40 120% of 33.33 = 33.33% of 120. a% of b = b% of a

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120% of 33.33?

120% of 33.33 = 33.33% of 120.

a% of b = b% of a

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Simple Interest = ?
Amount = ?

SI = nPR / 100

A = P + nPr% = P (1 + nR%) , after n years.

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Compound Interest = ?
Amount = ?

Here, n = 1 if compounded anually.

CI = A - P

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How to solve ratio questions?

Multiply or divide the ratio by some constant - ‘k’, and then you can assume their specific values.
Addition / Subtraction will change the ratio.

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a : b = 1 : 2 and b : c = 5 : 3. Find a : b : c.

Make b equal, by x2 and x5.
a : b : c
5 : 10 : 6

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Variation

*review it once

review

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About the work done questions, how to go about it?

If A does a piece of work in 20 days and B does the same work in 30 days, then how many
days will it take to complete the same work if both A and B do the work together ?

work done is proportional to people(m), rate(r) and time(t)

work = (k) * m * r * t

For example,
* take the work to be of 60 units (LCM or any multiple)
* rate of A is 3units/day and B is 2units/day
* Now, working together, they will do it in 5units/day
* So, 60/5 = 12 days.

Solve all questions using this concept.

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Average speed = ?

Average speed = total distance / total time

Never use speed, if speed is given then use that to find time or write time in the terms of speed.

But the formula is this ^.

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Set Theory (2 sets)

What is “at least 1”?
What is the total?

The region a + b + c is called (A or B) / Either A or B / A union B (A ∪ B) / at least 1.

=> a + b + c
=> A + B - (both ∪ A and B)
(where A & B are the actual numbers) and (a & b & c are the areas in the venn)

Total = none + at least 1 = n + A + B - ( both A and B )

Total = n + A + B - ( both A and B )

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In a survey of 200 people, 80 people read Magazine M and 96 people read magazine
R. If the number of people surveyed who do not read either magazine is 3 times the number who read both magazines, how many of the people surveyed read both the magazines?

Total = None + M + R - Both M and R
200 = 3*Both + 80 + 96 - Both
2 Both = 24
Both = 12

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Matrix method (Only for 2-Set)

(used in certain types of questions but very useful)

by example -
50% of the apartments in a certain building have windows and hardwood floors . 25%
of the apartments without windows have hardwood floors. If 40% of the apartments do not have hardwood floors , what percent of the apartments with windows have hardwood floors ?

Fill in the blanks as per the information provided.

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Set Theory (3 sets)
What is “at least 1” ?
What is “exactly 2” ?
What is total?

Total = none + at least 1 = n + (a + b + c + d + e + f + g)

At least 1 / (A or B or C) = (a + b + c + d + e + f + g)
=> A + B + C - (AandB) - (BandC) - (CandA) + (AandBandC)

Exactly 2 = (AandB) + (BandC) + (CandA) - (AandBandC)

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Each of the 59 members in a high school class is required to sign up for a minimum of
one and a maximum of three academic clubs. The three clubs to choose from are the poetry
club, the history club, and the writing club. A total of 22 students signed up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs ?

make sure to use the formula and also relate from the diagram, then you can derive a lot of formulas.

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If population increase by X% per annum, then after n year it will be?

Same for decrease too?

if increases,
P = p * [1 + (X/100)]^n

if decreases,
P = p * [1 - (X/100)]^n

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What is

1.08 ka square?
1.07 ka square?

1.1664
1.1449

Easy method ->

(a + b) ^ 2 = a^2 + 2ab + b^2
for 1.08
1 / 16 / 64

So, answer is 1.1664

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If a population increases by a factor of x, every t hours, what will be the value after T hours?

Like ->, initally 5, factor = 2, every 3 hours, in 15 hours?

Number of jumps (j) = Total time / time to multiply
Final = Initial * (factor)^j

J = 15/3 = 5
F = 5 * (2) ^5

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How to solve work related questions?

Take the LCM of time taken by all the given persons to finish the job alone, then you will be able to solve the problem without fractions.

But fundamentally,

If A alone takes X hours and B alone takes Y hours to do a piece of work, and if T is the total time taken when they work together, then we have: 1/X + 1/Y = 1/T or T = XY/(X + Y)
So, If A and B can do a piece of work in X & Y days respectively while working alone, they will together take XY / (X + Y) days to complete it.

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How to solve overlapping sets which consists of 2 sets, questions like -> male/female along with they are students/non-students?

Go with the matrix method, make the matrix for each of them and then put in the value.

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Average Speed Formula for two equal distances with different speeds:

2xy / (x + y)

where x is the first speed, and y is the second speed.

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Mean of combined series, what is the combined formula?

M = M1xN1 + M2xN2 / N1 + N2

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Sum of first N natural numbers

n ( n + 1) / 2

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Sum of squares of first N natural numbers

n(n+1)(2n+1) / 6

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The average of an odd number of consecutive integers is?
The average of an even number of consecutive integers is?

odd number -> integer
even number -> non-integer

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For consecutive terms / for an AP, what is the relation between mean and median?

Median = Mean = sum of first and last term / 2.

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Median of a continuous series?

What is range?

Range is defined as the difference between the two extreme observations of the distribution. Max - Min If range = 0, then all the numbers are equal, and also range can never be negative.

Standard Deviation?

SD = Square Root of (sum of (number - mean of all number) / N) It measures how much each value varies from the mean of all the values. If SD = 0, that means all the values are equal. Less SD means less variation, less spread, more compactness.

SD in relation to range?

SD <= Range / 2 SD is always smaller than range except when they both are 0 (when all the numbers are equal).

List of Prime Numbers Up to 100

n = 25 2, 3, 5, 7 11, 13, 17, 19 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

The number of factors of a given number of the type N = a^m * b^n * c^p... is ?

(m+1)x(n+1)x(p+1)....

LCM * HCF =?

LCM * HCF = Product of the two numbers

What is the formula for HCF of fractions?

HCF of numerators / LCM of denominators

What is the formula for LCM of fractions?

LCM of numerators / HCF of denominators

What is the divisibility rule of 2?

When the last digit is even or zero.

What is the divisibility rule of 3?

When the sum of digits is divisible by 3.

What is the divisibility rule of 4?

When the last 2 digits are divisible by 4.

What is the divisibility rule of 5?

When the unit digit is either five or zero.

What is the divisibility rule of 6?

When the number is divisible by both 2 and 3.

What is the divisibility rule of 7?

Double the last digit, subtract it from the remaining number; if the result is divisible by 7, then yes.

What is the divisibility rule of 8?

When the last three digits are divisible by 8.

What is the divisibility rule of 9?

When the sum of digits is divisible by 9.

What is the divisibility rule of 10?

The unit digit should be 0.

What is the divisibility rule of 11?

When the difference between the sums of digits in odd and even places is either 0 or a multiple of 11.

What is the divisibility rule of 12?

When the number is divisible by both 3 and 4.

What is the divisibility rule of 25?

When the number formed by the last two digits is divisible by 25.

What is the cyclicity of 2?

4 ## Footnote (2, 4, 8, 6)

What is the cyclicity of 3?

4 ## Footnote (3, 9, 7, 1)

What is the cyclicity of 4?

2 ## Footnote (4, 6)

What is the cyclicity of 5?

2 ## Footnote (5, 5)

What is the cyclicity of 6?

1 ## Footnote (Always remains 6)

What is the cyclicity of 7?

4 ## Footnote (7, 9, 3, 1)

What is the cyclicity of 8?

4 ## Footnote (8, 4, 2, 6)

What is the cyclicity of 9?

2 ## Footnote (9, 1)

Unit digit in powers -> Every digit has cyclitity of ?

4 The fifth power of any single digit number has the same right-hand digit as the number itself. So just divide your high powers by 4.

How do you convert a recurring decimal to a fraction?

Write the recurring figure once in the numerator and then write as many nines in the denominator as the number of repeating figures. ## Footnote Example: 0.666666666... = 6/9 Example: 0.61616161 = 61/99

How can you determine if a fraction results in a terminating decimal?

Express the fraction in the lowest form and then express the denominator in terms of Prime Factors. If the denominator contains powers of only 2 and 5, it is terminating. ## Footnote If the denominator contains any power of any other prime number, it is non-terminating.

Check whether (x + 1) is a factor of f(x) = 4x2 + 3x – 1.

Putting x + 1 = 0, i.e., x = –1 in the given expression we get f(–1) = 0. So, (x + 1) is a factor of f(x).

Remainder Theorem: If an expression f(x) is divided by (x – a), then the remainder is?

f(a)

Some properties of square numbers: (review)

* A square number always has odd number of factors. * A square number cannot end with 2, 3, 7, 8 or an odd number of zeroes. * Every square number is a multiple of 3, or exceeds a multiple of 3 by unity. * Every square number is a multiple of 4 or exceeds a multiple of 4 by unity. * If a square number ends in 9, the preceding digit is even.

Alligation Method (V.Imp) to calculate weighted average.

N1/N2 = (M2 - M) / (M - M1) Read the below image carefully

Find the number of terms in the following series : 1, 3, 5, 7, ...., 31

31 - 1 / 2 = 15 is the number of gaps So, n = 16 (gaps + 1)

Tell the factorial of 0, 1, 2, 3, 4

0! = 1 (Just remember) 1! = 1 2! = 2 3! = 6 4! = 24

Find the highest power of a prime number in a factorial. Find the highest power of 2 in 5!.

Step 1 : Divide 5 by 2 , Quotient 1 = 2 Step 2 : Divide the Quotient obtained in Step 1 by 2 , Quotient 2 = 1 Step 3 : Divide the Quotient obtained in Step 2 by 2 , Quotient 3 = 0 **(We should only stop this process when the final Quotient becomes 0)** So , Highest power of 2 in 5! will be ‘Sum of the Quotients’ obtained => 2 + 1 + 0 = 3

How many trailing 0s does 49! + 50! have ?

Relations between mean, median and mode?

3Median = 2Mean + Mode

What is 5! (5 factorial)?

120

What is 6! (6 factorial)?

720

What is 7! (7 factorial)?

5040

What is 8! (8 factorial)?

40320

What is 9! (9 factorial)?

362880

What is 10! (10 factorial)?

3628800

Sum of an AP =>

N/2 * (2a + (N-1) d)

Sum of an AP in terms of first and last terms.

a = first term l = last term S = n/2 (a + l)

x^3 - y^3 (x cube - y cube) = ?

(x-y) * (x^2 + xy + y^2)

What is the formula for x^3 + y^3?

(x + y) * (x^2 - xy + y^2)

What is the expansion of (x + y)^3?

x^3 + 3x^2y + 3xy^2 + y^3

What is the expansion of (x - y)^3?

x^3 - 3x^2y + 3xy^2 - y^3

f(x) = x Tell a few points about the graph (3)

- slope is 1. (45deg) - it passes through origin - it passes through 3rd and 1st quadrant.

Out of two linear equations, how would you find the values / coefficients so that they dont' have any solution?

They should be parallel, that means their slope will be equal or b1/a1 = b2/a2 and make sure that their constant terms are not equal.

Sum of roots of an quadratic equation -> ax^2 + bx + c?

-b/a

Product of an quadratic equation -> ax^2 + bx + c?

c/a

Roots of of an quadratic equation -> ax^2 + bx + c?

1st root = [-b + root(b^2 - 4ac) ] / 2a 2nd root = [-b - root(b^2 - 4ac) ] / 2a

Find the maximum and the minimum possible value of the expression (2x^2 - 12x + 24)

How to know if if an quadratic equation is always negative, positive or zero or any?

So, if b < a^2 / 4 , then always negative.

Distance between the points (x1, y1) and (x2, y2) is?

Area of a triangle whose vertices are - x,y ...

The points that divide the line joining two given points in the ratio m:n internally and externally are?

The coordinate of the mid-point of the line joining the points (x1, y1) and (x2, y2) | jo midpoint pe ho do points ke in the same line.

Centroid of a triangle whose vertices are (x, y, .....)

Slope of a line joining points -> (x1, y1) and (y1, y2) =?

(y2 - y1) / (x2 - x1) where x2 != x1. The slope is also called m.

If the slopes of two lines be m1 and m2, then the lines will be 1. parallel if =? 2. perpendicular if =?

parallel if m1 = m2 perpendicular if m1 * m2 = -1

Equation of a line -> y = mx + c, what are m and c?

m => slope c => y-intercept (where line cuts the y axis when x = 0.

Equation of line having slope=m and passing at (x1, y1) is?

y - y1 = m(x - x1).

The length of perpendicular of a given point to a given line is?

Equation of line when x-intercept and y-intercept are known?

x/a + y/b = 1. That's it. Can be used to solve various questions.

Sum of GP? (when -1 < r < 1)

S = a * (1 - r^n) / (1 - r) when r < 1, also a is the first term.

Sum of GP? (when r > 1)

S = a * (r^n - 1) / (r - 1) where a is the first term. This is also -> (nth term of GP * r) - a / (r - 1) Since nth term = a * r^(n-1)

Sum of GP? (when -1 < r < 1) and it's an infinite GP

S = a / (1 - r)

To make problems easier, what are the values that you should take for an AP.

for 3 terms -> a - d, a, a + d for 4 terms -> a - 3d, a - d, a + d, a + 3d

To make problems easier, what are the values that you should take for an GP.

for 3 terms -> a/r, a, ar for 4 terms -> a / r^3 , a/r, ar, ar^3

Sum of first n natural numbers?

n(n+1) / 2

Sum of squares of first n natural numbers?

n(n+1)(2n+1) / 6

Sum of cubes of first n natural numbers?

n^2(n+1)^2 / 4

In the equation, y = (x - h)^2 + k, what is the vertex of the graph?

The vertex of the graph (that is the lowest point) is (h, k).

If the axis of parabola created by an equation is y-axis (lies on y-axis), then?

Then, there is no x term. Means, only x^2 wala term hoga, no x wala term.

Identify the tenth, hundreth and thousandth digits in a decimal.

0.abc a = tenth b = hundreth c = thousandth

For a set of 'n' consecutive integers, the sum of integers is : when n is odd & when n is even? property hai

(a) always a multiple of 'n' when n is ODD (b) never a multiple of 'n' when n is EVEN

Distance formula in respect to acceleration

v^2 - u^2 = 2as s = ut + (half) at^2 s = vt - (half) at^2 2s = ut + vt | v = final speed u = initial speed t = time a = acceleration s = distance

Area of an EQUILATERAL triangle when height is given?

h^2 / root (3)

Area of an EQUILATERAL triangle when base is given?

root (3) * a ^2 / 4

Base / side of an EQUILATERAL triangle when height is given?

h = root (3) * a / 2

Number of ways of dividing 'n' identical objects into 'r' groups such that each group can contain any number of objects is given by

n+r-1 C r-1 matlab (n+r-1) combination (r-1) 5 apples among 4 people will be - => 5 + 4 - 1 C 4 - 1 => 8C3 = 56

What is the sum of all possible 7-digit numbers that can be constructed using the digits 1, 2, 3, 4, 5, 6 and 7, if each digit can be used only once in each number? A) 7! * 8888888 B) 3.5! * 8888888 C) 3.5!/2 * 88888888 D) 7!/2 * 8888888 E) 7 * 8888888 | V import concept.

Two important pricicples when adding 'k' to a fraction x/y, x > y x < y

when x > y, adding k to both num and deno reduces the fraction. when x < y, brings the fraction closer to 1, means increases the value.

𝑥 < 𝑦, both positive write down what happens when you add 'k' and when you subtract 'k' to both numerator and denominator, k > 0

adding k, x/y -> increases subtracting k, x/y -> descreases.

𝑥 > 𝑦, both positive write down what happens when you add 'k' and when you subtract 'k' to both numerator and denominator, k > 0

adding k, x/y -> decreases subtracting k -> increases

What is the decimal value of 1/1?

1.0

What is the decimal value of 1/2?

0.5

What is the decimal value of 1/3?

0.333...

What is the decimal value of 1/4?

0.25

What is the decimal value of 1/5?

0.2

What is the decimal value of 1/6?

0.1666...

What is the decimal value of 1/7?

0.142857...

What is the decimal value of 1/8?

0.125

What is the decimal value of 1/9?

0.1111

What is the decimal value of 1/10?

0.1

What is the decimal value of 1/11?

0.090909...

What is the decimal value of 1/12?

0.0833...

What is the decimal value of 1/13?

0.076923...

What is the decimal value of 1/14?

0.071428...

What is the decimal value of 1/15?

0.0666...

What is the decimal value of 1/16?

0.0625

What is the decimal value of 1/17?

0.058823...

What is the decimal value of 1/18?

0.055555...

What is the decimal value of 1/19?

0.052631...

What is the decimal value of 1/20?

0.05

x^2 = |x| means?

x = 0 or 1 or -1

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