Cezarys and Filips weekly salaries have just been raised b

Question

Cezary’s and Filip’s weekly salaries have just been raised by 4% and 5%, respectively. Each of their new weekly salaries is a multiple of 100 dollars, and each former weekly salary was an integer number of dollars. If Filip’s new weekly salary is lower than Cezary’s, what is the smallest possible sum of their new weekly salaries?

(A) 3,400 dollars
(B) 4,200 dollars
(C) 4,500 dollars
(D) 4,700 dollars
(E) 5,500 dollars

(A) 3,400 dollars
(B) 4,200 dollars
(C) 4,500 dollars
(D) 4,700 dollars
(E) 5,500 dollars

stne · Answer

If Cezary’s increased salary is  100  then his old salary was not an interger , it was  96.15 
If Cezary’s increased salary is  200  then his old salary was not an interger, it was  192.30 
......
Checking like this,we find that the lowest new salary for Cezary’s is  1300  for which his old salary is an integer  1250 .

If Filip’s increased salary is  100  then his old salary was not as interger, it was  95.23 
If Filip’s increased salary is  200  then his old salary was not as integer, it was  190.47

Checking like this , we find that the lowest new salary for Filip’s is  2100  for which his old salary is an integer  2000

However the question says that " Filip’s new weekly salary is lower than Cezary’s "

Now we have to find the next new salary for Cezary’s for which is old salary is an integer .

Previous old and new salary of Cezary were  1250  and  1300 , let's just double and check.

Next old and new salary of Cezary  2500  and  2600,  ( Fairly certain there would be nothing in between,as these follow a pattern)

So if Cezary's new salary is  2600  it meets all the conditions, including Filip's new salary being lower than Cezary's.

Thus the smallest possible sum of their new weekly salaries =  2600 +2100 = 4700

Ans D

Hope it helped.

kevincan · Answer

Better with fractions !

stne · Answer

True Sir !
Confession - after establishing the logic - Used calculator here and there !! 
Please share your method. 
Thank you.

kevincan · Answer

4% = 1/25

If Cezary’s old salary was c , his new one is 26c/25

As his new salary is a multiple of 100, 26c/25 =100k for some positive integer k

C =2500k/26= 1250k/13

We know that c is an integer , so k must be a multiple of 13. The smallest possible value of c is 2500/2, in which case his new salary would be 1300.

SolarNote · Answer

Consider Cezary's old salary as "x" and Filip's old salary as "y"

Since we know Cezary's new salary > Filip's new salary, its a better idea to start from Filip's salary calculations.

Considering Filip, a 5 % increase can be represented as,

y * (1 + 5/100) = y'  , where y' is his new salary

so  y = (20/21)* y' .

From this equation for y and also from given data we know that y ' is a multiple of 100 and y is an integer, hence y' must be the LCM of 21 and 100 to be an integer as well as multiple of 100 .

LCM ( 21, 100) = 2100

so y' = 2100 and y = 2000

Now, to represent a 4% increase in Cezary's old salary, the new salary " x' "

x * (1 + 4/100) = x'

Therefore,  x = (25/26)*x'

From the given question, we know that x' is a multiple of 100 and from the above equation we also know it must be a multiple of 26, so x' must be LCM of 26 and 100 or its multiple,

LCM ( 26,100) = 1300.

But from the question we know x' > y' so we must consider the next multiple of 1300 i.e 1300 * 2 = 2600.

So y' = 2600 and y = 2500

1) So 2600 > 2100 i.e x' > y'
2) 2600 and 2100 are multiples of 100
3) 2500 and 2000 are integers.

Hence, all three given conditions are satisfied as well.

Final answer, sum of the lowest possible values of the weekly salary = 2600 + 2100 = 4700

Answer D

kevincan · Answer

Amazing first post ! Welcome

Cezarys and Filips weekly salaries have just been raised b

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