0Hi,
In this lesson, I will demonstrate how to solve a tricky problem (asked in a non regular manner) using reverse methods.
Problem Statement:
2 Numbers x and y are in ratio 3:4 and there LCM is 48, Find the 2 numbers x and y.
Explanation:
This problem is different from a standard regular problem, which asks to calculate LCM of 2 numbers.
So, we will first start by solving a regular problem and see, how we can derive a formula to solve the reverse problem statement.
Regular problem is, Find LCM of 12 and 16.
We take factors of 12 = 2 x 2 x 3.
Then take factos of 16 = 2 x 2 x 2 x 2.
The LCM would be taking the longest occuring natural number, among all factors.
Here factor 2 occurs longest as 4 times, and factor 3 occurs longest once.
Hence LCM = 2 x 2 x 2 x 2 x 3 = 48.
Now, If we analyze the factors of 12 ( 2 x 2 x 3 ) and 16 ( 2 x 2 x 2 x 2 ), and take ratio of 12/16, then we remove the common factors (2 x 2) and are left with ( 3 / (2 x 2) ) = 3 / 4.
Doesn't this look like the original problem mentioned above, x and y are in ration 3:4 .... ?
So now, the common factors that we removed by taking ration of 12 and 16, lets call it 'k' where k = ( 2 x 2 ).
While taking LCM, we will calculate it as k x (remaining non common factors ) = k x (2 x 2 x 3).
hence, If LCM = 48 (LCM of x and y) then 48 = k x (2 x 2 x 3). Hence, k = 4.
Therefore, x / y = ( 3 x k ) / ( 4 x k). Which gives, x = 12, and y = 16.
This way through using the regular method of calculating LCM, we were able to derive a formula to calculate 2 numbers
whose ratio and LCM is already given.
x = k x (numerator of ratio)
y = k x (denominator of ratio)
