GMAT Math Questions | Interest Q20

Compound Interest | Equated Annual Installments | Word Problem

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?

  1. \\frac{(2.07)x}{(1.07)^{2}})
  2. \\frac{(2.07)x}{(1.07)^{3}})
  3. \\frac{(1.07^2+1)x}{(1.07)^{3}})
  4. \\frac{(2.07^2+1)x}{(1.07)^{3}})
  5. \\frac{x}{1.07}) + \\frac{x}{(1.07)^{2}})

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Explanatory Answer | GMAT Interest

Step 1 : Deduce the given data

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.

Step 2 : Compute the Present value of individual payment

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})

Step 3 : Compute the combined Present value of payments

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})

Choice B is the correct answer.

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