Question

you are prepared to make monthly payments of $250, at the beginning of the month, into an account that pays 8% interest compounded monthly. how many payments will you have made when your account balance reaches $50,000 (rounded)? a 127 b

Answer 1

If monthly installments of $250 are made, 127 payments will be required when the account balance reaches $50,000.

Given in the question:

Monthly payments made: $250.

Monthly compounded interest: 8%

Future account balance: $50,000

Since the payments begin at the end of the month, the formula for calculating the Future Value (FV) of an Ordinary Annuity is used as follows:

$FV = M * { \frac{[(1 + r)^{n} -1] }{r} }$     ..........................eq. (1)

Where,

FV = Future value of the amount = $50,000

M = Annuity payment = $250

r = Monthly interest rate = 8% ÷ 12 = 0.67% or 0.0067

n = number of periods the investment will be made = n

Substituting the values into equation (1), we have:

$FV = M * { \frac{[(1 \;+\; r)^{n} -1] }{r} }$

$50,000 = 250 * { \frac{[(1 \;+\; 0.0067)^{n} -1] }{0.0067} }$

$\frac{50,000}{250} = { \frac{[(1.0067)^{n} -1] }{0.0067} }$

$200 = { \frac{[(1.0067)^{n} -1] }{0.0067} }$

$200 * 0.0067 = (1.0067)^{n} -1$

$1.34 = (1.0067)^{n} -1$

$1.34 + 1 = (1.0067)^{n}$

$2.34 = (1.0067)^{n}$

By log-linearizing the above, we have:

ln(2.34) = n * ln(1.0067)

0.85015 = n * 0.00667

n = 0.85015 / 0.00667

n = 127.46, or 127 months approximately

Therefore, the number of payments to make is approximately 127 payments.

Visit the link below to learn more about the logarithmic equation:

brainly.com/question/7993920

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