Question

Tasha invests $5,000 annually at 6% and an additional $5,000 annually at 8%. Thomas invests $10,000 annually at 7%. Which statement accurately compares the two investments if interest is compounded annually?
Compound interest formula: mc020-1.jpg
t = years since initial deposit
n = number of times compunded per year
r = annual interest rate (as a decimal)
P = initial (principal) investment
V(t) = value of investment after t years
Each person will have exactly the same amount over time because each invested $10,000 at an average interest rate of 7%.
Tasha’s investment will yield more over many years because the amount invested at 8% causes the overall total to increase faster.
Thomas’s investment will yield more from the start because he has more money invested at the average percentage rate.
Tasha’s investment will be greater at first because she invested some at a higher rate, but Thomas’s investment will be greater over the long run.

Answer 1

Answer:

B is correct.

Step-by-step explanation:

Compound Interest Formula:

$V(t)=P(1+\frac{r}{n})^{nt}$

Tasha invests in two ways

Case 1:

• Principal P=$5,000
• Rate of Interest r=0.06
• Time t=t
• n=1

$V(t)=5000(1+0.06)^t$

$V(t)=5000(1.06)^t$

Case 2:

• Principal P=$5,000
• Rate of Interest r=0.08
• Time t=1
• n=1

$V(t)=5000(1+0.08)^t$

$V(t)=5000(1.08)^t$

Total amount yield by Tasha = $5000[(1.06)^t+(1.08)^t]$

Thomas Investment

• Principal P=$10,000
• Rate r= 0.07
• t=t
• n=1

Total amount yield by Thomas = $10000(1.07)^t$

Now we make table for Tasha and Thomas different value of t

    t                  Tasha                  Thomas

   1                  $10,700                 $10,700  

   2                 $11,450                  $11,449

   3                 $12,254                 $12,250

   4                 $13,115                  $13,108

In table we can see Tasha investment will yield more from Thomas.

Thus, Tasha’s investment will yield more over many years because the amount invested at 8% causes the overall total to increase faster.

Answer 2

Answer:

Option B is correct.

Step-by-step explanation:

We will compare the interest earned by both.

Tasha: p = $5000

r = 6% or 0.06

n = 1

So, Amount after a year will be = $5000(1+\frac{0.06}{1})^{1}$= $5300

And amount the next year with p = 5300: 5300*1.06= $5618

Additional $5000 at 8%

Here the amount will be = $5000(1+0.08)$=5400

Next year amount with p = 5400 : 5400*1.08 = $ 5832

Amount in total Tasha will have in 2 years = 5618+5832 = 11450

Thomas:

p = 10000

r = 7% or 0.07

n = 1

After a year the amount will be = $10000(1+\frac{0.07}{1})^{1}$=$10700

Amount Next year with p = 10700 : 10700*1.07 = $11449

*****Just after 1 year we can see that Tasha's total amount is high than Thomas. This means at the same consistent rate, each year Tasha's amount will always be higher than Thomas.

So, option B is correct. Tasha’s investment will yield more over many years because the amount invested at 8% causes the overall total to increase faster.