Decimal Fractions
Fractions in which denominators are powers of 10 are known as decimal fractions.
Thus,$\dfrac{1}{10}$ = 1 tenth = .1; $ \dfrac{1}{100}$ = 1 hundredth = .01;
$\dfrac{99}{100}$ = hundredths = .99; $\dfrac{7}{1000}$ = 7 thousandths = .007, etc.;
Conversion of a Decimal into Vulgar Fraction
Put 1 in the denominator under the decimal point and annex with it as many zeros as is the number of digits after the decimal point. Now, remove the decimal point and reduce the fraction to its lowest terms.
Example
0.25 = $\dfrac{25}{100}$ = $\dfrac{1}{4}$
Exercise
2.008 = $\dfrac{2008}{1000}$ = $\dfrac{251}{125}$
Annexing Zeros and Removing Decimal Signs
Annexing zeros to the extreme right of a decimal fraction does not change its value. Thus, 0.8 = 0.80 = 0.800, etc.
If numerator and denominator of a fraction contain the same number of decimal places, then we remove the decimal sign.
Thus,$\dfrac{1.84}{2.99}$ = $\dfrac{184}{299}$ = $\dfrac{8}{13}$
Operations on Decimal Fractions
i. Addition and Subtraction of Decimal Fractions
The given numbers are so placed under each other that the decimal points lie in one column. The numbers so arranged can now be added or subtracted in the usual way.
Example
21.3 + .213 + 3.21 + .021 + 2.0031 = ?
21.3 .213 3.21 .021 2.0031 --------------- 26.7471 ----------------
Exercise :
ii. Multiplication of a Decimal Fraction By a Power of 10
Shift the decimal point to the right by as many places as is the power of 10.
Example
5.9632 x 100 = 596.32
Exercise
iii.Multiplication of Decimal Fractions
Multiply the given numbers considering them without decimal point. Now, in the product, the decimal point is marked off to obtain as many places of decimal as is the sum of the number of decimal places in the given numbers.
Example
To find the product (.2 x 0.02 x .002).
Now, 2 x 2 x 2 = 8.
Sum of decimal places = (1 + 2 + 3) = 6.
$\therefore$ .2 x .02 x .002 = .000008
Exercise
Now, 68 x 79 = 5372.
Sum of decimal places = (2 + 2 ) = 4.
$\therefore$ .68 x .79 = 0.5372
iv. Dividing a Decimal Fraction By a Counting Number
Divide the given number without considering the decimal point, by the given counting number. Now, in the quotient, put the decimal point to give as many places of decimal as there are in the dividend.
Example
To find the quotient (0.0204 ÷ 17). Now, 204 ÷ 17 = 12
Dividend contains 4 places of decimal. So, 0.0204 ÷ 17 = 0.0012
Exercise
63 / 9 = 7
Decimal places in dividend = 2
∴ 0.63 / 9 = 0.07
v.Dividing a Decimal Fraction By a Decimal Fraction
Multiply both the dividend and the divisor by a suitable power of 10 to make divisor a whole number.
Example
$\dfrac{0.00066}{0.11}$ = $\dfrac {0.00066 \times 100}{0.11 \times 100}$
= $ \dfrac{0.066}{11}$ = .006
Exercise
$\dfrac{ 0.00042}{0.06}$ = $\dfrac{(0.00042 x 100 )}{ (0.06 x 100)}$
= $\dfrac{0.042 }{6}$
= 0.007
Comparison of Fractions
Suppose some fractions are to be arranged in ascending or descending order of magnitude, then convert each one of the given fractions in the decimal form, and arrange them accordingly.
Example
Let us to arrange the fractions $\dfrac {3}{5}$, $\dfrac{6}{7}$ and $\dfrac{7}{9}$ in descending order
Now, $\dfrac{3}{5}$ = 0.6, $\dfrac{6}{7}$ = 0.857, $\dfrac{7}{9}$ = 0.777...
Since, 0.857 > 0.777... > 0.6. So, $ \dfrac{6}{7}$ > $\dfrac{7}{9}$ > $\dfrac{3}{5}$.
Exercise
Now, $\dfrac {1}{2}$ = 0.5, $\dfrac {3}{4}$ = 0.75, $\dfrac {7}{8}$ = 0.875 , $\dfrac {5}{12}$ = 0.416...
Since, 0.875 > 0.75 > 0.5 > 0.416... . So $\dfrac {7}{8}$ > $\dfrac {3}{4}$ > $\dfrac {1}{2}$ > $\dfrac {5}{12}$
Recurring Decimal:
If in a decimal fraction, a figure or a set of figures is repeated continuously, then such a number is called a recurring decimal.
n a recurring decimal, if a single figure is repeated, then it is expressed by putting a dot on it. If a set of figures is repeated, it is expressed by putting a bar on the set.
Example
$\dfrac{1}{3}$.. = ?
$\dfrac{1}{3}$.. = $0.\overline{3}$
Exercise
$\dfrac{22}{7}$ = 3.142857142857.... = $3.\overline{142857}$
Pure Recurring Decimal
A decimal fraction, in which all the figures after the decimal point are repeated, is called a pure recurring decimal.
Converting a Pure Recurring Decimal into Vulgar Fraction
Write the repeated figures only once in the numerator and take as many nines in the denominator as is the number of repeating figures.
Example
0.$\overline{5}$ = ?
0.$\overline{5}$ = $\dfrac{5}{9}$
Exercise
0.$\overline{53}$ = $\dfrac{53}{99}$
Mixed Recurring Decimal:
A decimal fraction in which some figures do not repeat and some of them are repeated, is called a mixed recurring decimal.
Eg. 0.1733333.. = 0.170.$\overline{3}$
Converting a Mixed Recurring Decimal Into Vulgar Fraction:
In the numerator, take the difference between the number formed by all the digits after decimal point [taking repeated digits only once] and that formed by the digits which are not repeated. In the denominator, take the number formed by as many nines as there are repeating digits followed by as many zeros as is the number of non-repeating digits.
Example
0.1$\overline{6}$ = ?
0.1$\overline{6}$ = $\dfrac{16-1}{90}$ = $\dfrac{15}{9}$ = $\dfrac{1}{6}$
Exercise
0.22$\overline{73}$ = $\dfrac{2273 - 22}{9900}$ = $\dfrac{2251}{9900}$
The century divisible by 400 is a leap year.
$\therefore$ The year 700 is not a leap year.
22 Apr 2222 = [2221 years + period from 1-Jan-2222 to 22-Apr-2222]
We know that number of odd days in 400 years = 0
Hence the number of odd days in 2000 years = 0 [Since 2000 is a perfect multiple of 400]
Number of odd days in the period 2001-2200
= Number of odd days in 200 years
= 5 x 2 = 10 = 3
As we can reduce perfect multiples of 7 from odd days without affecting anything
Number of odd days in the period 2201-2221
= 16 normal years + 5 leap years
= 16 x 1 + 5 x 2 = 16 + 10 = 26 = 5 odd days
Number of days from 1-Jan-2222 to 22 Apr 2222
= 31 Jan + 28 Feb + 31 Mar + 22Apr = 112
112 days = 0 odd day
Total number of odd days = 0 + 3 + 5 + 0 = 8 = 1 odd day
1 odd days = Monday
Hence 22 Apr 2222 is Monday.