Problem based on Direct and Inverse Variation

35 cattle can graze on a field for 18 days. After 10 days, 15 cattle are move to a different field. For how long can the remaining cattle graze on the field? b) 14.5 days c) 14 days • d) 12 days a) 15 days ​ 

Answer:

If one increases the number of cattle, then the number of days will decrease to graze the same field.

If we increase the number of days, then less cattle will be needed to graze the same field.

So the variables, cattle and number of days are inversely proportional.

Now, if we consider the variables cattle and field size ( keeping the number of days fixed), it is easy to see that they are directly proportional. ( as if you increase the field size, then the number of cattle must be increased to graze in the same day and vice versa)

Let the corresponding quantities for cattle, days and field size be C, D and F.

Then C ∝  ......(i)

⇒C = k , where k is some non-zero constant.

For the initial data, C= 35 and F = 1 ( taking field size as 1 unit ) and D = 18

∴

Now in 1 day, 35 cattle will graze

unit of the field.

Thus in 10 days, they will graze,

unit of the field.

Thus remaining unit of field to be grazed

As 15 cattle were removed, remaining cattle

So the question, boils down to finding the number of days (D) in which 20 cattle (C) will graze

unit (F) of the field.

From (i), C = k

⇒ 

⇒   days

(c) is the correct option

Bayes' Theorem Problem from ISC 2023 Maths Paper

In a company, 15% of the employees are graduates and 85% of the employees are non-graduates. as per the annual report of the company, 80% of the graduate employees and 10% of the non-graduate employees are in the administrative positions. find the probability that an employee selected at random from those working in administrative positions will be a graduate. 

Answer:

Step-by-step explanation:

Let G be the event that the selected employee is a graduate, and NG be the event that the selected employee is non-graduate.

Clearly, G and NG forms a mutually exclusive and exhaustive set of events.

Further, let A be the event that the selected employee works in administrative office.

According to the problem, we have to find, the selected employee is a graduate given he/she works in the administrative position P(G/A).

By Bayes' theorem,

Now, Probability of an employee to be graduate =

Probability of an employee to be non-graduate =

Probability of an employee working in administrative office given he is a graduate =

( as 80% of the graduate employee works in the administrative positions)

Probability of an employee working in administrative office given he is a non-graduate =

( as 10% of the non-graduate employee works in the administrative positions)

Substituting the values in

, we get,

Prove that cos pi/11 + cos 3pi/11 + cos 5pi/11 + cos 7pi/11 + cos 9pi/11...

A elegant solution for a difficult trigonometric problem using complex numbers. Many problem of trigonometry and even algebra can be beautifully solved using complex numbers. The use of different formulas from trigonometry can be avoided. #CBSE #ISC #IIT #ISI #wbjee #wbchse #jeeadvanced Prove that cos pi/11 + cos 3pi/11 + cos 5pi/11 + cos 7pi/11 + cos 9pi/11 =1/2