1) What single payment today would replace a payment of $2,600 in 1 years and a payment of $5,700 in 5 years if the interest rate is 6

1) Computation of Single Payment Today:
A=P(1+r/400)^4n
where
A=future value
P=present value
r=rate of interest
n=time period.

2600=P1(1+0.0685/4)^(4*1)

P1=2600/(1+0.0685/4)^(4*1)

=(2600/1.07028)

=$2429.27

Also:

5700=P2(1+0.0685/4)^(4*5)

5700/(1+0.0685/4)^(4*5)=P2

P2=4,058.71

Hence single payment today =$2429.27+$4058.71 = $6,487.98 

2) Computation of Single payment 3 years from now:

Interest rate r = 4.32% compounded quarterly.

$2,170 was due 3 years ago and $650 is due today.

So, value of these dues after 3 years can be calculated using compounding formula

Single equivalent payment = 650*(1+0.0432/4)^(4*3) + 2,170*(1+0.0432/4)^(4*6) = $3,547.61

So, Single payment 3 years from now is $3,547.61

3) Computation of Borrowing Amount or Present Value:

Present value = Future Value / (1+r)^n

r = 0.0393 / 12 = 0.3275%

n = 3 years * 12 months = 36 months

Present Value = 57908.29 / (1+0.3275%)^36 = 51,477.93

4) Computation of Amount at the End of the Period:

Amount = Principal*(1+Rate)^Time

Here,

Principal = $69,000

Rate = 3.67%/4 

Time = 1 Year 6 Months or 4 quarters + 2 quarters = 6 quarters

Amount = $69,000*(1+(3.67%/4))^6 = $72,886.65

5) Computation of Payment required in 30 Months:

A payment of $1250 in 1 year and another $2,900 in 4 years to settle a loan are to be rescheduled with a payment of $1,050 in 18 months and the balance in 30 months

Interest rate = 4.5% compounded semiannually

So, PV of payment of $1250 in 1 year and another $2,000 in 4 years using 

PV = FV/(1+r/n)^(n*t) 

PV = 1250/(1+0.045/2)^(12*1) + 2900/(1+0.045/2)^(2*4) = $3,384.21

Comparing this with PV of rescheduled payments is

3,622.71 = 1050/(1+0.045/12)^18 + A/(1+0.045/12)^30

3,622.71 - 981.59 = A/(1+0.045/12)^30

2,641.13 * (1+0.045/12)^30 = A

A = $2954.99

So, Payment required in 30 months is $2688.14

6) Computation of Amount of Investment or Principal:

Amount = Principal*(1+Rate)^Time

$415,000 = Principal*(1+4.77%/12)^(12*13)

$415,000/1.85683 = Principal 
Principal = $223,499.17

So, She need to invest $223,499.17 today.