How to convert a decimal into a fraction

Take the following steps to convert a decimal into a fraction:

1. Write down the decimal number.
2. Convert it to a fraction by putting the number without decimal in the numerator and expressing the denominator in the powers of ten.
3. Convert the numerator and denominator in terms of the factors of their primes.
4. Cancel out all the possible prime factors from numerator and denominator.
5. Multiply the primes, and write down the number as a fraction.

The above mentioned method is used in the example given below:

1. Take a number, for example, 0.36.
2. It can be expressed as 36/100.
3. Expressing in terms of prime factors, we may write the fraction as follows: 36/100 = (2* 2 * 3 * 3)/(2 * 2 * 5 * 5)
4. In the expression (2* 2 * 3 * 3)/(2 * 2 * 5 * 5), cancel the primes from numerator and denominator.
5. We finally get the following expression: (3 * 3)/(5 * 5) = 9/25

Difference between Linear Permutation and Circular Permutation

Linear Permutation: In Linear Permutation, the objects are arranged in a line. A linear permutation is simply known as permutation.

Circular Permutation: A permutation is said to be a Circular Permutation if the objects are arranged along a circle.

Circular Permutation of n distinct object = P_c = (n-1)!

Important values of permutation and combination

Following are the important values of permutation and combination:

• 0! = 1
• ⁿC_r = ⁿC_n-r
• ⁿC₀ = 1
• ⁿC_n = 1
• ⁿC_r = ⁿP_r/r!

Probability

Probability theory is the branch of mathematics concerned with analysis of random phenomena. In other words, probability defines the possibility between happening of an event and non-happening of that event. For example: There are 50 percent chances of getting head on tossing a coin. Probability is a measure of uncertainty.

Let E be any event. If m is the favorable number of outcomes and n is the number of total outcomes.

Probability of occurrence of the event E = P(E) = favorable number of outcome/total number of outcomes = m/n

Example: If a coin is tossed, probability of getting head is 1/2.

Conditional Probability

Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written as P(A|B), and read as 'the probability of A, given B'. It is defined by

P(A|B) = P(A ∩B)/P(B)

If P(B) = 0 then P(A|B) is undefined.

Important rule of percentage change in dimensional figure.

Note: If all the measuring sides of any two dimensional figure are changed (increased or decreased) by x%, then perimeter also changes by same x%.

Percentage change in area of two dimensional figure

If all the measuring sides of any two dimensional figure is changed by x%, then its area changes by (2x + x²/100) %

Eg. If all the sides of a hexagon is increased by 5%, then the % change in area is:

x = 5

(2x + x²/100) %

(2 x 5 + 5²/100) %

= 10.25%

Rule of H.C.F and L.C.M

Relationship between H.C.F and L.C.M of any two numbers:

1st no. x 2nd no = H.C.F x L.C.M

H.C.F of fractions = H.C.F Of numerator/L.C.M of denominator

L.C.M of fractions = L.C.M Of numerator/H.C.F of denominator

Formula of squares of numbers

If you have to find square of two digit number beginning with five.

Rule: write the square of last number and then the square of 5 + the last number.

54 ² = (25+4)16
= 2916

Calculate cube of any number

Rule: If you are asked to compute cube of any number whose square is known is known to you, then multiply the square with that number and obtain the answer. Multiplication should be done by easy method.

Eg. 13 ³ = 169 x 13
                                  = 2197

Short cut to solve two digit number questions

Formula: Difference in 2 digit of that number = Given difference/9

Note: In such questions it is better to proceed from back whenever feasible but if not so then only follow the conventional direct method.

Solving questions on two digit numbers

The general way of expressing any two digit no is 10 x+y (where y is unit place digit and x is tens place digit)

Question. The sum of 2 digits of a number is 9 if the digits are reversed the number is increased by 63. Find the number.

Solution: Let the digit at unit place be x, so the digit at tens place be 9-x
The number = 10(9-x)+x

According to the question:
[10x+(9-x)] - [10 (9-x) + x ] = 63
9x+9 - (90-10x+x) = 63
9x + 9 -90 + 9x = 63
18x - 81 = 63
18x = 144
x = 8

Avoid Careless Mistakes

Show Your Work!


Show every step of your work. Once you're in the habit of showing your work, the process will actually go faster. You'll spend less time re-reading questions and trying to juggle seven steps in your head. Instead, your pen will always be moving, and you'll make steady progress toward the answer.


When you write down all those steps, you'll be able to better isolate where you are making mistakes. Perhaps it's long division--in which case you should focus on incorporating some mental math tricks into your arsenal.