A certain sum amount to $1400 in 2 yrs and $2000 in five years on cert

Question

A certain sum amount to $1400 in 2 yrs and $2000 in five years on certain fixed simple interest. In how many yrs would the amount be $1440 had it been put at the rate of interest compounded annually?

1. 4.5
2. 3.5
3. 3
4. 2
5. 1

KarishmaB · Accepted Answer

The sum amounts to 1400 in 2 yrs and 2000 in 5 yrs. It increases by 600 in 3 yrs. Since simple interest remains the same every year, SI = 200.
Then the original sum must have been 1400 - 400 = 1000 and rate of interest must have been 20%.

If compounded annually, a sum of 1000 at 20% will become a little more than 1200 in 1 year, more than 1400 in 2 yrs and more than 1600 in 3 yrs...
Answer must be 2 yrs.

Note: Even these calculations are excess. At SI, if sum is 1400 in 2 years, same sum at same rate of interest will become something more than 1400 in 2 yrs and more than 1600 in 3 years.
The only option that fits is 2 yrs.

soumanag · Answer

We can form equations as following:

Initial amount = P, Total amount after interest = T, Interest per year = I

T = P+I

Given T1 = P+2I = 1400 & T2 = P+ 5I = 2000

From this we get, P = 1000 & I = 200 $

Now we know that simple interest is  <  Compound interest ( CI = SI only after 1st year if CI is compounded annually)

consider simple interest - after 1st yer it is 200$ , after 2nd years it is 400$ and after 3rd year it is 600$.

To get a 440 $ compound interest the duration have to be > 1 year and < 3 years.

Only option fitting : Option D.

generis · Answer

Simple interest, SI, per year
Years (5-2) = (2,000-1,400) = $600 SI in 3 years
SI per year is $600/3 = $200

Original amount?
Year 2 - 2(200): 1,400- 2(200) = $1,000

Simple interest rate:  \frac{200}{1000} *100  = 20%

How many years for $1,000 to be $1,440 if interest rate were compounded annually?

First year only: simple interest = interest compounded annually

End Year 1: (1,000 + 200) = $1,200

End Year 2: (1,200 * 1.2) = $1,440 
OR
1,200+(.2)(1,200)=(1,200+240)= $1,440

Answer D

Alternatively,
  A = P(1 + r)^{t}  
P = principal, r = decimal interest rate, t = time

The number "$1,440," and compound interest rate "1.2" are big hints.

(1.2)^2 = 1.44 ; principal = $1,000

Mentally I just did
  $1,000(1.2)^2 = $1,000(1.44) = $1,440  
 
The long way:

$1,440 = 1,000(1 + .20)^{x} 
 $1,440 = 1,000(1.20)^{x} 
 \frac{1,440}{1,000} = (1.2)^{x} 
 1.44 = 1.2^{x} 
 (1.2)^1(1.2)^1 = 1.2^{x} 
 1 + 1 = x 
 x = 2

Answer D

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A certain sum amount to $1400 in 2 yrs and $2000 in five years on cert

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