Respuesta :
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)
[tex]e^{ln(1 + im)} = e^{8.33x10^{-5}}[/tex]
1 + im = 1.00083
im = 0.00083 = 0.083%
For a investimenting, the final amount (A) can be calculated by:
[tex]A = R*(\frac{(1+i)^n-1}{i} )[/tex]
Where R is the amount invested per month, i is the interest, and n the number of months:
160000 = 400 *[tex]\frac{(1 +0.00083)^n - 1}{0.00083}[/tex]
[tex]\frac{(1 +0.00083)^n - 1}{0.00083}[/tex] = 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:
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.
[tex]r(e.m)=\frac{0.01}{12} =0.000833[/tex]
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.
[tex]Future Value=\frac{A((1+r)^{n} -1)}{r}[/tex]
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.
[tex]160,000=\frac{400((1+0.000833)^{n} -1)}{0.000833}[/tex]
[tex]\frac{160,000*0.000833}{400} =(1.000833)^{n}[/tex]
[tex]0.3332=1.000833^{n} -1[/tex]
[tex]0.3332+1=1.000833^{n}[/tex]
[tex]1.3332=1.000833^{n}[/tex]
[tex]Ln(1.3332)=n*Ln(1.000833)[/tex]
[tex]n=\frac{Ln(1.3332)}{Ln(1.000833)} =346[/tex]
So, it will take 346 months (nearly 29 years) for your savings account to reach $160,000.
Best of luck.
