Question

compound interest:

two friends, ben and jerry, made investments of $500 each where they earned 5% interest. bens investment was simple interest and jerrys was compounded monthly. fill in the blanks below to find which investment is worth more in 25 years.

a) similarities:

p = ____, r = __, t = ____

differences: jerrys investment has n = ______ compounds per year.

b) in 25 years, bens investment is worth $ ______ which means he earned $ ______ in interest.

in 25 years, jerrys investment is worth $ ______ which means he earned $ ______ in interest. (round to the nearest cent)

blank 1:

blank 2:

blank 3:

blank 4:

blank 5:

blank 6:

blank 7:

blank 8:

Explanation:

Step1: Identify the values for \(P\), \(r\), \(t\) and \(n\)

• \(P\) (principal) is the initial investment. Both Ben and Jerry invested \(\$500\), so \(P = 500\).
• \(r\) (annual interest rate) is \(5\%=0.05\).
• \(t\) (time in years) is \(25\) years.
• For Jerry's investment (compounded monthly), \(n = 12\) (number of compounding periods per year).

Step2: Calculate Ben's investment (simple - interest formula \(A=P(1 + rt)\))

• \(A=500(1+0.05\times25)\)
• First, calculate \(0.05\times25 = 1.25\).
• Then \(1 + 1.25=2.25\).
• \(A = 500\times2.25=\$1125\).
• Interest earned \(I=A - P\), so \(I=1125 - 500=\$625\).

Step3: Calculate Jerry's investment (compound - interest formula \(A=P(1+\frac{r}{n})^{nt}\))

• \(A = 500(1+\frac{0.05}{12})^{12\times25}\)
• First, \(\frac{0.05}{12}\approx0.004167\).
• \(nt=12\times25 = 300\).
• \((1 + 0.004167)^{300}\approx3.50558\).
• \(A=500\times3.50558\approx\$1752.79\).
• Interest earned \(I=A - P\), so \(I=1752.79 - 500=\$1252.79\).

Answer:

a)

• Blank 1: \(500\)
• Blank 2: \(0.05\)
• Blank 3: \(25\)
• Blank 4: \(12\)

b)

• Blank 5: \(1125\)
• Blank 6: \(625\)
• Blank 7: \(1752.79\)
• Blank 8: \(1252.79\)