answer the following (write your solution at the back of this page)\n6. find the interest if php987,650 is…

answer the following (write your solution at the back of this page)\n6. find the interest if php987,650 is deposited in a bank at 4.15% compounded monthly for 39 months.\n7. find the maturity value f given p = 36,960, t = 5.75 years, r = 8\\frac{3}{5}%, m = 1.\n8. if money is worth 6\\frac{4}{5}% compounded semi - annually, what is the present value (p) of php500,500 due at the end of 5 years and 3 months?\n9. how long will it take php7,350 to amount to php18,500 if invested in a bank at 8% compounded monthly?\n10. at what interest rate will php2,050 amounts to php3,875 in 4 years and 6months, if interest is compounded semi - annually?

answer the following (write your solution at the back of this page)\n6. find the interest if php987,650 is deposited in a bank at 4.15% compounded monthly for 39 months.\n7. find the maturity value f given p = 36,960, t = 5.75 years, r = 8\\frac{3}{5}%, m = 1.\n8. if money is worth 6\\frac{4}{5}% compounded semi - annually, what is the present value (p) of php500,500 due at the end of 5 years and 3 months?\n9. how long will it take php7,350 to amount to php18,500 if invested in a bank at 8% compounded monthly?\n10. at what interest rate will php2,050 amounts to php3,875 in 4 years and 6months, if interest is compounded semi - annually?

Answer

Explanation:

Step1: Recall compound - interest formula

The compound - interest formula is $A = P(1+\frac{r}{m})^{mt}$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (in decimal), $m$ is the number of compounding periods per year, and $t$ is the number of years. The interest $I=A - P$.

Question 6

  1. First, convert the annual interest rate to a decimal and find $m$ and $t$:
    • The annual interest rate $r = 4.15%=0.0415$. Since it is compounded monthly, $m = 12$. The time $t=\frac{39}{12}=3.25$ years.
    • Use the compound - interest formula $A = P(1+\frac{r}{m})^{mt}$:
      • Substitute $P = 987650$, $r = 0.0415$, $m = 12$, and $t = 3.25$ into the formula.
      • $\frac{r}{m}=\frac{0.0415}{12}\approx0.00345833$, and $mt=12\times3.25 = 39$.
      • $A=987650(1 + 0.00345833)^{39}$.
      • Calculate $(1 + 0.00345833)^{39}\approx1.14077$.
      • So, $A=987650\times1.14077\approx1126035.49$.
      • The interest $I=A - P=1126035.49-987650=\text{Php}138385.49$.

Question 7

  1. First, convert the mixed - number interest rate to a decimal:
    • $r = 8\frac{3}{5}%=\frac{43}{5}% = 8.6%=0.086$, $m = 1$, $P = 36960$, and $t = 5.75$ years.
    • Using the compound - interest formula $A = P(1+\frac{r}{m})^{mt}$:
      • Since $m = 1$, the formula simplifies to $A=P(1 + r)^{t}$.
      • Substitute the values: $A=36960(1 + 0.086)^{5.75}$.
      • Calculate $(1 + 0.086)^{5.75}\approx1.6077$.
      • $A=36960\times1.6077\approx59420.59$.

Question 8

  1. First, convert the mixed - number interest rate to a decimal and find $m$ and $t$:
    • $r = 6\frac{1}{5}%=\frac{31}{5}%=6.2% = 0.062$, $m = 2$ (semi - annual compounding). The time $t = 5+\frac{3}{12}=5.25$ years.
    • The compound - interest formula can be rewritten for present value as $P=\frac{A}{(1+\frac{r}{m})^{mt}}$.
    • $\frac{r}{m}=\frac{0.062}{2}=0.031$, and $mt=2\times5.25 = 10.5$.
    • $P=\frac{500500}{(1 + 0.031)^{10.5}}$.
    • Calculate $(1 + 0.031)^{10.5}\approx1.3777$.
    • $P=\frac{500500}{1.3777}\approx\text{Php}363289.17$.

Question 9

  1. First, use the compound - interest formula $A = P(1+\frac{r}{m})^{mt}$:
    • $P = 7350$, $A = 18500$, $r = 0.08$, $m = 12$.
    • Substitute into the formula: $18500=7350(1+\frac{0.08}{12})^{12t}$.
    • First, divide both sides by 7350: $\frac{18500}{7350}=(1+\frac{0.08}{12})^{12t}$.
    • $\frac{18500}{7350}\approx2.517$, and $1+\frac{0.08}{12}\approx1.00667$.
    • Take the natural logarithm of both sides: $\ln(2.517)=12t\ln(1.00667)$.
    • $\ln(2.517)\approx0.923$ and $\ln(1.00667)\approx0.00665$.
    • Then, $t=\frac{\ln(2.517)}{12\ln(1.00667)}$.
    • $12\ln(1.00667)\approx0.0798$.
    • $t=\frac{0.923}{0.0798}\approx11.57$ years.

Question 10

  1. First, convert the time to the number of compounding periods:
    • $t = 4.5$ years, $m = 2$ (semi - annual compounding), so $n=mt = 9$ compounding periods. Let the interest rate per period be $i$.
    • $P = 2050$, $A = 3875$.
    • Using the formula $A = P(1 + i)^{n}$, we have $3875=2050(1 + i)^{9}$.
    • Divide both sides by 2050: $\frac{3875}{2050}=(1 + i)^{9}$.
    • $\frac{3875}{2050}\approx1.89$.
    • Take the 9th root of both sides: $1 + i=\sqrt[9]{1.89}\approx1.075$.
    • The interest rate per period $i\approx0.075$.
    • The annual interest rate $r = 2i=15%$.

Answer:

  1. Interest: Php138385.49
  2. Maturity value: Php59420.59
  3. Present value: Php363289.17
  4. Time: Approximately 11.57 years
  5. Interest rate: 15%