You deposit $10,000 in an account that pays 4.5% interest compounded quarterly. Using the future value $10,457.65, what is the interest rate if simple interest was used? (I am struggling with formulas)​

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  4.5765%

Step-by-step explanation:

Apparently, you have to figure out how long it would take to accumulate a balance of $10,457.65 with interest compounded, then use that time period to compute the equivalent simple interest rate. The relevant formulas are ...

  A = P(1 +r/n)^(nt) . . . . amount of P at rate r compounded n times per year for t years

  A = P(1 +rt) . . . . amount of P at rate r simple interest for t years

We need to use the first formula to find t, then use the second formula to find r. In both cases A=10457.65 and P=10000. Quarterly compounding means n=4. The interest rate is r=0.045 for the first formula.

Looking for t:

  A/P = (1 +r/n)^(nt) . . . . . . . . . . divide by P

  log(A/P) = nt·log(1 +r/n) . . . . . take logs

  t = log(A/P)/(n·log(1 +r/n)) . . . divide by the coefficient of t

Filling in the given values, we find t to be ...

  t = log(10457.65/10000)/(4·log(1 +0.045/4)) = 0.999998 ≈ 1.0

The compound interest is applied for 1 year (t=1), so now we can find r using the second formula.

  A = P(1 +rt)

  A/P -1 = rt

Filling in the known values, we have ...

  1.045765 -1 = r = 0.045765 = 4.5765%

The equivalent simple interest rate on the deposit is 4.5765%.