Question

The banker's discount on Rs. 1600 at 15% per annum is the same as true discount on Rs. 1680 for the same time and at the same rate. The time is:

Answer 1

The answer is 4 months.

Step by step solution:-
Method 1

Let Present Worth of Bill  be P.W.
True Discount  be TD
Banker's Discount be BD

S.I. on Rs. 1600 = T.D. on Rs. 1680.
Rs. 1600 is the P.W. of Rs. 1680,
=>Rs. 80 is on Rs. 1600 at 15%.
=>Time = 100x80 / 1600x15 = 1/3 yr = 4months


Method 2 :

Banker's Discount = BD
 Face Value 1 of bill (F1) = F1 = Rs. 1600.

True Discount = TD
Present Worth of Bill = PW
Face Value (F2) of P.W. = Rs. 1680.

According to the question,
BD = TD......1
As BD = F1*T*R /100.
=> BD = 1600*T*15/100.
=> BD = 16*15*T ...........2

We know that
PW = F2/{1+(T*R/100)}
=> PW = 1680/{1+(15*T/100)}...........3

We also know that,
 TD = PW*T*R /100.
=> TD = PW*T*15/100.
putting the value of PW in (3) in (4),we get;
=> TD = [1680/{1+(15*T/100)}]*15*T/100  ......4


Solving equation (1), (2) and (4),we get;

16*15*T = [1680/{1+(15*T/100)}]*15*T/100.

=> 16*15*T*100/(15*T) = [1680/{1+(15*T/100)}].

=> 16*100 = [1680/{1+(15*T/100)}].

=> {1+(15*T/100)} = 1680/1600.

=> (15*T/100) = (1680/1600)-1.

=> (15*T/100) = (1680-1600)/1600.

=> (15*T/100) = 80/1600 = 1/20.

=> T = 100/(15*20) = 1/3 years.

=> T = (1/3)*12 months.

=> T = 4 months.