Question

Nick bought a new motorbike for £14500 The value, £V, of Nick's motorbike at the end of n years is given by the formula V 14 500 x (0.88)" a) At the end of how many years was the value of Nick's motorbike first less than 50% of its original value? Optional working Answer: A savings account pays interest at a rate of R% per year. Nick invests £8 500 in the account for one year. At the end of the year, Nick pays tax on the interest at a rate of 30%. After paying tax, he gets £166.60 b) Work out the value of R. Optional working Answer: R= years (2) (3) Total marks: 5​

Answer 1

a) the answer is 6 (6 years to get below 50% of original value)

b) the answer is 2.8 (the value of R is 2.8 so interest rate is 2.8%)

Explanation:

a) We have to find the number of years n after which his bike's value is less than 50%

Now, Originally, his bike's value was,

£14500

and 50% of that is,

(0.5)(14500) = 7250 = V

Putting this value of V into the given formula and solving for n,

`7250 = 14500*(0.88)^n\\7250/14500 = 0.88^n\\1/2 = 0.88^n\\Taking \ the \ log \ on \ both\ sides,\\log(1/2) = log(0.88^n)\\log(1/2)=nlog(0.88)\\n = log(1/2)/log(0.88)\\which \ gives,\\n = 5.422`

Since we only look at the end of years,

so we round up to get,

6 years,

After 6 years, The value becomes less than 50% of the original

b) Work out the value of R

The invested amount = 8500

he invests for n = 1 year.

and pays 30% tax on the amount he gets due to interest.

Now, without tax, the amount he gets is,

A = (8500)(R%)

After paying 30% of A as tax, he gets £166.60

so, 70% of A is £166.60

or,

(0.7)A = 166

A = 238

Using this to find R, since

A = (8500)(R%)

238 = (8500)(R%)

238/8500 = R%

0.028 = R%

Hence multiplying by 100 on both sides to get R,

R = 2.8

The interest rate is 2.8%