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Question

Find value of (99^99 - 98^98)/(99^98 + 98^98)? 1) 1 2) <1 3) >1 4) 0

Answer 1

You can either apply the a^n - b^n expansion and proceed OR : Try for smaller values like 4 and 3. If it holds true for 99 and 98, it should hold true for 4 and 3 respectively

Answer 2

Here n is constrained, hence write m in terms of n (the constrained variable). Now use the constraints to try out all possible values of n and find corresponding m. This way you only have to cycle through limited no. of possible values

Answer 3

Find equation whose roots are 2 more than the roots of the equation x^2 -6x +4 =0 Ans : Let the root of this equation be a and b. the new equation will have root x = a + 2. Using this a = x -2. We know a is a root of given equation. Hence put x -2 in the given equation to get required equation. Similarly if it was 2 less than a and b, we would have put a = x +2

Answer 4

1) Prime factorize x into a ^ b x c^d x e^f ... The no. of factors = (b+1)(d+1)(f+1).. NOTE : YOU CAN ONLY DO THIS IF YOU KNOW THAT a,c,e... are PRIMES. DON'T APPLY THIS IF ANY ONE OR MORE OF THEM CAN BE COMPOSITE. MADE AN ERROR ON DS QUESTION ASSUMING THEY ARE PRIME WHEN THEY WERE ACTUALLY NOT. 2) Odd, use the above formula

Answer 5

b - a + 1 irrespective of any scenario

Answer 6

Multiply the inner expression by (5-4) which is fine since it is equal to 1 to get ((5-4)(5+4)(5^2+4^2)(5^4+4^4)(5^8+4^8) + 4^16)^1/3 Now repeatedly apply (a+b)(a-b) formula

Answer 7

1 )Prime factorize 900 = 2^2 x 3^2 x 5^2 Ignore even terms and calculate no. of factors using only the odd terms : (2+1)(2+1) = 9 Odd factors = 9 2) Even factors = total no. of factors - odd factors

Answer 8

Prime factorize 100 : 2^2 * 5^2 | Sum of prime factors : (2^0 + 2^1 + 2^2) (5^0 + 5^1 + 5^2) (you go up till max power in prime factorization)

Answer 9

Instead of brute forcing, see that 5x and 55 have 5 in common. So, 8y = 5(11-x) y = 5(11-x)/8 Now, 8 cannot divide into 5, so 11-x has to be a multiple of 8. Given the constraints, x can only be 3 and so y = 5. In a similar vein : A store sells card a and card b. card a is for 5 and card b is for 7. a customer spent 60 buying at least 1 of each kind. What are the total no. of cards bought. Here, if you use data as is, you get 5a + 7b = 60 (with a and b >=1 )and you can take 5a to the RHS and proceed. HOWEVER, if you re-write it as a 5(a+1) + 7(b+1) = 60 where a and b can be >=0, you run into trouble as it simplifies to 5a + 7b = 12 and nothing is in common anymore. So, don't be oversmart.

Answer 10

When cancelling terms OR EXPRESSIONS, always make sure that they cannot be zero. Also, be on the lookup that multiplying by a negative number will change the inequalities up. This changes a little if it's across numerator and denominator. In that case, you can cancel even if you're not sure if it's zero or not AS LONG AS it's present in both the numerator and denominator.

Answer 11

If the product of two integers or two expressions of integers which doesn't convert them to non-integers is equal to 1 then either both are 1 or both are -1. NOTE : if one of the expressions is an even power, it can be both -1 and 1 as long as the remaining expression is 1. (Made silly mistake on the above)

Answer 12

1) undefined. 2) 0 is the only number equal to its OPPOSITE (0 = -0) 3) Yes 4) Yes 5) 0 is not a factor of any number EXCEPT itself NOTE : 'a' is a factor of 'b' if b = a x k (for some integer k) 6) 0 is an even number 7) Yes 8) Yes

Answer 13

1) Yes | 2) No, reason in (1)

Answer 14

Something that helps is factorizing it to a product of expressions since applying odd even logic to products is easier than applying it to sum/difference (for product, if one subexpression is even, the entire expression is even, for the entire expression to be odd, all the subexpressions have to be odd) Hence, write the given expression as n(m + n). Now is this is odd then n has to be odd and so does m +n. Hence, m has to be even. Use this to solve further.

Answer 15

The absolute value of a number is it's distance from 0 on the number line. Signed numbers can be positive or negative.

Answer 16

When a nonzero base is raised to an even exponent, the result will always be positive. When a nonzero base is raised to an odd exponent, it retains it's original sign. What this means is that, if we know the sign of the result of an expression with even exponent, we CANNOT find the sign of the original expression. But, if we know the sign of the result of an expression with an odd exponent, we can find out the sign of the original expression.

Answer 17

Number of prime factors and the number of unique prime factors refer to different things. For 32 = 2^2^2^2^2 (i.e. 2^5), the total number of prime factors is 5 while the total number of unique prime factors is 1 For 12, total = 3, unique = 2

Answer 18

For a set of positive integers, the LCM will always be greater than or equal to the largest number in the set and the GCF will always be less than or equal to the smallest number in the set. If y divides evenly into x, GCF(x, y) = y and LCM(x, y) = x To find GCF of 3 or more no.s : Here, for a prime factor to be repeated, it needs to be present IN ALL of the no.s. For the repeated root, find the lowest exponent and pick that. Do this for all the repeated roots and then multiply them all. GCF(a, b, c) = GCF(GCF(a, b), c) If the LCM of x and y is p and the GCF of x and y is q, then xy = pq. The LCM of a set of positive integers provides us with all the unique prime factors of the set. Thus, it provides all the unique prime factors of the product of the no.s in the set. It also tells us the max power of the prime factors in any of the set's no.s I.e if LCM of x, y and z is a^b x c^d x e^f. Then if c divides x, the max power of c in x will be <=d

Answer 19

'x is a dividend of y' means that y is a factor of x 'y divides into x evenly' : same meaning as above. If z is divisible by both x and y, z must also be divisible by the LCM of x and y

Answer 20

``` sale price should be divisible by 4. Reason : let n be the sale price n = 16/100 * x = 4/25 *x (25/4)*n = x Since we know that x is a whole number, n must be divisible by 4. ```

Answer 21

``` It is divisible by 2 and 3. If the last 2 digits are divisible by 4 If the last 3 digits are divisible by 8 If sum of all digits is divisible by 9 If sum of odd numbered place digits - sum of even numbered place digits is divisible by 11 (numbering starts with 1) If number is divisible by 4 and 3 ``` The divisibility rules apply EVEN IF THE NUMBERS ARE IN DECIMAL FORM

Answer 22

We first take the last digit and multiply it by 2. So,(9 × 2 = 18). Subtract 18 with the rest of the number, which is 45840. So, 45840 -18 = 45822. We are not sure if 45822 is a multiple of 7. We repeat the same process again with 45822. Multiply the last digit by 2. So, (2 × 2 = 4). Subtract 4 with the rest of the number, which is 4582. So, 4582 - 4 = 4578. We are not sure if 4578 is a multiple of 7. Let us repeat the process again with 4578. Multiply the last digit by 2. So, (8 × 2 = 16). Subtract 16 with the rest of the number, which is 457. So, 457 - 16 = 441. We are not sure if 441 is a multiple of 7. Let us repeat the process again with 441. Multiply the last digit by 2. So, (1 × 2 = 2). Subtract 2 with the rest of the number, which is 44. So, 44 - 2 = 42. 42 is the sixth multiple of 7. Therefore, we can confirm that 458409 is divisible by 7.

Answer 23

Product of ANY n consecutive even integers will always be divisible by (2^n) * n! No special rule for Odd consecutive integers. IMPORTANT : You can use these two rules in conjunction : If you know that you have 3 consecutive no.s and 2 of them are even. Then in addition to being divisible by 3! = 6, it will also be divisible by (2^2)*2! = 8

Answer 24

Super smart way : it can only be 30. If it was 15, it would've been 1, 3 and 5 also. It cannot be 3 and not 15/30 and so on. Normal way : n/15 = q + 2/15 : n = 15q + 2 n = 17z +2 i.e. 17z = 15q. Since q is an integer : 17z/15 is an integer. Hence, z is div by 15. I was getting stuck since I was thinking only z is an integer and writing z = 15q/17 which was leading me nowhere.

Answer 25

Because, all we know is x/y = 9.48 =9 + 48/100. Now we cannot say for certain that it is 48/100 (which means remainder =48 and y=100). It can also be 4800/10000 (r=4800 y=10000 which is an entirely different division altogether). It can also be 24/50. So, an infinite number of remainders are possible. All we know is that since the simplest form of 48/100 is 12/25, whatever the remainder is, it is divisible by 12. To find r, we need to know the value of x or y or both.

Answer 26

Remainders IN FRACTIONAL FORM can be multiplied but we need to correct for excess remainders in the end. Suppose we need to find remainder when 12 x 13 x 17 is divided by 5 : 12 = 2 2/5 13 = 2 3/5 17 = 3 2/5 ONLY the remainders can be multiplied to get 2*3*2 = 12. Fraction = 12/5 (Don't straight up multiply the fractions, you'll get 12/125 in that case. Only take the numerators) After correcting for excess remainder (remove the excess in the numerator by dividing by the denominator and removing multiples of the denominator until we get a proper fraction), the remainder is 2/5 (Check : 12*13*17 is 2652 which when divided by 5 gives remainder 2) If the result is negative (eg : 12*-13*17 => 2/5 , -3/5, 2/5) : -12 then correct for negative remainder. -2 should be corrected to 3. Remainder is 3.

Answer 27

REMAINDERS can be added or subtracted as well. We need to correct for excess remainder in the case of addition and negative remainder in the case of subtraction. (A remainder must be a non negative integer less than the divisor) Eg : Remainder when 12 + 13 + 17 is divided by 5 Remainders : 2/5 , 3/5 and 2/5 Adding them : 2 + 3 + 2 (as in the case of multiplication of remainders, only take the numerators) = 7. Fraction = 7/5 After correction : 2/5. Hence, remainder is 2 Eg : Remainder when 17 - 13 is divided by 5 Remainder = 2/5 and 3/5. Subtracting : -1. Fraction : -1/5 After correction (add denominator to numerator until it becomes positive) : 4/5 Hence, remainder is 4 Consider the timing payoff between simply adding the numbers and then finding remainder vs doing using these methods.

Answer 28

(5x2) pairs give us the number of ZEROES only. for the digits to the left of the zeroes, you need to see if there are any other factors left, then take their product and pad with the zeroes (from 5x2 pairs) to get final number. Else, USE 1 and pad 1 with the zeroes. DO THIS CAREFULLY

Answer 29

1) If X is an integer with k digits, and if X is NOT A perfect POWER of 10, then 1/X will will k-1 leading zeros Eg : 1/10001 will have 4 leading zeros in its decimal form. 2) If X is an integer with k digits, and if X is a perfect POWER of 10, then 1/X will have k-2 leading zeros.

Answer 30

To determine the largest number of a NON-PRIME NUMBER x that divides into y! (if the prime factorization of x = a x b x c .... where all of a, b, c ... have power 1) , we perform the following steps: 1) Break x into prime factors 2) Using the largest prime factor of x, apply the factorial divisibility shortcut to determine the quantity of that prime factor. The quantity determined represents the largest number of x that divides into y!

Answer 31

if 100!/8^n is an integer, what is the largest possible value of integer n. Expression : 100!/(2^3)^n. We determine the quantity of 2 in 100! = 97 Now we solve the inequality 3n <=97 (n has to be int) n = 32. Greatest possible value of n is 32

Answer 32

A perfect square OTHER THAN 0 and 1, is a number such that all of its prime factors have even exponents. A perfect cube OTHER THAN 0 and 1, is a number such that all of its prime factors have exponents that are divisible by 3

Answer 33

The GCF of two consecutive integers is always 1. When a whole number is divided by 10, the remainder will be the units digit of the dividend. When by 100, the last two digits of the dividend will be the remainder and so on. When integers with the same units digit are divided by 5, the remainder is constant. I.e remainder when 1{7} is divided by 5 is same as when 342332{7} is divided by 5 since both have the same units place.

Answer 34

If we know the GCF and the LCM of 2 no.s, that is NOT enough to find the 2 no.s, we can only find their product with this information. HCF won't even give you all the factors of the no.s, it'll only give you the factors that it has in common with the lowest possible power among them. If there are two fractions and both can be expressed as a terminating decimal, then their sum can be expressed as a terminating decimal. If there are two fractions and one can be expressed as a terminating decimal while the other can't, then their sum can't be expressed as a terminating decimal. It there are two fractions and both can't be expressed as terminating decimal, then their sum MAY BE a terminating decimal.

Answer 35

Very TRICKY if you solve normally, 500 factorizes to 5^3 * 2^2. Now you find the max power of 5 in 500 which comes to 124. NOW IF YOU SOLVE 3x <= 124, you get 41 and 41 x3 =123 WHICH IS WRONG Instead understand that the numerator is (500!)^3. So the inequality to solve is 3x <= 3*124 So, x is 124 HENCE, always find the final expression in numerator BEFORE solving the equality.

Answer 36

ALWAYS think of 1 as a factor when you factorize (especially during word problems)

Answer 37

Two consecutive integers cannot share the same factors since GCF = 1. So, 40! and 40! + 1 will not have any common factors except 1. And since we know 11, 19 and 37 divide 40!, they will NOT divide 40! + 1

Answer 38

If you want to find no. of multiples of x between y and z (both inclusive), find nearest multiple of x >=y and nearest multiple of x<=z let them be p and q respectively. Then, the no. of multiples of x is : ((q-p)/x) + 1 The only integers with exactly 3 factors are squares of prime no.s.

Answer 39

It's C because, if add the inequalities, you get a^3 < b^2 < a^2 solution interval for a : 0

Answer 40

Using 1) you can find that n is a multiple of 36. Now, there is a nuance to this. What you did was 'if a number is divisible by a (18) and b (12), it is divisible by its LCM'. Now, you can say that if a number is divisible by m and n, it must be divisible by the LCM of 36 and 30 but that's not what you want here. Had you been asked about the characteristics of a number divisible by m and n, you would use the LCM to draw conclusions. Here, you are straight up given the number which is mn. Hence, mn will be 36p x 30q = 1080pq. While it is obvious that it will be divisible by 180, it is also evident that it is divisible by 120. The point is don't just find the LCM of 36 and 30 and since that'll be 180 conclude that mn need not be divisible by 120. THAT IS WRONG. That conclusion would have been right if you had not been told that the no. being considered is mn. Since you know the number (mn), you can draw additional conclusions.

Answer 41

We know that for a number to be a perfect square, it has to be of the form n^2. Note that if n is a perfect square in itself, then n^k where k is >= 0 will always be a perfect square so you NEED NOT HAVE THE CONDITION THAT k SHOULD BE EVEN

Answer 42

In questions like this, put values quickly. x will be 2,5,8,11... y will be 4,11,18,25,32.... A and B are easy to eliminate since we won't know anything about x + y since each concern with only x or y C ) Now, x = 2 and 5 satisfy condition 1 and y = 4, 25 satisfies 2) x + y = 6, 9, 27, 30. Divided by 21, these leave diff remainders hence E. Using algebra to solve this would've taken ages.

Answer 43

NOT ALWAYS ONE, if s = k, GCF(s,k) = 1 MADE A SILLY MISTAKE ON THIS FFS ALWAYS THINK 0? EQUAL? INTEGER? NEGATIVE? FRACTION? OPPOSITE? ``` FZONE : Fraction (both +ve and -ve) Zero (one or more) Opposite Negative (one or more) Equal (both +ve and -ve) ```

Answer 44

The square root of a variable squared is equal to the absolute value of that variable which is ALWAYS positive except in the case of 0. (x^2)0.5 = |x|

Answer 45

``` 512 and 729 are perfect cubes of 8 and 9 respectively. Approx values : (2)^0.5 = 1.4 (3)^0.5 = 1.7 (values for 5 to 8 are in gaps of .2) (5)^0.5 = 2.2 (6)^0.5 = 2.4 (7)^0.5 = 2.6 (8)^0.5 = 2.8 (10)^0.5 = 3.15 ```

Answer 46

Find closest 4th power immediately less and greater than the given number and apply crude interpolation. 3^4 is 81, 2^4 is 16. (80)^0.5 must be very close to 3 but a smidge less (9)^0.25 : 1^4 =1 , 2^4 = 16. 9 must be somewhere in between tending towards 2, so 1.6 (actually 1.7)

Answer 47

Two or more radicals are like radicals if they both have the same root index and the same radicand (expression under the radical) A simplified fraction will never have a radical in the denominator.

Answer 48

If a =/= -1, 0 or 1, and a^x = a^y, then x = y. Also, if a =/= -1, 0 or 1, and a^x * a^y = a^z, then x + y = z x^a * y^a = (xy)^a If y =/= 0, x^a/y^a = (x/y)^a

Answer 49

How to compare : (3)^1/2, (4)^1/3, (5)^1/4, (7)^1/6? Compute LCM of powers : 2,3,4 and 6 => 12, raise them all to the 12th power : 3^6, 4^4, 5^3 and 7^2. Compare these no.s, and their order will give you the order of the initial terms How to compare : 3^100, 4^75, 5^50, 6^25 Compute GCF of the powers : 100, 75,50 and 25 => 25 Raise all the terms to their 1/GCF th power to get 3^4, 4^3, 5^2 and 6. Compare these no.s, and their order will give you the order of the initial terms.

Answer 50

If base is greater than 1 and exponent is a positive proper fraction => result is smaller If base is positive proper fraction and exponent is a positive proper fraction => result is larger

Answer 51

If a decimal with a finite number of decimal places is a perfect square, its square root will have EXACTLY HALF the number of decimal places. Thus, a perfect square decimal MUST HAVE AN EVEN NUMBER OF DECIMAL PLACES. NOTE : We're talking about decimal places NOT DIGITS here. Extrapolating this, the square of a decimal number will have TWICE AS MANY DECIMAL PLACES

Answer 52

The cube root of a perfect cube decimal has exactly one third the number of decimal paces as the original perfect cube. Extrapolating this, the cube of a decimal number will have 3 TIMES AS MANY DECIMAL PLACES

Answer 53

Square of 5 is 25. The number of decimal places is 6, so the square will have 12. So, fill enough zeroes so that the total no.s of digits is 12 including '25' so basically 10 zeros 0.000000000025

Answer 54

1) Estimate if the options are discernably apart | 2) Square it and simplify to get exact answer. (Rationalization is not always the only approach)

Answer 55

9^(-x) * 16^(-10y) = 18^-30 You factorize both LHS and RHS to common prime factors on both sides to get : 3^(-2x) * 2^(-40y) = 3^(-80) * 2(-40) NOW, you can equate -2x to -80 and -40y to -40 ONLY IF YOU KNOW THAT x and y are integers. Otherwise, we will have infinitely many solutions. For eg : set x to 0 and y will be approx 4.17 etc.

Answer 56

When the signs face the same direction, multiple inequalities can be added in the same way that we add multiple equations. A compound inequality, or a three part inequality, such as a<= x + 5 <= b can be manipulated in the same way as an equation can. Make sure that whatever you do to one part of the compound inequality, you do to all the parts. When a compound inequality is multiplied or divided by a negative number, both of the inequality signs must be reversed.

Answer 57

If two absolute values/equations are equal, it must be true that the expressions within the absolute value bars are either equal or opposite

Answer 58

| a - b | >= | a | - | b | They (RHS and LHS) can be equal only if : 1) the second quantity (b) is 0 OR 2) Both quantities are of the same sign and the absolute value of the first quantity is greater than or equal to the absolute value of the second quantity i.e. | a | >= | b |

Answer 59

If an absolute value equation has a variable on both sides of the equation, WE MUST CHECK the solutions of the equation for extraneous solutions (another case where you need to plug and check is when radical sign is involved). The extraneous solutions happen because for some solutions, the equation becomes of the form | ax + b | =-k where k is a +ve constant. the above equation form has no solutions. Hence, we need to check to prevent this. ``` Eg: | 2x + 3 | = x We solve as normal, 2x + 3 = x and -(2x + 3) = x x = -3 and x = -1 Plugging them back into the initial equation, we see that both do NOT satisfy the equation. This is because, we have modulus on one side and negative values on the RHS which will obviously not have any solutions. ```

Answer 60

a /(b+c) = b/(c+a) = a/(b+c) = r. What values can r take? When you put smart values, try more than 1 for each case to make sure that it holds for them all. Case 1 : r = 1/2 a = b = c = 1 1/2 = 1/2 = 1/2 = r = 1/2 Case 2 : r = -1 Now, solving using r = -1 gives you a + b+ c =0 What I did wrong : Take 1, 0 and -1 and see that it does not satisfy the initial equation hence -1 cannot be a value. What I did not notice : If I take -1,0 and -1, then r does not have a consistent value of -1 in the first place. What I could have done : Take another value 2, -1, -1 and see that it satisfies all the required equations.

Answer 61

If after solving an equality equation (any order) you get the roots are (1, 3) and (3,5) for each of the CASES respectively, consider 3 as the same root and not a repeated root. I.e. the sum of the roots will be 1 + 3 + 5 and NOT 1 + 3 + 3 +5. Similarly, product will be 1 * 3 *5 and NOT 1*3*3*5

Answer 62

If x > y > 0, is x > 3? 1) 2/x + 1/y = 11/15 2) 2x + 3y = 6 It's easy to see that 2) is enough For 1, we solve to get y = 15x/11x-30 and we put it into the given inequality (x>y) Now, here's the neat part, since it is given that x and y are positive, we know that y is positive and 15x is positive. So, 11x - 30 HAS TO BE positive. So, x > 15x/11x-30. Now, you can cancel x since you know it is positive and MULTIPLY by 11x-30 without altering the inequality sign SINCE YOU KNOW IT IS POSITIVE TOO. You get 11x - 30 > 15 x > 45/11 1) is enough too Answer is D

Answer 63

Tricky question : A certain airplane travels 26 miles per gallon of gasoline. if the plane carries 100 cubic feet of gasoline, how many miles can it travel. 1 gallon = 0.13 cubic feet. Note that the conversion factor is ALREADY IN CUBIC FEET. so (100 * 1/0.13 )*26 is the answer I was INCORRECTLY doing : (100 * (1/0.13)^3)*26

Answer 64

For age problems, start with a variable for present day age and then add/subtract from it as needed.

Answer 65

I did the following mistake : Morning : x/4 + 2 Afternoon : (2/5)*(x - x/4) THIS IS WRONG It should be : (2/5)*(x - x/4 - 2)

Answer 66

``` If you solve normally, you would get original fraction as 2/8 DO NOT SIMPLIFY this to 1/4 Reason, 2+7/8+7 = 9/15 = 3/5 BUT 1+7/4+7 = 8/11 =/= 3/5 VERY IMPORTANT ```

Answer 67

Simple Interest = principal x rate x time. Note : If rate 5%, write as 5/100. Convert rate and time to same terms. Month-month or year-year. Compound Interest : A = P(1 + r/n)^(nt) A :future value P : prinicpal r : ANNUAL INTEREST RATE (if 5%, write as 5/100) t : time IN YEARS n : NUMBER OF COMPOUNDING PERIODS PER YEAR

Answer 68

HERE 10% IS THE ANNUAL INTEREST. IT IS JUST THAT THE INTEREST IS BEING PAID SEMI ANNUALLY

Answer 69

Without knowing the growth driver/growth patter, we cannot find the nth element of a series. Eg : 2, 4, 6, 8, ? We CANNOT say 10 until we are told that this follows an evenly spaced pattern etc. Similarly, 200, 400, 800, ? We CANNOT say 1600 until WE ARE TOLD that there is exponential growth of factor 2 going on

Answer 70

A mixture matrix helps a lot in a wet mixture question. Eg : Alcohol solution 1 contains 60 percent alcohol and solution 2 contains 20 percent alcohol. If 1 and 2 are mixed together to produce 80 L of solution of 50 percent alcohol, how many L of 2 were used? Mixture matrix looks like this : | Concentration | volume | Weight of compound A | .6 | x | .6 * x B | .2 | (80-x) | .2 * (80 - x) Final | 0.5 | 80 | .5 * 80 Now solve the last column to get x The above can be extended to any number of mixtures.

Answer 71

If one is the mth person in a line counted from the beginning and nth person counted from the end, the number of people in the line is (m + n - 1) Be careful when 2 or more people are involved and you are not told who comes first. Eg : Adam and Ben are standing in a line, if Adam is 10th from beginning and Ben is 15th from the end and there are 5 people between Adam and Ben, there can be BOTH 30 AND 18 PEOPLE IN THE LINE DEPENDING ON WHO COMES FIRST BETWEEN Ben and ADAM i.e 3 B 5 A 8 = 18 or 9 A 5 B 14 = 30

Answer 72

What I did : 24 x (3/4)^n < 10 : (3/4)^n = 5/12 and got stuck Better solution : Calculate iteratively, manually. 1% increase : 24 - 1/4x24 = 18 2% increase : 18 - 1/4*18 = 13.5 3% increase : 13.5*3/4 = 10.125 4% increase will make it sub 10

Answer 73

In questions like these, you can decouple the individual payments/installments. 1st installment : d(1 + .08/2)^2 2nd installment : d(1 + .08/2)^1 The sum of the two installments will be total money at the end of the year. YOU NEED NOT DO : Find money at 6 months, add d and then calculate the final for that amount after another 6 months. This gives some complex numbers.

Answer 74

I mistakenly selected (B). The answer is (C) The reason is that there can be more than one person in between John and Maria even when Maria is in front of John while keeping the 5th position of both intact. Eg : | | M | J | | or in extreme case M | | | J I mistakenly thought only | | | M J | | | is possible else Maria and John would not be 5th from their respective ends. If John is before Maria, the arrangement is | | | | J x M | | | | where x can be any number of people including 0.

Answer 75

Answer is B) Easy to see that (1) alone is not sufficient In the case of 2, see that if the distance from A to B is 100 (can be anything for that matter). Then average speed is 200/{(100/y) + (100/50)} where y is the speed from A to B => 200/{100/y + 2} => Hence, no matter what y is, the fraction will be less than 100. Hence, (2) alone is sufficient

Answer 76

If it's y hours, then it is 3600y seconds. NOT y/3600 seconds *facepalm*

Answer 77

Useful rate equality : x workers/combined rate of x workers = y workers/ combined rate of y workers

Answer 78

let the time taken be f and f + 20. so, the rates are 1/f and 1/(f+20). further, 1/f + 1/(f+20) = 1/24 Simplify this to get : t^2 - 28t - 480 Here comes the beautiful part. Factorizing the above is time taking. Obviously one root will have to be negative and the other positive. Check the options, if you pick 12, it'll have to be +12 so in middle term split, -12x. So, the other root will have to be positive which won't work since we want sum as -28 pick 40, it'll have to -40x, the other root can be +12x which works perfectly, sum is -28x and product is -480. The other options will need a big positive term to get sum to -28x which will blow up the product term hence they won't work. Use this instead of time taking factorization

Answer 79

Answer is B) Important things: 1) Understand the rate variable you create. You write rate as n/x and n/(x+20). THIS MEANS that to mow n lawns, he needs x mins and x+20 mins respectively. SO, TO ANSWER THIS QUESTION, YOU JUST NEED x NOT n. 2) Further, when you get a quadratic equation like : x^2 - 28x -480 which is what you get here, note that this might give you ONLY 1 possible solution since we have a real variable like time for x which can only take positive values. Solving the above gives x = 40 and -12 and obviously only 40 can be an actual value for x.

Answer 80

Before putting smart values, become smart. See that you'll need to test more than 6 and fewer than 7 both. SO PICK VALUES BETWEEN THE TWO, THAT WAY YOU WON'T NEED TO CREATE MORE VALUE AGAIN AND AGAIN. Pick 6.1 and 6.9 for example.

Answer 81

We know that if we have a fraction a/b and we add k to both a and b, we move it closer to 1. For subtraction of k, think of it this way. We know that : 1) if a/b is less than 1, adding k to both num and denom makes it closer to 1 hence greater. So, a+k/b+k > a/b Observe that a/b is k subtracted from both num and denom of a+k/b+k and since we know that a/b is smaller than a+k/b+k so WE KNOW THAT SUBTRACTING k from both num and denom reduces the number. 2) if a/b is greater than 1, adding k to both num and denom makes it closer to 1 hence smaller. so a+k/b+k < a/b. Observe that a/b is k subtracted from both num and denom of a+k/b+k and since we know that a/b is larger than a+k/b+k so WE KNOW THAT SUBTRACTING k from both num and denom increases the number. NOTE : WE ASSUME THAT EVEN AFTER SUBTRACTING, THE NUMBER IS STILL POSITIVE

Answer 82

If the ratio of A to B is a to b and x is the ratio multiplier, then ax represents the actual amount of quantity A and bx represents the actual amount of quantity B

Answer 83

Realize that if you add 2, 3 and 7, you would get 12 CUPS of sauce by default. You WON'T get only 1 CUP of sauce by mixing multiple CUPS of sauce. Answer: (5/7)*12

Answer 84

Two equal ratios are known as proportion. | If a/b = c/d then we have a proportion and it must be true that ad = bc

Answer 85

Here, DO NOT ASSUME THAT MUFFINS CANNOT INCREASE OVER THE COURSE OF THE DAY. IT IS GIVEN THAT THEY ARE BAKED AND SOLD OVER THE DAY WHICH MEANS THEY CAN INCREASE IF MORE MUFFINS ARE BAKED THAN ARE SOLD. Made mistake by assuming the above FFS

Answer 86

Direct variation can be between higher order terms as well. If y varies directly with the square of x, we write y = k * x^2 where k is a constant. If x^2 [NOT x] increases or decreases by any factor, y will increase or decrease by the same factor

Answer 87

Inverse variation can be between higher order terms as well. If y varies inversely with the square of x, we write y = k / x^2 where k is a constant.

Answer 88

A single variable can vary directly with two other variables simultaneously. if y varies directly jointly with x and z then y = kzx It can also vary inversely with two, y = k/zx Language : a varies inversely jointly with the square of B and fourth root of c : a = k/(b^2 * c^(1/4)) v varies directly jointly with j and square of r. v = kj(r^2)

Answer 89

A pack has 12 A cards and 8 B cards. What is the lowest number of A cards that would have to be removed from the pack so that no more than 40 percent of the pack would be A cards. In such questions where you have terms like 'no more, less than, up to' etc. , it's always better to put in the corresponding inequality while solving. (12 - x)/(20 - x) [<=] 2/5 Solving this would automatically give you the correct inequality to pick the answer : 6.666 <= x The '<=' is what I'm talking about. Normally you would put equal to 2/5 and solve to get x = 6.66 and think whether x should be greater than 6.66 (7) or less than (6). This step is eliminated by solving in the way mentioned above.

Answer 90

Tricky (C) Cheerleader : coach :: 13x : x Cheerleader : teacher :: 8y : y It is clear that the number of cheerleaders would have to be a multiple of LCM of 13 and 8 which is 104. Hence, it is clear that there are more than 104 cheerleaders at HU Usually when given only two ratios, we cannot find the total number of entities. But here, we don't need the total, we just need to know if it is greater than 100

Answer 91

Start off with the usual amount of F, M and S being 10x, 5x and 2x Ratio of F to S is halved, (10x/2x) x 1/2 = 5x/2x Ratio of F to M is tripled, (10x/5x) x 3 = 6x/x DON'T CANCEL x IN THE PREVIOUS STEPS NOTICE THAT F is common to both the fractions/rations. Make the numerator/denominator corresponding to F same in both the ratios and combine to form 3 part ratio. You can also make the F num/denom equal to 1. This is what we'll do here. New ratio of F to M to S is 1 : 1/6 : 2/5 i.e. 30x 5x and 12x. Given, S is 72, x = 6 So, Total : 47 x 6 = 282

Answer 92

You can use the above method for stuff like '100 is what percent of 50' etc. => x/100 * 50 = 100, x will be the answer is percentage terms. The percent figure will always be larger than its equivalent decimal figure (5% > 5/100 since 5 > 0.05 :P )

Answer 93

3 percent greater than z = 103 percent of z 450 percent greater than z = (100+450)/100 *z = 5.5*z = 550 percent of z 100 percent greater than z = 2z = 200 percent of z 800 percent greater than z = 9z = 900 percent of z

Answer 94

I purchased 10 shares of a stock for x dollars each. I later sold the 10 shares for y dollars each and paid z percent tax on the profit. After tax, what was the profit I made? 1) y - x = 5000 2) If I had paid 10 percent more tax, my profit would have been 30000 dollars For 2) If the tax you paid before was z/100, now IT WOULD BE 1.1z/100 and NOT (z+10)/100 To make it (z+10)/100, the language would say something like 'percentage points/basis points of the tax were increased by 10%/were 10% greater'

Answer 95

Super Smart way What you need is 100x(1.4)^5 Realize that the power of 4 will follow cycle of 4,6,4... and the only option that satisfies this is 5)

Answer 96

Answer 97

Simplified overlapping set setup, Draw three horizonal lines, draw three vertical lines intersecting the 3 horizontal lines Also the table headers (A and B) need not be single character long, you can use full words or 2 or more characters also for clarity :P | A | A' | Total B | B' | Total |

Answer 98

For triple overlapping sets, we need 3 intersecting circle Venn diagram and a 4th circle depicting 'neither a nor b nor c' group. Total number of unique members : A only + B only + C only + A and B only + B and C only + C and A only + A and B and C + Neither A nor B nor C (last term v important) Total number of unique members : A + B + C - A and B only - B and C only - C and A only - 2(A and B and C) + Neither A nor B nor C (Easy to understand if you visualize) Pick which of the above two formulas to use based on the data given. Further, if you don't have the two pairs (combos of 2 of A, B and C) separately, you can group them into one group of 'exactly two of 3' or 'two of 3' based on which formula you use. NOTE : if you are not given how many are A and B etc. then subtract the triple overlap from the double overlap numbers to get only double overlap numbers and then proceed from there. Else, draw Venn diagram and proceed.

Answer 99

Notice that it is 'at least 500', take as 500 + x and solve, answer is 1200.

Answer 100

Ans : 10 NOTE : If it is given that 5 play guitar and sing, then IT INCLUDES THE PEOPLE WHO DO ALL THREE since it is not '5 ONLY play guitar and sing'

Answer 101

Very important Find the number of people who play all 3 instruments and subtract it from the total (120) so it's 105. The number of people who play neither of the 3 instruments here is 0.

Answer 102

A or B = A only + B only + both A and B A or B = A + B - (A and B) If you have trouble visualizing : A or B = (not A and not B)' A or B = Total - (not A and not B)

Answer 103

HACK : Make the diagonals as you go and not at the beginning. That way it gets easier when you want to apply both 1 and 2 together. If you're not able to understand, just shut up and do it.

Answer 104

Super tricky questions. If you use both S1 and S2 (stem 1 and stem2), you'll get the equation T - F = 80 (T : variable for total students and F variable for females). Now, I thought this is not enough to find male students and did E but if you look carefully, you'll see that T-F is the number of male students hence the answer is C. Be more alert!!!!

Answer 105

An evenly spaced set is one in which the numbers in the set increase or decrease by a fixed amount. The elements of an evenly spaced set share a common difference.

Answer 106

To count the number of consecutive integers in a set that includes the first and last numbers, use the formula : highest number - least number + 1 (works for all combos : 1) negative 2) some positive, some negative, 0 etc. as well). If only one of the endpoints is included, no. of consecutive integers : highest - lowest.. If both are excluded i.e. we need to find no. of consecutive integers BETWEEN two numbers : highest - lowest - 1

Answer 107

You can use simple AP formula to find number of multiples. An = A1 + (n-1)d Since the terms are 1-indexed, n is the total number of terms or in this case, multiples. A1 is the lowest number div by given number An is the highest number div by given number d is the number whose multiples we want to count

Answer 108

Average of an evenly spaced set of numbers. Method 1 : Add first (lowest) and last (highest) number and divide by 2 Method 2 : It is equal to the midpoint of the set i.e. median. (the set has to be sorted for this to work) Note that in case you have an odd number of terms in your evenly spaced set, the average exists in the set but if you an even number of terms, the average DOES NOT exist in the set.

Answer 109

The number of multiples of A or B but NOT BOTH = no. of multiples of A + no. of multiples of B - 2 x (number of multiples of LCM(A, B))

Answer 110

The weighted average of two points will be closer to the data point with the greater no. of observations or with the greater weighted percentage.

Answer 111

If a set of numbers has n terms where n is odd, the median is the value at the (n+1)/2 position when the numbers are in numerical order. If n is even, the median is the average of the numbers are n/2 and {(n/2) + 1} position In an evenly spaced set, the mean of the set is equal to the median of the set.

Answer 112

IMP : If each number in the data set occurs the same number of times as the others, then there is no mode.

Answer 113

Standard deviation measures how far the data points are from the mean of the data set. Range based on mean and no. of standard deviations apart the points are : High = mean + n*(sd) Low = mean - n*(sd)

Answer 114

If we multiply or divide the elements of a data set by a constant amount, the standard deviation will also be multiplied or divided by that amount.

Answer 115

IMPORTANT : To compare the standard deviation among data sets that have an equal number of data points, perform the following steps : 1) Determine the mean of each set 2) For each individual set, determine the abs difference between the mean of that set and each data point in the set 3) Sum the differences obtained from each individual set The set with the greater sum has the greater standard deviation.

Answer 116

IMPORTANT : Instead of brute forcing to find the mean, check if they are evenly spaced. If so, you have wayyy quicker methods to find the mean

Answer 117

If we are given that either the largest value or the smallest value in a data set is equal to the mean, all the data points in the set are the same, and thus the sd must be zero.

Answer 118

On the first even day of the month, Steve saves $5 and on each following even day, he saves $5 more than he saved the previous even day of the month. If there were 30 days in the certain month, how much money did Steve save by the end of the month? So on 2nd, he saves $5 and last day is 30th. Calculate the number of days : (30-2/2) + 1 = 15 Now, IMPORTANT : find average (DON'T FIND SUM OF FIRST AND LAST NUMBER AND FIND HOW MANY SUCH PAIRS EXISTS AND SUM THEM AND THEN ADD MIDDLE TERM, DONT DO THIS). Last term : 5 + (15-1)5 = 75 Average = 75+5/2 = 40 since evenly spaced Sum is simply : average x number of terms : 40*15 = 600 This is much simpler, using pairs and finding middle term etc. is much more cumbersome and error prone.

Answer 119

I always fumble in this. Understand the core funda Initial ratio : 7 : 9 : 4 Dollar value = 7x, 9x and 4x Now, the bonds to metals is cut in half : 7/4 * 1/2 = 7/8 metals to stonks : 4/9 * 2 = 8/9 Now, find what is common to the two ratios : metals is common. MAKE THE COEFFICIENT OF METALS (COMMON THING) TO BE THE SAME ACROSS BOTH THE RATIOS. here it is already the same (8). So, you can simply convert this to 3 part ratio : 7 : 9 : 8

Answer 120

Fundamental principle of counting : If there are m ways to perform task 1 and n ways to perform task 2 and the tasks are independent, then there are m x n ways to perform both the tasks together.

Answer 121

For rubies, it's 5C3. Now, for the remaining two jewels, you can make cases : 1 E and 1 D, 2 E 0 D or 0 E 2 D. But note that, you don't need to differentiate between the jewels as long as you're picking two. So, just do 7 (4 E + 3 D) C2 Saves a lot of time.

Answer 122

If you know that events are mutually exclusive and completely exhaustive, you can use 'total - something' to get required amount as long as 'something' is easier to compute that 'required amount'. This ends up saving a lot of time.

Answer 123

Realize that there are two possible high level arrangements : one starting with orange and one with apple : 1) OAOAOAOA 2) AOAOAOAO Ans : 1152

Answer 124

When the probabilities are not equiweighted, you can calculate the prob this way : Actual prob = no. of outcomes producing the event x prob of 1 outcome Eg : prob of rain on a given day : 2/5. What is the prob of rain for 3 out of 4 days? Now, if you calculate no. of possible success events and divide by total events, that will be WRONG. Instead, do this : no. Of success events = 4!/3! = 4 (arrangements of R R R NR) Prob of 1 success event = 2/5 x 2/5 x 2/5 x 3/5 Multiply : 4 x 24/625 = 96/625

Answer 125

Further, the previous concept can be extended to circular arrangements as well : The only thing that changes is in the numerator, instead of n!, we put (n-1)!. Denom will still be repeated characters' repetition counts.

Answer 126

b,d,f,x => 3! where x = (a,c,e). We do 3! and not 4! since it is circular arrangement Total : 5! (and NOT 6!) Prob = 3! x 3!/5!

Answer 127

Here, DO NOT USE prob that none of the three stand next to each other = 1 - the three stand next to each other. This will be wrong since we need to remove cases where only a and b are together but c is not together which DOES NOT get removed in the above equation. The correct equation would be : required prob = 1 - (a is next to b + b is next to c + c is next to a) - 2(a is next to b is next to c) INSTEAD, use the simpler A X B Y C approach : X can be filled in 2 ways and Y in 1. Total success : 2 x (3!) = 12 Total : 5! = 120 Final answer : 1/10 => Much easier

Answer 128

1 = P(A) + P(B) - P(A and B) + P(neither A nor B) To decide how to choose A and B, create a set matrix form and use the column header as A and row header as B or vice versa

Answer 129

Note that while 1) will change the answer, it is not necessary to answer on a standalone basis. 2) alone will allow you to answer (you will include Dec as well since only 1 tells you to ignore it) Hence, the answer is B and NOT C.

Answer 130

NOTE : To apply formula you need double overlap numbers on a pure double overlap sense and not including triple overlap areas. Hence, convert the double overlap to double overlap ONLY. bass and trout only = 16 - 7 bass and salmon only = 12 - 7 salmon and trout only = 18 - 7 Now, you can apply formula 100 = 41 + 44 + 46 - 9 - 5 - 11 - 2x7 + x solve for x to get 8

Answer 131

Wasted a lot of time on this idk why. Realize that guessing weight and height are independent. Prob of guessing height and weight correctly : 2/3 x 3/5 = 2/5 Prob of missing = 3/5 (This includes missing just one of weight/height and missing both) Hence, required prob = 1 - (3/5)^3

Answer 132

In these types of questions, always consider that you can change one of the two variables (beads) to cause a failure. 1) Trap choice: 1 - Prob(2 red beads) is NOT prob of not picking any red beads since we are not accounting for cases like picking 1 red and 1 white which is also not a success event. 2) Sufficient since this is what we want. Answer : B

Answer 133

Now you know that for a number to be a multiple of 4, the last 2 digits need to be div by 4. After you find the suitable combos satisfying this, DON'T FORGET TO MULTIPLY BY THE POSSIBLE CASES FOR THE HUNDRED'S AND THOUSAND'S PLACE SINCE THOSE TWO PLACES' DIGITS CAN BE ANYTHING SINCE WE ONLY CARE ABOUT THE TENS AND ONES PLACE DIGITS. this will increase the number of multiples. I FORGOT TO DO THIS AND MADE AN ERROR. A quicker way to do this would be realizing that only the last two digits matter. So final prob = suitable last two digits/total combos of LAST TWO DIGITS only. We can practically ignore the remaining 2 digits (thou and hundreds) since they will be common across num and denom.

Answer 134

My approach was give each face a name of A, B, C ,D so we want prob of A, B, C, D in any combo. So 4!/4^4 (4^4 since sides can repeat) Alternate neat solution which can be used for a surprisingly large number of questions is : Prob that a unmarked side comes up in 1st spin =1 in 2nd spin : 3/4 3rd spin : 1/2 4th spin : 1/4 Total : 1 x 3/4 x 1/2 x 1/4

Answer 135

Even though it seems like the selection is happening as a single event, we can still use the turn by turn approach as highlighted in previous card : For 1st R, prob = 44/100 For 2nd R, 43/99 Final : 44/100 x 43/99

Answer 136

An angle is the union of two rays sharing a common endpoint. The common endpoint is called the vertex of the angle. The two rays are called the sides of the angle. Obtuse angle : more than 90 but less than 180 Straight angle : angle whose measure is exactly 180 For an angle, the vertex is the middle of the three points and must be in the middle of the angle's designation.

Answer 137

An angle is the union of two rays sharing a common endpoint. The common endpoint is called the vertex of the angle. The two rays are called the sides of the angle. Obtuse angle : more than 90 but less than 180 Straight angle : angle whose measure is exactly 180 For an angle, the vertex is the middle of the three points and must be in the middle of the angle's designation. Diagrams are not necessarily drawn to scale on GMAT. Whenever a degree measure is used on the GMAT, it is accompanied by the degree symbol. If the symbol is not present then it not a measure, it is instead a name for that angle/angle measure. To distinguish a straight angle from a straight line, we put a semicircular arc on the straight line. We put a full circle to show a 360 rotation about end point.

Answer 138

When a transversal cuts a pair of parallel lines, ANY acute angle + ANY obtuse angle will sum to 180 degrees. Vertical angles are equal when a transversal cuts a pair of parallel lines. Vertical angles are angles that are diagonally oriented to each other. _a/b_ : a & d and b & c c/d

Answer 139

When two parallel lines are cut by a transversal, corresponding angles are equal. When a pair of parallel lines are cut by a NON PERPENDICULAR transversal, both obtuse and acute angles are necessarily created.

Answer 140

Given any polygon, when taking one exterior angle at each vertex, the sum of the measures of the exterior angles will always equal 360 degrees. VERY VERY IMPORTANT : Exterior angle is the angle created by ONE SIDE OF A TRIANGLE and the EXTENSION of AN ADJACENT SIDE. Any angle that follows this definition will sum up with all the other exterior angles chosen in any combo to sum up to 360 (as long as it is one angle per vertex). It doesn't matter if you pick the angles clockwise/anticlockwise/whatever. They will ALWAYS sum up to 360 as long as they SATISFY the definition of the exterior angle.

Answer 141

To determine the height of a triangle, we might need to draw ALTITUDE lines which may be inside/outside the triangle and perpendicular to the base/extension of the base of the triangle. The difference of lengths of any two sides of the triangle is LESS than the length of the third side. Even if either the sum or the difference property holds, we can draw a triangle. We do not need to satisfy both conditions (One implies the other anyway).

Answer 142

If a triangle with side A, B and C has C^2 = A^2 + B^2 then angle opposite side C must measure 90 degrees and thus the triangle must be a right triangle. 3-4-5 Right Triangle : If we know that the two smaller sides of a triangle are of the form 3x and 4x, then we know that it is a right triangle and the third side it the hypotenuse of length 5x. This is true for all x >= 1. 5-12-13 Right Triangle : If we know that the two smaller sides of a triangle are of the form 5x and 12x, then we know that it is a right triangle and the third side it the hypotenuse of length 13x. This is true for all x >= 1.

Answer 143

Area of a equilateral triangle : ((s^2) x (3)^0.5)/4 Dropping an altitude from upper vertex to the base of an equilateral triangle produced two identical 30-60-90 triangles.

Answer 144

Parallelogram : Opposite sides are equal in length, opposite angles are equal in measure, the diagonals bisect each other and each diagonal divides the parallelogram into 2 congruent triangles. Any two consecutive angles within a parallelogram are supplementary. Area : base x height Square : The diagonals of a square are perpendicular to each other (not the case in rectangles) VV Important : Given a rectangle with a fixed perimeter, the rectangle with the max area will be a square.

Answer 145

Tricky, I wrongly chose B If I choose B, i do not know the shape of X. If X is a rectangle then X cannot have an area greater than Y but if X is a circle, it can. MAKE SURE OF THE SHAPE BEFORE YOU APPLY LOGIC. IMPORTANT : A circle gives the maximum area for a given perimeter. So the triangle that gives the maximum area for a given perimeter is an equilateral triangle. The quadrilateral that gives the maximum area for a given perimeter is a square. The pentagon that gives the maximum area for a given perimeter is a regular pentagon. The n-gon that gives the maximum area for a given perimeter is a regular n-gon. Additionally, as the number of sides, n, grows in a regular n-gon, the area will increase for a given perimeter. Thus of a square and an equilateral triangle with the same perimeter, the square will have more area. But of a square and a regular pentagon with the same perimeter, the pentagon will have the larger area.

Answer 146

A trapezoid is a quadrilateral in which one pair of opposite sides called the bases are parallel (never of equal length) but the other pair of opposite sides are not parallel. If the two non parallel sides are equal in length, the trapezoid is referred to as an isosceles trapezoid. In that case, the angle they make with the bases are equal. Area of trapezoid = (base1 + base2)height/2

Answer 147

In a regular polygon, all the SIDES are equal in length along with all the ANGLES.

Answer 148

A chord is a line segment that connects ANY two points on the circle. If a chord passes through the center of the circle, it is called a diameter of the circle.

Answer 149

central angle/360 = arc length/circumference = area of sector/area of circle Very important : If points A and B are two points on a circle and arc AB is not a semicircle, arc AB refers to the shorter portion of the circumference between A and B. This shorter portion is also known as the minor arc. The longer portion of the circumference between A and B is known as the major arc but is usually referred as arc ACB, C representing some third point on this longer portion of the circumference.

Answer 150

A triangle is inscribed in a circle when all the three of the triangle's vertices lie on the circle's circumference. When a triangle is inscribed in a circle, if one side of the triangle is also the diameter of the circle, then the triangle is a right triangle with the 90 degree angle opposite the diameter. When an equilateral triangle is inscribed in a circle, the center of the triangle coincides with the center of the circle. A line segment drawn from the center of the triangle to a vertex of the triangle would bisect that interior angle. The triangle divides the circumference into three arcs of equal length.

Answer 151

When a rectangle is inscribed in a circle, a diagonal of the rectangle is also a diameter of the circle. When a circle is inscribed in a square, each point at which the circle touches the square is the midpoint of the side of the square (The above is when each side of the square is tangential to the circle and it touches the square at exactly 4 points and the circle is the largest circle that can fit inside the square) When a regular polygon is inscribed in a circle, the polygon divides the circle into arcs of equal length. So they subtend equal angles at the center equal to 360/n where n is the number of sides

Answer 152

When a rectangle is inscribed in a semicircle and a radius is drawn from one of the vertices of the rectangle to the center of the circle, a right triangle is created in which the base of the triangle is equal to half the side of the rectangle.

Answer 153

On the GMAT, cylinder = right circular cylinder. Other forms of cylinder are elliptical cylinder (base is not circular) and oblique cylinder (bases (top and bottom) are not aligned) Surface area of right circular cylinder : 2*pi*r*h + 2*pi*r^2 Volume of a right circular cylinder : pi*r^2*h

Answer 154

In inscribed figure questions, it is sometimes easier and more error proof to calculate the diameter rather than the radius Don't assume that the border is always of equal width. Inscribed angle : Angle formed by two chords of a circle with the vertex of the angle on the circumference. VERY VERY IMPORTANT : The degree measure of an inscribed angle is equal to half of the degree measure of the arc that it intercepts. This is very important. This concept is independent of the central angle principle. The Central Angle Theorem is always true unless the vertex of the inscribed angle lies on the minor arc instead of the major arc. In this case, the inscribed angle is the supplement of half the central angle. As a formula: angle = 180 −central angle/2 An extension of the above is : When an inscribed angle shares the same endpoints as the central angle, the degree measure of the central angle is twice the degree measure of the inscribed angle.

Answer 155

Just knowing the volume or the TOTAL SURFACE AREA of an object does not tell us anything about its dimensions/shape unless we know it is a regular object (in which case, all sides will be equal in length). Even if we know it's a rectangular cube (not regular cube), that still does not tell us anything about the dimensions. We can have wonky dimensions. I knew about volume but wasn't sure about total surface area

Answer 156

Lines with positive slopes always intersect quadrants 1 and 3. A positively sloped line intersects quadrant 4 only if its x intercept is positive and it intersects quadrant 2 only if x intercept is negative. If its x intercept is 0, then the line will intersect only quadrants 1 and 3. Lines with negative slopes always intersect quadrants 2 and 4. A negatively sloped line intersects quadrant 1 only if its x intercept is positive and it intersects quadrant 3 only if x intercept is negative. If its x intercept is 0, then the line will intersect only quadrants 2 and 4. We use x intercept to apply logic in the above and not y intercept since y intercept can be theoretically +inf/-inf. A horizontal line intersects quadrants 1 and 2 if the line's y intercept is positive and intersects quadrants 3 and 4 if the line's y intercept is negative. If the y-intercept of a horizontal line is zero, the line is coincident with the x axis and does not intersect any of the quadrants. A line with an undefined slope i.e. vertical line intersects quadrants 1 and 4 if the x intercept of the line is positive and intersects quadrants 2 and 3 if the x intercept of the line is negative. If the x intercept of a line with undefined slope is zero, the line is coincident with the y axis and DOES NOT INTERSECT ANY OF THE QUADRANTS

Answer 157

In y = mx + b, b is the y intercept. Standard/General form of equation of a line : Ax + By = C where A, B and C are all constants. You can define a line if you know a point on that line in addition to one of the following : 1) The slope of the line, or the slope of a line that is parallel or perpendicular to the line 2) A second point on the line Parallel lines have the same slope and different y intercepts and as a result, they never intersect. The slopes of two perpendicular lines are negative reciprocals, negative reciprocals multiply to -1

Answer 158

To reflect a line segment over the x axis, y axis or origin, reflect the endpoints of the line segment. Once we have the two new endpoints, we can draw a line segment connecting them, and this new line segment is the reflection of the original.

Answer 159

The reflection of (x, y) : Just memorize this, tough to do it logically under time constraints 1) over the line y = x is (y, x) 2) y = -x is (-y, -x) 3) y = b is (x, 2b -y) 4) x = a is (2a -x, y) 5) (x = a, y = b) is (2a-x, 2b-y) -> IMPORTANT

Answer 160

Before sketching inequalities, rewrite to get y isolated on LHS before proceeding. 'Greater than' inequality (y > mx + c) graph has a dashed line to signify that the points on the line DO NOT satisfy the equation. The shaded region lies ONLY ABOVE THE LINE. -> VERY IMPORTANT In 'less than' equality (y < 6x + 8), the shaded region lies ONLY BELOW THE LINE. 'Greater than or equal to' inequality (y >= mx + c) graph has a solid line to signify that the points on the line DO satisfy the equation. The shaded region lies ABOVE THE LINE and INCLUDES THE POINTS ON THE LINE.

Answer 161

If you have one point (x,y) and slope is for p/q and you are given that the points of a certain figure are integers then to try out points you can simply do (x + q, y + p), (x + 2q, y + 2p).... Equation of line is y = (-W/Z)*x + C and WZC =/= 0. If (a, b) is a point on this line, what is the value of b? 1) -W/Z = 5/2 2) y intercept is 4 Now even though by a,b it seems like you know the value of a and b, it is not so. If you're asked value, it cannot be in terms of 'a' or something else unless explicitly mentioned. So, E Similar example : if point (x, y) is on line L. what is the slope of L given that (8x, 6y) also lies on L. Ans : We cannot determine 5y/7x even though IT MAY SEEM that x and y are given as fixed values. IMPORTANT : Circle X has center at (10, 10) and diameter d. Circle Y has center at (5, -8) and radius r. Does Y intersect X? If one circle is inside the other, THAT DOES NOT MEAN THAT IT IS INTERSECTING. MADE A MISTAKE ON THIS FFS. INTERSECTION MEANS THERE HAS TO BE CROSSOVER. x^2 + y^2 =2 does NOT INTERSECT x^2 + y^2 = 4 Even if the radius of one circle is larger than the distance between the centers of the two circles (which is a good metric to use), is is possible that one circle has a radius so small that it is contained in the other and doesn't intersect.

Answer 162

Ratio of area of similar triangles = (side of triangle 1/side of triangle 2)^2 Use hero's formula to understand the derivation of the above

Answer 163

Every sequence has a rule or a formula that dictates how the sequence works. Without knowing this rule, we cannot make any conclusions about the value of any term in the sequence. If the first few terms of a sequence are given, but the rule is not given, we cannot determine any further terms in the sequence. Example : If x is the fifth term of some sequence an, what is the value of x? 1) The first four terms of an are 2,4,16 and 256 2) Each term of an from the 2nd term on is obtained by squaring the term preceding it. ANSWER IS 'C' even though the logic of the sequence is obvious on using 1), IT IS NOT 'A'

Answer 164

VERY IMPORTANT : If the formula of a sequence is given explicitly, we can find any term of the sequence without knowing the term or terms prior to it. However, if the formula is given recursively, we must know ALL the terms prior to it. ``` AP = an = a1 + (n-1)*d Sn = (a1 + an)n/2 ``` GP = an = a1 * r^(n-1) You can use some heuristic based methods to calculate mid elements i.e. if we have GP with terms as an then a4 will be (a2*a6)^0.5 if AP then a4 = (a2 + a6)/2

Answer 165

5) Know when to put values. THIS IS A SKILL I CONSTANTLY GET TRIPPED UP ON. And before putting values, think of an optimized strategy and then start. 6) In counting questions, take a second to check if it will be easier to count normally on a case by case basis or if it would quicker to do total - complement. Sometimes, counting misses out on edge cases which the latter approach captures. 7) In word problems, when factorizing a number, always note that 1 IS ALSO A FACTOR, say you're dividing 10 toffees among n kids such that each gets an equal number, the possibilities of the no. of toffees one gets is 5, 2 AND 10!!!! DON'T FORGET 10 which happens when there is only 1 kid. 8) In DS, before putting values, LOOK AT BOTH THE OPTIONS TO PICK VALUES SMARTLY. 9) When you have a uniform border, the length and breadth increase by 2x and NOT x.

Answer 166

In some questions, use mathematical solving for some parts and put values for the rest, THIS IS VVVVV HELPFUL. Could have saved tons of time on this : https://gmatclub.com/forum/if-d-0-and-0-1-c-d-1-which-of-the-following-must-be-true-168429.html This one straight up fucked me up : https://gmatclub.com/forum/the-table-above-shows-the-results-of-a-survey-of-100-voters-who-each-89187.html

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