Single-Sum Problems: Find the Interest Rate or Time
Single-sum problems are time value of money problems built around one amount of money, either today or at a future date. They use one equation, FV = PV × (1 + i)n. If you know any three of FV, PV, i, and n, you can solve for the fourth, including an unknown rate or time.
Most textbooks start with two types: finding an unknown future value and finding an unknown present value. This lesson covers the next step. When you already know both dollar amounts, divide FV by PV, then use a table, a formula, or Excel to answer one of two questions: “How long must one deposit remain invested?” or “What compound rate must it earn?”
Watch: solving for the unknown rate or time (starts at 11:06)
This clip starts at 11:06 of Mansour Farhat’s future value lesson, where he rearranges the formula to solve for the number of periods or the interest rate. That part runs to 16:06.
How do you set up a single-sum problem?
Use positive dollar magnitudes for the algebra below. Present value (PV) is the starting amount; future value (FV) is the ending amount. The interest rate per period is i, and the number of periods is n. Assume a constant rate, retained interest, and no additional deposits or withdrawals.
| Unknown | Known information | Question answered |
|---|---|---|
| FV, the future value | PV, i, and n | How much will this amount grow to? |
| PV, the present value | FV, i, and n | What is a future amount worth today? |
| n, the number of periods | PV, FV, and i | How long does the deposit need to grow? |
| i, the rate per period | PV, FV, and n | What rate produces the required growth? |
If one dollar amount is missing instead, use the future value of a single sum formula or the present value of a single sum formula. Identify the unknown before selecting a table or entering spreadsheet arguments.
Picture a simple timeline. PV sits at today, FV sits at the end, and n periods at rate i connect them. Each unknown has its own version of the same equation:
FV = PV × (1 + i)n
PV = FV ÷ (1 + i)n
n = ln(FV ÷ PV) ÷ ln(1 + i)
i = (FV ÷ PV)1/n − 1
Exam tip: before you open a table, write down all four variables and put a question mark next to the missing one. Then label n and i with the same time unit, like 8 quarters and a rate per quarter.
Single-Sum Problems: Interactive Concept Check
Before solving for an unknown interest rate or number of periods, review how present value and future value relate. This six-question exercise checks the underlying concepts; the worked examples below apply them to rate and time calculations. Assume a positive interest rate over one or more periods.
Practice distinguishing present-value and future-value problems and the method used to solve each one.
What is a single-sum problem? It involves a single amount of money that exists now or will exist in the future.
Unknown future value: You know what you have today, the rate, and how long it stays invested. Accumulation carries that amount forward, and interest makes it grow.
Unknown present value: You know the amount due later, the rate, and the time. Discounting carries that amount back to today, which makes it smaller.
Choose the correct answer
Complete all six statements, then select “Check answers.”
Review: future value versus present value
- What is known in a future-value problem?
- The single sum invested now is known. Its future value is unknown. The solution accumulates the amount to a future point.
- What is known in a present-value problem?
- The single sum in the future is known. Its present value is unknown. The solution discounts the future amount to the present.
- How do accumulation and discounting differ?
- Accumulation moves an amount forward in time. Discounting moves a future amount back to the present.
How do you find the number of periods in a single-sum problem?
Hypothetical example. Maple Workshop invests $10,000 at 10% compounded annually. It wants $14,641. How many years must the funds remain invested?
In a future value of 1 table, move down the known 10% column until you find 1.4641. It appears at four periods. Since compounding is annual, the answer is four years.
| Periods at 10% | Future value factor | Value of $10,000 |
|---|---|---|
| 2 | 1.21000 | $12,100 |
| 3 | 1.33100 | $13,310 |
| 4 | 1.46410 | $14,641 |
A formula gives the same result: n = ln(FV ÷ PV) ÷ ln(1 + i). The symbol ln means natural logarithm, a calculator function that helps isolate an exponent. Here, ln(1.4641) ÷ ln(1.10) = 4. You can use the table method without manually evaluating logarithms.
How do you find the interest rate in a single-sum problem?
Hypothetical example. Juniper Lab has $18,000 today and needs $20,837.25 in three years. What constant annual compound rate would produce that ending balance?
Move across the known three-period row of a future value of 1 table. The factor approximately 1.15763 appears in the 5% column. Therefore, the required rate is 5% annually.
i = 1.1576251/3 − 1 = 0.05 = 5%
The exponent 1/3 means take the cube root. Subtract one to remove the original principal component. Check the answer by calculating $18,000 × 1.053 = $20,837.25. The total three-year increase of 15.7625% is not the annual rate.
Should you use FV ÷ PV or PV ÷ FV?
| Approach | Calculate this ratio | Unknown time | Unknown rate |
|---|---|---|---|
| Future value of 1 | FV ÷ PV | Search down the known rate column. | Search across the known period row. |
| Present value of 1 | PV ÷ FV | Search down the known rate column. | Search across the known period row. |
Both approaches work because the corresponding factors are reciprocals, meaning each one equals 1 divided by the other. For Maple Workshop, PV ÷ FV = $10,000 ÷ $14,641 ≈ 0.68301. That present value factor also identifies four periods at 10%.
Match the ratio to the table heading. Under positive compound growth, FV ÷ PV exceeds one and PV ÷ FV is below one. The California State Board of Equalization’s time-value lesson explains this reciprocal relationship. If the exact factor isn’t in your table, don’t pretend it is. The next section shows what to do.
What if the factor falls between two table values?
Real problems often miss the table factors. When that happens, you can estimate between the two closest factors. This is called interpolation. A formula or Excel gives a more precise answer.
Hypothetical example. Birch Studio invests $5,000 at 6% compounded annually and wants $8,000. How many years does it need?
FV ÷ PV = $8,000 ÷ $5,000 = 1.6
| Periods at 6% | Future value factor | Value of $5,000 |
|---|---|---|
| 8 | 1.59385 | $7,969.24 |
| 9 | 1.68948 | $8,447.39 |
The factor 1.6 sits just above the eight-period factor. Interpolating gives 8 + (1.6 − 1.59385) ÷ (1.68948 − 1.59385), or about 8.06 periods. The formula gives n = ln(1.6) ÷ ln(1.06), or about 8.07 periods.
Now read the question carefully. After eight years, Birch is still $30.76 short. If interest is added only once a year, the balance first reaches $8,000 at the end of year nine.
How do you solve single-sum problems in Excel?
| Required result | Original example | Excel entry |
|---|---|---|
| Number of annual periods | Maple Workshop | =NPER(10%,0,-10000,14641) |
| Rate per annual period | Juniper Lab | =RATE(3,0,-18000,20837.25) |
NPER returns 4; RATE returns 0.05, which displays as 5% when formatted as a percentage. The periodic payment input is zero. Enter the starting investment as negative and the ending receipt as positive to represent opposite cash-flow directions.
Microsoft documents the NPER function and the RATE function. RATE returns a rate per period. If the inputs represent quarters, its result is a quarterly rate, not an annual rate.
How do you find the rate when compounding isn’t annual?
The steps stay the same, but n counts compounding periods, not years.
Hypothetical example. Alder Supply has $8,000 today and needs $10,000 in two years. Interest compounds quarterly. What rate does it need?
Start with the periods: 2 years × 4 quarters = 8 periods. Then find the factor: $10,000 ÷ $8,000 = 1.25.
In the eight-period row, 1.25 falls between the 2.5% column (1.21840) and the 3% column (1.26677). So use the formula:
i = 1.251/8 − 1 = about 0.028286, or about 2.83% per quarter
That’s a quarterly rate, not an annual one. The nominal annual rate is the stated yearly rate before compounding, so multiply by four to get about 11.31%. The effective annual rate shows one full year of compound growth: 1.0282864 − 1 = about 11.80%.
In Excel, =RATE(8,0,-8000,10000) returns about 2.83%. Review compounding periods if you want more practice converting rates.
Where do single-sum problems show up in accounting?
You’ll use this skill when a contract doesn’t state the true interest rate. A zero-interest-bearing note is a good example. It states no interest, but the borrower repays more than it received.
Under FASB ASC 835-30-25-4, a note issued solely for cash, with no other right or privilege exchanged, is presumed to have a present value equal to the cash proceeds. So the cash received is your PV, the face amount is your FV, and the rate is your unknown.
Hypothetical example. On January 1, Cedar Tools issues a three-year, $13,310 zero-interest-bearing note and receives $10,000 cash.
FV ÷ PV = $13,310 ÷ $10,000 = 1.331
In the three-period row of a future value of 1 table, 1.33100 appears in the 10% column. The effective rate is 10%.
Journal entry at issuance:
| Account | Debit | Credit |
|---|---|---|
| Cash | $10,000 | |
| Discount on notes payable | $3,310 | |
| Notes payable | $13,310 |
Discount on notes payable is a contra-liability account, which means it reduces the note’s carrying amount. Under the interest method, the $3,310 discount becomes interest expense over the three years.
| Year | Beginning carrying amount | Interest expense at 10% | Ending carrying amount |
|---|---|---|---|
| 1 | $10,000 | $1,000 | $11,000 |
| 2 | $11,000 | $1,100 | $12,100 |
| 3 | $12,100 | $1,210 | $13,310 |
Journal entry for Year 1 interest:
| Account | Debit | Credit |
|---|---|---|
| Interest expense | $1,000 | |
| Discount on notes payable | $1,000 |
Total interest of $3,310 equals the full discount. Time value of money questions also appear in the FAR section of the CPA Exam, and the same calculations are used to value bonds and leases.
Common mistakes with unknown rates and time
Dividing total growth by years. That gives a simple average increase relative to the original balance; it does not generally recover a compound rate. Juniper’s 15.7625% total growth divided by three is 5.2542%, not the correct 5%.
Calling every period a year. Eight semiannual periods represent four years. Match the units before interpreting the answer. Review compounding periods and rate conversions when needed.
Rounding time down too soon. A formula may return a fraction of a period. If a question asks for the first whole compounding date at which a target is reached, round up to the next whole period. A partial-period withdrawal requires the problem’s stated interest convention.
Three practice questions with answers
All questions are hypothetical and assume a single deposit with retained interest.
1. Find the time
A $6,000 deposit grows to $7,260 at 10% compounded annually. How long does it take?
Two years. $7,260 ÷ $6,000 = 1.21, and 1.102 = 1.21. Two annual compounding periods equal two years.
2. Find the annual rate
A $12,000 deposit becomes $13,891.50 after three years of annual compounding. What is the rate?
5% annually. The factor is 1.157625. Its cube root is 1.05; subtracting one gives 0.05. The 15.7625% increase is the total return for all three years.
3. Interpret periods
A table calculation gives n = 6 for an investment earning 4% per quarter. How many years does that represent?
1.5 years. Six quarters ÷ four quarters per year = 1.5 years. Do not call the answer six years or treat 4% as an annual rate.
Key takeaways
- A single-sum problem involves one amount of money, either today or at a future date.
- FV, PV, i, and n are linked by FV = PV × (1 + i)n. Know any three and you can find the fourth.
- To find a rate or time, start with FV ÷ PV.
- For time, search down the known rate column. For the rate, search across the known period row.
- The rate you find is a rate per period. It’s annual only when each period is one year.
- If a target must be reached on a compounding date, round up to the next whole period.
Frequently asked questions
What is a single-sum problem?
A single-sum problem is a time value of money problem with one amount of money rather than a series of payments. You move that amount forward to find its future value, move it back to find its present value, or find the rate or time that connects the two.
What is the difference between a single sum and an annuity?
A single sum is one amount at one point in time. An annuity is a series of payments, usually equal, made at equal intervals. Annuity problems use different tables and formulas.
How do you find the interest rate in a single-sum problem?
Use i = (FV ÷ PV)1/n − 1. You can also divide FV by PV and search the n-period row of a future value of 1 table, or use RATE in Excel.
How do you find the number of periods in a single-sum problem?
Divide FV by PV and search the known rate column of a future value of 1 table. You can also use n = ln(FV ÷ PV) ÷ ln(1 + i), or NPER in Excel.
What do you do if the factor isn’t in the table?
Interpolate between the two closest factors for an estimate, or use a formula or Excel for a more precise answer.
Can either interest table find an unknown rate?
Yes. Use FV ÷ PV with a future value table, or PV ÷ FV with a present value table. Keep the number of periods fixed while searching across the row.
Why does Excel require opposite signs?
A deposit paid out now and a receipt collected later move in opposite directions, so one is negative and the other is positive. If you give both the same sign in a single-sum problem, Excel typically returns the #NUM! error.
Is an interest rate per period always an annual rate?
No. It is annual only when each period is one year. A semiannual, quarterly, or monthly calculation produces the corresponding periodic rate.
For more worked examples and practice, explore the Intermediate Accounting course. Studying for CPA FAR? Test yourself with the FAR Test Bank.
About the author: Mansour Farhat, CPA, CIA, DBA, teaches accounting and finance at Farhat Lectures, helping CPA, CMA, and EA candidates and accounting students understand the material step by step. His Farhat Lectures YouTube channel has about 2,700 videos. Read more on his about page.
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