Answer:
The initial amount invested into the account on 1st January 2014 was £23,360
Step-by-step explanation:
In order to calculate the initial deposit in question, we will have to make use of the formula for calculating compound interest:
Fv = Pv × (1 + r)^t ------ since it was compounded annually.
Where Fv = future value
Pv = present value
r = interest rate
t = time (years)
Here, r = 2.5% or 0.025
t = 1 year
Let x represent this initial deposit. Then at the end of the year(31st Dec 2014), we will have:
Fv = x × (1 + 0.025)^1
Fv = x × (1.025)
Fv = £1.025x
Therefore, the amount that will be in the account as at 31st Dec 2014 is = £1.025x.
Now, if the owner of the account then withdraws a sum of £1,000 from the account on the 1st of Jan 2015, then the balance that will be in the account after the withdrawal will be = £1.025x - 1000. This will now be the principal or present value that will accrue interest for the year - 2015.
Again, we were told that that the future value or the balance on the 1st of Jan 2016 will be £23,517. We will then once again use the compound interest formula, making the present value (1.025x - 1000) the subject of the formula so that we can solve for x. Once "x" is determined, then we have exposed the initial deposit that was made into the account on the 1st of January 2014.
Fv = Pv × (1 + r)^t
Here,
Fv = 23,517
Pv = 1.025x - 1000
r = 0.025
t = 1
23,517 = 1.025x - 1000 × (1 + 0.025)^1
23,517 = 1.025x -1000 × (1.025)
23,517 = 1.050625x - 1025
1.050625x = 23,517 + 1025
1.050625x = 24,542
Then making "x" the subject of the formula:
x = 24,542/1.050625
x = £23,360
Therefore, the initial deposit that Carole made into the account on 1st January 2014 was £23,360
