A sum of money increased at the rate of per 6 months in a certain time. If the principal is Rs 4000 and compound interest is Rs 1324 annually. Find rate and time.
Answers
Gɪᴠᴇɴ :-
- Principal = Rs.4000
- Compound interest = Rs.1324.
- Interest is compounded Half yearly . (6 Months.)
ᴛᴏ ꜰɪɴᴅ :-
- Rate of interest & Time ? .
Sᴏʟᴜᴛɪᴏɴ :-
→ Amount = Principal + CI
→ Amount = 4000 + 1324
→ Amount = Rs.5324 .
Now, Let us Assume That, Rate is R% Per annum & Time is T years.
→ Amount = Principal[ 1 + (R/100) ]^T
→ 5324 = 4000 [ 1 + (R/100) ]^T
→ (5324/4000) = [ 1 + (R/100) ]^T
→ (1331/1000) = [ 1 + (R/100) ]^T
→ (1331/1000) = ( 100 + R)^T / 100^T
Multiply & Divide by 1000 the LHS now,
→ (1331000) / (1000000) = ( 100 + R )^T / 100^T
→ ( 100 + 10)³ / (100)³ = ( 100 + R )^T / 100^T
Comparing Now, we get,
→ Rate = 10% Per Annum.
→ Time = 3 Years.
But Since Rate of interest was Compounded Half Yearly,
Hence,
→ Actual Rate = 10 * 2 = 20% (Ans.)
→ Actual Time = (3/2) = 1(1/2) Years. (Ans.)
Note :-
When rate is compounded Half-Yearly :- We know that Half yearly Means Half of a year, when we Multiply it with 2 we will get a year .
So we can say that in this case :-
→ Rate is Divided by 2.
→ Time is Multiply by 2 . ( To make it a complete year).
____________________
ANSWER:
Given
- Principal = 4000₹
- Compound Interest (C.I) = 1324₹
- Rate of Interest is compounded 6 month
To find:-
- Rate of interest
- Time
SOLUTION:
★ Amount = Principal + (C.I.)
★ Amount = 4000 + 1324
★ Amount = 5324₹
.°. Amount = 5324₹
♦ Amount = Principal [1 + (R/100)]^T
Amount/Principal = [1 + (R/100)]^T
♦ 5324/4000 = [1 + (R/100)]^T
♦ 1331/1000 = [100 + R/100]^T
♦ 1331/1000 = (100 + R)^T/100^T
Multiplying & dividing with 1000 only with LHS
→ 1331000/1000000 = (100 + R)^T/100^T
→ (100 + 10)³/100³ = (100 + R)^T/100^T
By Comparison:
→ Rate of interest = 10% } Per ½ year
→ Time taken = 3 } Per ½ year
But rate of interest compounded for ½ years
Hence,
→ Actual Rate of interest : 2 × 10 = 20%
→ Actual time : 3/2 = 1 (1/2)
Hence , Interest = 20% , Time = 1(1/2) yrs