- Mock and Flt
- English Zone
- Students Corner
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
|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
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).
How many divisors does 1200193 have and what is the sum of all those divisors?
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 = am * bn * co * ..... where a,b,c are prime.
thrn the number of divisors = (m+1)(n+1)(o+1).....
the sum of divisors = [(am+1 - 1)(bn+1 - 1)(co+1 - 1)......] / [(a-1)(b-1)(c-1).....]