Question

76. Currently, your savings for each month is capped at $400. If you start investing all of this into a savings plan earning 1% interest annually, compounded monthly, then how long will it take to save $160,000? (Hint: Use logarithms.)

Answer 1

Ans. It wil take 346 months for you to save $160,000.

Explanation:

Hi, well, first, we need to find the equivalent effective rate (effective monthly) of 1% compounded monthly, that is, dividing it by 12.

`r(e.m)=(0.01)/(12) =0.000833`

That is 0.0833% effective monthly, but we need to use the decimal representation in order to solve this problem. Now, we need to solve for "n" the following equation.

`Future Value=(A((1+r)^(n) -1))/(r)`

Where:

Future Value= 160,000

A=400

r= 0.000833

The answer is going to be in months, since the annuity (periodic saving) is monthly and the rate is effective monthly. Let´s solve this equation.

`160,000=(400((1+0.000833)^(n) -1))/(0.000833)`

`(160,000*0.000833)/(400) =(1.000833)^(n)`

`0.3332=1.000833^(n) -1`

`0.3332+1=1.000833^(n)`

`1.3332=1.000833^(n)`

`Ln(1.3332)=n*Ln(1.000833)`

`n=(Ln(1.3332))/(Ln(1.000833)) =346`

So, it will take 346 months (nearly 29 years) for your savings account to reach $160,000.

Best of luck.

Answer 2

10 years and 10 months.

Explanation:

The annually interest rate (ia) can be converted by monthly (im) by the equation:

(1 + im)¹² = 1 + ia

(1 + im)¹² = 1 +0.01

(1 + im)¹² = 1.001 (putting ln in both sides)

ln(1 + im)¹² = ln1.001

12*ln(1 + im) = 1.0x10⁻³

ln(1 + im) = 8.33x10⁻⁵(applying "e in both sides)

`e^(ln(1 + im)) = e^{8.33x10^(-5)}`

1 + im = 1.00083

im = 0.00083 = 0.083%

For a investimenting, the final amount (A) can be calculated by:

`A = R*(((1+i)^n-1)/(i) )`

Where R is the amount invested per month, i is the interest, and n the number of months:

160000 = 400 *
`((1 +0.00083)^n - 1)/(0.00083)`

`((1 +0.00083)^n - 1)/(0.00083)` = 400

1.00083ⁿ - 1 = 0.332

1.00083ⁿ = 1.332 (applying ln in both sides)

n*ln1.00083 = ln1.332

8.3x10⁻⁴n = 0.2867

n = 345.4 months

345.4 months *1 yea12 months = 10 years and 10 months.