Miscellaneous Topics
Coin Problems
The key to these problems is to keep the quantity of coins distinct from the value of the coins. An example will illustrate.
Example : Laura has 20 coins consisting of quarters and dimes. If she has a total of $3.05, how many dimes does she have?
(A) 3 (B) 7 (C) 10 (D) 13 (E) 16
Let D stand for the number of dimes, and let Q stand for the number of quarters. Since the total number of coins in 20, we get D + Q = 20, or Q = 20 - D. Now, each dime is worth 10 cents, so the value of the dimes is 10D. Similarly, the value of the quarters is 25Q = 25(20 - D). Summarizing this information in a table yields
Dimes | Quarters | Total | |
Number | D | 20 - D | 20 |
Value | 10D | 25(20 - D) | 305 |
Notice that the total value entry in the table was converted from $3.05 to 305 cents. Adding up the value of the dimes and the quarters yields the following equation:
10D + 25(20 - D) = 305
10D + 500 - 25D = 305
-15D = -195
D = 13
Hence, there are 13 dimes, and the answer is (D).
Maximum Value of an expression
(A^x ) (B^y) (C^z) will be maximum when A/x = B/y = C/z
Ex (a+x)^3 (a-y)^4 will have the maximum value when
(a+x)/3 = (a-y)/4
i.e 4a+4x=3a-3y i.e a=3y+4x
Age Problems
Typically, in these problems, we start by letting x be a person's current age and then the person's age a years ago will be x - a and the person's age a years in future will be x + a. An example will illustrate.
Example : John is 20 years older than Steve. In 10 years, Steve's age will be half that of John's. What is Steve's age?
(A) 2 (B) 8 (C) 10 (D) 20 (E) 25
Steve's age is the most unknown quantity. So we let x = Steve's age and then x + 20 is John's age. Ten years from now, Steve and John's ages will be x + 10 and x + 30, respectively. Summarizing this information in a table yields
Age now | Age in 10 years | |
Steve | x | x + 10 |
John | x + 20 | x + 30 |
Since "in 10 years, Steve's age will be half that of John's," we get
(x + 30)/2 = x + 10
x + 30 = 2(x + 10)
x + 30 = 2x + 20
x = 10
Hence, Steve is 10 years old, and the answer is (C).
Prime Numbers
How many divisors does 1200193 have and what is the sum of all those divisors?
Ans: (From the post http://www.cat4mba.com/node/3533#comment-1255 - user rajorshi)
The big pain being to fing the prime factors of 1200193 which i wont take, who knows the number could be prime and iwould have wasted half a day.
let me give the formula generally :
if p = a^{m} * b^{n} * c^{o} * ..... where a,b,c are prime.
thrn the number of divisors = (m+1)(n+1)(o+1).....
the sum of divisors = [(a^{m+1} - 1)(b^{n+1} - 1)(c^{o+1} - 1)......] / [(a-1)(b-1)(c-1).....]