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Engineering
Grade 11-12
Engineering Economics & TVM Reference Cheat Sheet
A printable reference covering time value of money, interest factors, present worth, annual worth, future worth, and benefit-cost analysis for grades 11-12.
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Engineering economics helps engineers compare design choices using money, time, and risk. This cheat sheet covers the time value of money formulas used to move cash flows to the present, future, or equal annual values. Students need these tools to evaluate projects, equipment purchases, loans, savings plans, and replacement decisions. The focus is on clear formulas, correct notation, and choosing the right method for a cash flow diagram.
Key Facts
- Future value of one present amount is F = P(1 + i)^n, where P is present value, i is interest rate per period, and n is number of periods.
- Present value of one future amount is P = F / (1 + i)^n.
- Future value of a uniform series is F = A[((1 + i)^n - 1) / i].
- Present value of a uniform series is P = A[((1 + i)^n - 1) / (i(1 + i)^n)].
- Capital recovery converts present cost to annual cost using A = P[i(1 + i)^n / ((1 + i)^n - 1)].
- The effective annual interest rate for m compounding periods per year is i_eff = (1 + r/m)^m - 1.
- Net present worth is NPW = present value of benefits - present value of costs, and a project is acceptable when NPW >= 0.
- Benefit-cost ratio is B/C = present value of benefits / present value of costs, and a project is usually acceptable when B/C >= 1.
Vocabulary
- Time Value of Money
- The principle that money available now is worth more than the same amount received later because it can earn interest.
- Present Value
- The equivalent value today of a future amount or series of cash flows.
- Future Value
- The value at a later date of money invested or borrowed today after interest is applied.
- Uniform Series
- A set of equal cash flows that occur at regular time intervals.
- Discount Rate
- The interest rate used to convert future cash flows into present value.
- Net Present Worth
- The total present value of all benefits minus the total present value of all costs for a project.
Common Mistakes to Avoid
- Mixing time periods and interest rates is wrong because i and n must use the same period, such as monthly interest with number of months.
- Using the future value formula when a present value is needed gives the opposite cash flow movement and usually makes the answer too large.
- Forgetting the sign convention makes project comparisons confusing because costs and benefits must be assigned consistent positive and negative signs.
- Treating annual payments as if they occur at time zero is wrong because ordinary uniform series formulas assume payments occur at the end of each period.
- Comparing alternatives with different lifetimes without adjustment is misleading because projects should be compared using a common study period or annual worth.
Practice Questions
- 1 A project requires 10,500 in 4 years. At i = 6% per year, what is the net present worth?
- 2 What equal annual payment A is equivalent to borrowing $12,000 for 5 years at 7% interest per year?
- 3 A machine costs 6,500 per year for 5 years. At i = 8%, should the machine be accepted using present worth?
- 4 Two projects have the same net present worth, but one has higher first cost and lower yearly operating cost. Explain what non-cost factors an engineer should consider before choosing.