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

Answer:

10 years and 10 months.

Step-by-step 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:

$A = R*(\frac{(1+i)^n-1}{i} )$

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

160000 = 400 *$\frac{(1 +0.00083)^n - 1}{0.00083}$

$\frac{(1 +0.00083)^n - 1}{0.00083}$ = 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.

Answer 2

Answer:

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

Step-by-step explanation:

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

$r(e.m)=\frac{0.01}{12} =0.000833$

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.

$Future Value=\frac{A((1+r)^{n} -1)}{r}$

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.

$160,000=\frac{400((1+0.000833)^{n} -1)}{0.000833}$

$\frac{160,000*0.000833}{400} =(1.000833)^{n}$

$0.3332=1.000833^{n} -1$

$0.3332+1=1.000833^{n}$

$1.3332=1.000833^{n}$

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

$n=\frac{Ln(1.3332)}{Ln(1.000833)} =346$

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

Best of luck.