The given question is a GMAT problem solving question from the topic Interest. It tests the concept of finding present value of future payments. A medium difficulty, GMAT 650 level sample question in interests.

Question 20: John is set to receive two equated annual payments of $x each. He will receive the first of his payments two years from today. Which of the following expressions provides the present value of the two payments if John uses 7% p.a. rate to compute present value?

- \\frac{(2.07)x}{(1.07)^{2}})
- \\frac{(2.07)x}{(1.07)^{3}})
- \\frac{(1.07^2+1)x}{(1.07)^{3}})
- \\frac{(2.07^2+1)x}{(1.07)^{3}})
- \\frac{x}{1.07}) + \\frac{x}{(1.07)^{2}})

From INR

John is set to receive two equated annual payments of $x each. The first one, two years from now. The second payment, three years from now.

Each of the **Annual payments will be $x**

Rate of interest, r = 7% p.a = \\frac{7}{100}) = **0.07**

**Investment Period:**

John will receive the first payment of $x after two years.

Let the present value of first payment be $y.

John will receive the second payment of same $x one year after the first payment.

Let the present value of the second payment by $z.

**Formula for the future value:** FV = PV(1 + r)^{n}

PV is the present value; FV is the future value; r is the rate of interest and n is the number of years.

**Present value of the First payment:**

FV = x; PV of the first payment = y; r = 0.07; n = 2

Therefore, y(1 + 0.07)^{2} = x

Or y = \\frac{x}{(1 + 0.07)^2})

y = \\frac{x}{(1.07)^2})

**Present value of the Second payment:**

FV = x; PV of second payment = z; r = 0.07; n = 3

Therefore, z(1+ 0.07)^{3} = x

Or z = \\frac{x}{(1 + 0.07)^3})

z = \\frac{x}{(1.07)^3})

**Present value of the two payments:**

PV = y + z

PV = \\frac{x}{(1.07)^2}) + \\frac{x}{(1.07)^3})

PV = \\frac{1.07x}{(1.07)^3}) + \\frac{x}{(1.07)^3})

Present value of two payments = **\\frac{2.07x}{(1.07)^3}) **

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