Fractions Formulas, Important Facts, Shortcuts, Questions and Answers
FRACTION : A fraction is a quantity which expresses a part of the whole.
Type of Fractions :
1. Proper Fraction : A fraction, whose numerator is less than its denominator, is called a proper fraction.
2. Improper Fraction : A fraction, whose numerator is greater than or equal to its denominator, is called an improper fraction.
3. Mixed Fraction : A mixed fraction consists of two parts: (i) an integer and (ii) a proper fraction.
4. Like and Unlike Fractions : Fraction having the denominator but different numerators are called unlike fractions.
If denominator of the given fractions are not same, the fractions are called unlike fractions
5. Equivalent Fractions : If two or more fractions have the same value, they are called equivalent or equal fractions.
Conversion of Fractions:
(i) Mixed Fraction into an Improper Fraction the integral part by the denominator and to this product add the numerator
(ii) Improper Fraction into Mixed Fraction :— Divide numerator by the denominator. The quotient Of this division is the integral part and the remainder obtained is numerator of the required mixed fraction.
(iii) Unlike Fraction into Like Fractions:
(1) Find L.C.M. of the denominators of all given fractions.
(2) Divide LCM. by the denominator and multiply the quotient to numerator and denominator of fraction.
Decimal Fractions Formulas
Decimal Fractions: Fractions in which denominators are powers of 10 are known as decimal fractions.
Number of Decimal Places: The number of digits in the decimal part of a number is the number of decimal places in it.
When the given number has only decimal part in it. It is always written 0 before it as 0.7, 0.55 are written as 0.7, 0.55.
Conversion of a Fraction into a Decimal Fraction :
- When the denominator is 10,100,1000, 10,000 etc. : Counting from right to left of the numerator of the given fraction, mark the decimal point after as many digits as the number of zeroes in it denominator
- When the denominator is not, 10, 100, 1000, 10,000 etc.
Multiply both, the numerator and denominator of the given fraction, by a suitable number to get the denominator 10 or a power of 10 and then proceed as above, e.g.
- Conversion of a given Decimal Fraction into a Non-Decimal Fraction : Remove the decimal point and at the same time write 1 in the denominator, as many zeroes to the right of 1 as there are digits in the decimal part e.g.,
Zero or zeores written at the right of a decimal number does not change its value, e.g. 3.4 is the same as 3.40, 3.400, 3.4000 etc.
Rule for converting 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.
Remark (1) : Annexing zeros to the extreme right of a decimal fraction does not change its value.
Remark (2) : If numerator and denominator of a fraction contain the same number of decimal places, then we remove the decimal sign.
Addition & 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 a usual way.
Multiplication of a Decimal Fraction by a power of 10:
Shift the decimal point to the right by as many places of decimal as is the power of 10.
Multiplication of Decimal Fractions:
Multiply the given numbers considering them without the 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.
Dividing a decimal fraction by a Counting Number:
Rule: 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 are there in the dividend.
Dividing a Decimal Fraction by a Decimal Fraction:
Rule: Multiply both the dividend and the divisor by a suitable power of 10 to make division a whole number. Now proceed as above.
H.C.F & L.C.M of Decimal Fractions:
Rule : In given numbers, make the ‘same number of decimal places by annexing zeros in some numbers, if necessary. Considering these numbers without decimal point, find H.C.F. or L.C.M. as the case may be. Now, in the result, mark off as many decimal places as are there in each of the given numbers.
If in a decimal fraction, a figure or a set of figures is repeated continuously then such a number is called a recurring decimal.
In 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.
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-tor as is the number of repeating figures.
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,
e.g. 0.273333 = 0.273
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.