$\Rightarrow a + b = 6\,.\,431 + 0\,.\,569 = 7$ Answer
LCM and HCF of Fractions and Decimal Fractions:
For finding LCM and HCF of Fractions and Decimal Fractions use the following formula:
$\therefore \text{LCM} \Rightarrow \dfrac{\text{LCM of numerators}}{\text{HCF of denominators}}$
$\therefore \text{HCF} \Rightarrow \dfrac{\text{HCF of numerators}}{\text{LCM of denominators}}$
Eg. Find LCM and HCF of $\dfrac{5}{9}$, $\dfrac{4}{15}$, and $\dfrac{25}{21}$
$\Rightarrow \text{LCM of numerators: } 5\text{, }4\text{, }25 = 100$
$\Rightarrow \text{HCF of denominators: } 9\text{, }15\text{, }21 = 3$
$\therefore \text{LCM of given fraction } = \dfrac{100}{3} = 33\dfrac{1}{3}$ Answer
$\text{Now, HCF of numerators: } 5\text{, }4\text{, }25 = 100 = 1$
$\text{and, LCM of denominators: } 9\text{, }15\text{, }21 = 315$
$\therefore \text{HCF of given fraction } = \dfrac{1}{315}$ Answer
When finding LCM and HCF of Decimal Fractions, first equal the number of decimal places, then find LCM and HCF of given numbers ignoring the decimal points, then in the result place the decimal point before as many numbers as in the original.
Eg. Find LCM and HCF of $2.22$, $33.3$, and $37$
$\Rightarrow \text{Making decimals equal} = 2.22\text{, }33.30\text{, }37.00$
$\Rightarrow \text{Removing the decimal point} = 222\text{, }3330\text{, }3700$
$\Rightarrow \text{LCM of given numbers} = 74$
$\therefore \text{LCM} = 0.74$ Answer
$\Rightarrow \text{HCF of given numbers} = 33300$
$\therefore \text{HCF} = 333.00 = 333$ Answer