An amount of (P + 3000) is invested on C.I. at the rate (R + 2)% for two years. If total interest obtained on principal is 56 1/4 % then find the value of R?
Answers
Answer:23%
Given :-
An amount of (P + 3000) is invested on C.I. at the rate (R + 2)% for two years. If total interest obtained on principal is 56 1/4 % .
To find :-
Find the value of R?
Solution :-
Given that :
Principle = (P+3000)
Rate of interest = (R+2)%
Time = 2 years
Total Interest = 56 1/4 % of principle
We know that
A = P[1+(R/100)]^n
=> A = (P+3000) [1+{(R+2)/100}]²
=> A = (P+3000)[(100+R+2)/100]²
=> A = (P+3000)[(102+R)/100]²
=> A = (P+3000)(102+R)²/10000---(1)
We know that
Amount = Principle+Interest
=> Interest = Amount - Principle
[(P+3000)(102+R)²/10000]-(P+3000)
=> (P+3000)[{(102+R)²/10000}-1]
=> (P+3000)[(102+R)²-10000)/10000]
=> (P+3000)[(102+R)²-100²]/10000]
=> (P+3000)[(102+R+100)(102+R-100)]/10000
Since (a+b)(a-b)=a²-b²
Where a = (100+R) and b = 100
=>CI=(P+3000)(202+R)(2+R)/10000----(2)
According to the given problem
Compound Interest = 56 1/4 % of principle
=> CI = 56 1/4% of (P+3000)
=> CI = 225/4% of (P+3000)
=>CI = [(225/4)/100]×(P+3000)
=> CI = (225/400)×(P+3000)
=> CI = (9/16)×(P+3000)----(3)
(2)&(3) are same since they are CI
=> (P+3000)(202+R)(2+R)/10000 = (9/16)×(P+3000)
On cancelling (P+3000) both sides then
=> (202+R)(2+R)/10000 = 9/16
=> (202+R)(2+R) = (9/16)×10000
=> (202+R)(2+R) = 90000/16
=> (202+R)(2+R) =5625
=> (202+R)(2+R) = 9×625
=> (202+R)(2+R) = 9×25×25
=> (202+R)(2+R) = 225×25
=> (202+R)(2+R) = (202+23)(2+23)
On Comparing both sides then
We have
R = 23
Therefore, R = 23%
Or
(202+R)(2+R) =5625
=> 404+202R+2R +R² = 5625
=> 404+204R+R²=5625
=> R²+204R+404-5625=0
=> R²+204R-5221=0
=> R²-23R+227R-5221=0
=> R(R-23)+227(R-23)=0
=> (R-23)(R+227) = 0
=> R-23 = 0 or R+227 = 0
=> R=23 or R=-227
R can not be negative since it is a rate of interest
So, R = 23%
Answer:-
Rate of Interest for the given problem is 23%
Used formulae:-
- A = P[1+(R/100)]^n
- Amount = Principle + Interest
- (a+b)(a-b)=a²-b²
Given : An amount of (P + 3000) is invested on C.I. at the rate (R + 2)% for two years. total interest obtained on principal is 56 1/4 %
To find : the value of R
Solution:
amount of (P + 3000)
Let say P' = P + 3000
R' = R + 2
n = 2
Interest obtained on principal is 56 1/4 % = 225/4
=> A = P' ( 1 + 225/400) = P' ( 1 + 9/16) = P'(25/16)
A = P'(1 + R'/100)ⁿ
=> P'(25/16) = P'(1 + R'/100)²
=> 25/16 = (1 + R'/100)²
=> 5/4 = 1 + R'/100
=> 1/4 = R'/100
=> R' = 25
R' = R + 2
=> 25 = R + 2
=> R = 23
value of R is 23
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