Decimal Fractions

Decimal Fractions :

Fractions in which denominators are powers of 10 are called decimal fractions.
1/10, 1/100, 1/1000, … etc., are respectively the tenth, the hundredth and the thousandth part of 1.

5/10 is 5 tenths, can be written as 0.5, called decimal five;

4/100 is 4 hundredths, can be written as 0.04, called decimal zero-four;

15/1000 is 15 thousandths, can be written as 0.015, called decimal zero-one-five; and so on.

Rule – Converting a Decimal into a 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. Remove the decimal point and reduce the fraction to its lowest terms.

Note 1 : Annexing zeros to the extreme right of a decimal fraction does not change its value.
(i.e.)0.2 = 0.20 = 0.200 etc.,
Note 2 : If numerator and denominator of a fraction contain the same number of decimal places, can remove the decimal sign.
(i.e.) 2.5/1.5 =25/15

Rule –
For Addition and Subtraction of Decimal Fractions :
The given Numbers are so placed under each. Other than the decimal point lies in one column. The numbers so arranged can be Added or Subtracted in a usual way.

For Multiplication :
Shift the decimal point to the right by as many places of decimal, as is the power of 10.
(i.e) 25.20369 X 100 = 2520.369

Multiply the given numbers considering them without the decimal point. 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.

For Division :
Dividing a decimal fraction by a counting number :
Divide the given decimal numeral without considering the decimal point, by the given counting number. 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 :
Multiply both the dividend and the divisor by a suitable multiple of 10 to make divisor a whole number. Then proceed as decimal fraction by a counting number.

Recurring Decimal :
In a decimal fraction, If a figure or set of figures is repeated continuously, such a number is called a Recurring Decimal.

If a single figure is repeated, it is expressed by putting a dot on it. If a set of figures is repeated, it is expressed by putting a bar on it.

Pure Recurring Decimal : A Decimal fraction in which all the figures after the decimal point are repeated, is called a Pure Recurring Decimal.

For Converting a Pure Recurring Decimal into a Vulgar fraction, write the repeated figure only once in the numerator, and take as many nines in the denominator as is the number of repeating figures.

Mixed Recurring Decimal : A Decimal fraction in which some figures do not repeat and some of them repeat, is called a Mixed Recurring Decimal.

For Converting a Mixed Recurring Decimal into a Vulgar fraction, in the numerator, take the difference between the number formed by all the digits after decimal point (taking the repeated digits only once) and the formed by non – repeating digits. 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.

H.C.F & L.C.M OF NUMBERS

Factors And Multiples :

If a number x divides another number y exactly, we say that x is factor of y. Also, in this case, y is called a multiple of x.

Highest Common Factor (H.C.F. Or G.C.D. or G.C.M) :

The H.C.F. of Two or more than Two numbers is the greatest number that divides each one of them exactly.
The Highest Common Factor is also known as Greatest Common divisor or Greatest Common Measure.

H.C.F. by Factorization :

Express each one of the given numbers as the product of prime factors. Now, choose common factors and take the product of these factors to obtain the required H.C.F.

For Example ;
Find the H.C.F. of 96, 528, 1584, and 2016.
Solution,96 = 2⁵ × 3 ; 528 = 2⁴ × 3 × 11 ; 1584 = 2⁴ × 3² × 11 ;
2016 = 2⁵ × 3² × 7
H.C.F. = 2⁴ × 3 = (16 × 3) = 48

H.C.F. By Division Method :

Suppose we have to find the H.C.F. of two given numbers. Divide the larger number by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding divisor by the remainder last obtained, till a remainder zero is obtained. The last divisor is the H.C.F. of given numbers.

Suppose we have to find the H.C.F. of three numbers. Then H.C.F. of (H.C.F. of any two and the third number) gives the H.C.F. of given three numbers.

Similarly, The H.C.F. of more than three numbers may be obtained.

Lowest Common Multiple (L.C.M.) :

The least number which is exactly divisible by each one of the given numbers, is called their L.C.M.

FORMULA :

Product of Two Numbers = ( Their H.C.F.) × (Their L.C.M.).

L.C.M. By Factorization :

Resolve each one of the given numbers into prime factors. Then, L.C.M. is the product of highest powers of all the factors.

Note : L.C.M. of three numbers = L.C.M. of (L.C.M. of any two & third).

H.C.F. and L.C.M. of Fractions :

H.C.F. of Fractions = H.C.F. of Numerators / L.C.M. of Denominators.
L.C.M. of Fractions = L.C.M. of Numerators / H.C.F. of Denominators.

BASIC FORMULAE & IMPORTANT RESULTS

1. (a + b)² = a² + b² + 2ab

2. (a – b)² = a² + b² – 2ab

3. (a + b)² – (a – b)² = 4ab

4. (a + b)² + (a – b)² = 2(a² + b²)

5. (a² – b²) = (a + b) (a – b)

6. (a + b + c)² = a² + b² + c² + 2 (ab + bc + ca)

7. (a³ + b³) = (a + b) (a² – ab + b²)

8. (a³ – b³) = (a – b) (a²+ ab + b²)

9. (a³ + b³ + c³ – 3abc) = (a + b + c)( a² + b² + c² – ab – bc – ca)

10. If a + b + c = 0, then a³ + b³ + c³ = 3abc

11. ( 1 + 2 + 3 + … + n ) = [n ( n + 1 )] divided by 2.

12. ( 1² + 2² + 3² + … + n²) = [ n (n + 1) (2n + 1)] divided by 6.

13. ( 1³ + 2³ + 3³ + … + n³) = [ n² ( n + 1 )²] divided by 4.

Distributive Law: For any three numbers a, b, c , we have

• a × (b + c) = a × b + a × c

• a × (b – c) = a × b - a × c

Division Algorithm Or Euclidean Algorithm: If we divide a given number by another number then,
DIVIDEND = ( DIVISOR × QUOTIENT ) + REMAINDER.


i. [(x)^n - (a)^n] is divisible by (x-a) for all values of n.

ii.[(x)^n - (a)^n] is divisible by (x+a) for all even values of n.

iii.[(x)^n + (a)^n] is divisible by (x+a) for all odd values of n.

LINEAR EQUATIONS : An equality containing an unknown number is called a linear equation, such as x + 4 = 6 , 3x – 5 = 4, etc.

In a linear equation , we can

• Add same number on both sides,

• Subtract same number on both sides,

• Multiply both sides by a same non – zero number,

• Divide both sides by a same non – zero number.

Transposition :
In a linear equation, we can take any number on the other side with its sign changed from + (positive) to – (negative) and from – (negative) to + (positive). This process is called Transposition.

Mental Subtraction

Here We see Mental Subtraction.

To Subtract Mentally, try and round off the number you are subtracting and then correct the answer.

(i.e) To subtract 9, take 10 and add 1; To subtract 8, take 10 and add 2; and so on.
For example, 47 – 9 = ?

The easiest and fastest method is to subtract 10, (37) and add 1 (38).
So the answer is 47 – 9 = 38.

To Subtract 27 from 44. Take 30, (14) and add 3, answer is 17.

Similarly to subtract a number near 100, take 100 and add the remainder.

• Subtracting One Number Below a Hundreds Value from Another which is just Above the same Hundreds Number.

This is easy method for mental subtraction. For example 124 – 86 = ?

Here 86 is 14 lower than 100, 124 is 24 higher than 100. So Add 14 to 24 for an easy answer of 38.

The Same principle applies for numbers above and below 10. For example, 15 – 6 = ?
(i.e.) 5 + 4 = 9 answer.

Similarly this works for any three digit subtraction.

For example, 445 – 286 = ?

286 is 14 lower than 300, 445 is 145 higher than 300. So the answer is 145 + 14 = 159.

Try these for yourself by applying the above method :-

39 – 7 = ?
54 – 38 = ?
436 – 84 = ?
134 – 76 = ?
152 – 88 = ?
14 – 9 = ?
834 – 275 = ?


Yes, Hope You have done all the above very easily and quickly.