MATH 114: Test 2 Hardest Problems

How to use this page

The bar under each heading shows the share of students who got that type right on the first try in past Test 2 results. Read the method, open the worked example, then cover it up and do the practice problem on your own before checking.

Keep every decimal in your calculator until the last step. Most "close but wrong" answers come from rounding in the middle.
Write the APR as a decimal (5.5% → 0.055) and put the whole exponent in parentheses: (1 + 0.055/12)^(12*20).
Several of these are two-part questions. Answer both parts; that is where most partial credit was lost.

Ranked hardest first

Problem types on this page

How long until an account reaches a goal
How much to deposit now (present value)
Total and annual return on an investment
Net cost of a class vs. working
P/E ratio and "overpriced or underpriced"
Monthly loan payment
Net monthly cash flow
Spending vs. the national average
Savings plan: balance or monthly deposit
Monthly credit card interest

Not sure which type a problem is? Use the 10-second "which formula?" check.

Hardest #1 · Compound interest

How long until an account reaches a goal?

36% · About 1 in 3 right on the first try

You know the starting deposit, the APR, how often it compounds, and the target balance. The unknown is the exponent, so you need logarithms.

Y = log(A ÷ P)n × log(1 + APR/n)

Identify P (starting deposit), A (goal), APR, and n (annually 1, quarterly 4, monthly 12, weekly 52, daily 365).
Top: divide A by P, then take log.
Bottom: compute 1 + APR/n, take log, then multiply by n.
Divide and round to the nearest year at the very end.

Common traps: using n = 4 for weekly (it's 52); taking log(A) − P instead of log(A ÷ P); forgetting to multiply the bottom by n; typing log(1 + 0.048)/12 instead of log(1 + 0.048/12).

Using the formula sheet: the sheet writes this as Years = log(1 + total return) ÷ log(1 + annual return). Here 1 + total return = A ÷ P and 1 + annual return = (1 + APR/n)ⁿ, so both versions give the same answer.

Worked example: $6,000 grows to $250,000 at 4.8% compounded monthly

P = 6,000, A = 250,000, APR = 0.048, n = 12.

Top: log(250,000 ÷ 6,000) = log(41.667) ≈ 1.61979

Bottom: 12 × log(1 + 0.048/12) = 12 × log(1.004) ≈ 12 × 0.0017337 ≈ 0.020804

Y ≈ 1.61979 ÷ 0.020804 ≈ 77.86, so about 78 years.

Calculator entry: log(250000/6000)/(12*log(1+0.048/12))

Excel: =NPER(0.048/12, 0, -6000, 250000)/12 (NPER counts months, so divide by 12).

$12,500 at 6% compounded quarterly. How many years to reach $400,000?

Y = log(400,000 ÷ 12,500) ÷ (4 × log(1 + 0.06/4)) = log(32) ÷ (4 × log(1.015)) ≈ 58.19, so about 58 years.

Hardest #2 · Compound interest

How much do I deposit now? (Present value)

44% · Fewer than half right on the first try

A single deposit today, no more deposits later, and a target amount in the future. You are running the compound interest formula backward: divide instead of multiply.

P = A(1 + APR/n)^{n × Y}

Find the growth factor (1 + APR/n)^nY first and write it down.
Divide the goal A by that factor.
Sanity check: P must be smaller than A. If your answer is bigger, you multiplied.

Common traps: multiplying by the growth factor; using the savings-plan formula (that is for monthly deposits, not one deposit now); exponent n × Y not in parentheses; using n = 360 for daily (use 365).

Worked example: $95,000 in 25 years at 4.2% compounded daily

A = 95,000, APR = 0.042, n = 365, Y = 25, so nY = 9,125.

Growth factor: (1 + 0.042/365)⁹¹²⁵ ≈ 2.85748

P = 95,000 ÷ 2.85748 ≈ $33,246.09

Calculator entry: 95000/(1+0.042/365)^(365*25)

Excel: =PV(0.042/365, 365*25, 0, -95000)

You want $60,000 in 20 years at 5.5% compounded monthly. Deposit now?

P = 60,000 ÷ (1 + 0.055/12)²⁴⁰ ≈ $20,022.52

Hardest #3 · Investments

Total return and annual return

45% · Fewer than half right on the first try; many got only one of the two parts

You buy for one price and sell years later. The question asks for two answers. They are not the same thing, and the annual return is not the total return divided by the number of years.

Total return = sell − buybuy × 100% Annual return = [ (sell ÷ buy)^(1/Y) − 1 ] × 100%

Compute the ratio sell ÷ buy and keep it in your calculator.
Total return: ratio − 1, times 100.
Annual return: ratio raised to the power (1/Y), minus 1, times 100.
Round each to one decimal place only at the end.

Common traps: dividing the total return by Y (that ignores compounding); typing ^1/8 instead of ^(1/8); dividing by the sell price instead of the buy price. If the investment lost money, both answers are negative.

Worked example: Bought for $4,800, sold for $7,350 eight years later

Ratio: 7,350 ÷ 4,800 = 1.53125

Total return: (1.53125 − 1) × 100% = 53.1%

Annual return: (1.53125^1/8 − 1) × 100% ≈ (1.054704 − 1) × 100% ≈ 5.5%

Notice 53.1 ÷ 8 ≈ 6.6% is wrong; compounding means a smaller yearly rate builds to the total.

Worked example (a loss): Bought for $12,000, sold for $9,600 three years later

Ratio: 9,600 ÷ 12,000 = 0.8

Total return: (0.8 − 1) × 100% = −20.0%

Annual return: (0.8^1/3 − 1) × 100% ≈ −7.2%

$2,500 grows to $6,100 in 12 years

Ratio 2.44. Total return 144.0%. Annual return 2.44^1/12 − 1 ≈ 7.7%.

Hardest #4 · Budgeting

Net cost of taking a class instead of working

46% · Fewer than half right on the first try

The class costs you twice: the tuition you pay and the wages you give up by not working those hours. Add them.

Net cost = (credits × tuition per credit) + (weeks × hours per week × hourly wage)

Tuition: multiply credits by the price per credit-hour.
Hours of time: multiply hours per week by the number of weeks.
Lost wages: multiply those hours by the hourly pay.
Add tuition and lost wages.

Common traps: using only one week of lost wages; subtracting the wages from tuition (you lose them, so they add to the cost); multiplying credits by weeks; reading the numbers in the wrong order when they appear in a sentence. Label every number before you calculate.

Worked example: 16-week, 4-credit course, 10 hours/week, $380 per credit, job pays $14/hour

Tuition: 4 × $380 = $1,520

Hours given up: 16 weeks × 10 hours = 160 hours

Lost wages: 160 × $14 = $2,240

Net cost: $1,520 + $2,240 = $3,760

Related: least expensive option. $360 parking pass, or $3/day parking 4 days a week for 30 weeks?

Daily option: $3 × 4 × 30 = $360. The two cost the same, so neither is cheaper. Always put both options on the same time period before comparing.

15-week, 3-credit course, 8 hours/week, $425 per credit, job pays $16/hour

Tuition 3 × 425 = $1,275. Lost wages 15 × 8 × 16 = $1,920. Net cost $3,195.

Hardest #5 · Stocks

P/E ratio: earnings per share, then "overpriced or underpriced?"

53% · About half fully right; most others got only partial credit

Almost half the class lost points on the second part of this question, the judgment call. Do both parts.

P/E = share priceearnings per share so earnings per share = share priceP/E

Part (a): divide the share price by the P/E ratio. Round to the nearest cent.
Part (b): compare the current P/E (not the earnings) to the historical range.
- Current P/E above the range → overpriced (you pay more per dollar of earnings than usual).
- Current P/E inside the range → about right.
- Current P/E below the range → underpriced.

Common traps: multiplying price × P/E; comparing the earnings per share (a dollar amount) to the P/E range (ratios); leaving part (b) blank.

Worked example: Price $62.40, P/E 22.5, historical P/E range 15 to 18

(a) Earnings per share = 62.40 ÷ 22.5 ≈ $2.77

(b) 22.5 is above the 15–18 range, so the stock seems overpriced.

Price $38.75, P/E 9.5, historical range 12 to 16

(a) 38.75 ÷ 9.5 ≈ $4.08. (b) 9.5 is below 12–16, so underpriced.

Hardest #6 · Loans

Monthly loan payment

54% · About half right on the first try

Student loans, car loans, and mortgages all use the same installment-loan formula. The exponent is negative.

PMT = P × (APR/n)1 − (1 + APR/n)^−nY

P is the amount borrowed, n = 12 for monthly payments.
Top: P × APR/12.
Bottom: 1 − (1 + APR/12)^−12Y. Compute this separately and check it is between 0 and 1.
Divide top by bottom.

This formula is not on the formula sheet. Copy it from the study guide, or use Excel's =PMT( as the sheet shows.

Common traps: dropping the minus sign in the exponent (gives a negative or huge answer); using the savings-plan formula; dividing P by the number of months (that ignores interest and is always too low).

Worked example: $24,500 borrowed at 6.4% APR for 10 years

APR/12 = 0.064/12 ≈ 0.0053333; 12Y = 120

Top: 24,500 × 0.0053333 ≈ 130.667

Bottom: 1 − (1.0053333)⁻¹²⁰ ≈ 1 − 0.52819 ≈ 0.47181

PMT ≈ 130.667 ÷ 0.47181 ≈ $276.95 per month

Calculator entry: 24500*(0.064/12)/(1-(1+0.064/12)^(-120))

Excel: =PMT(0.064/12, 120, -24500)

Bonus: total paid ≈ 276.95 × 120 ≈ $33,233.71, so total interest ≈ $8,733.71.

$18,000 at 5% APR for 15 years

PMT = 18,000 × (0.05/12) ÷ (1 − (1 + 0.05/12)⁻¹⁸⁰) ≈ $142.34

Hardest #7 · Budgeting

Net monthly cash flow

57% · A little over half right on the first try

Every income and expense has to be converted to per month before you add anything. Use the conversions the problem gives you (usually 1 month = 4 weeks).

weekly × 4 · yearly ÷ 12 · semiannually ÷ 6 · quarterly ÷ 3 Net monthly cash flow = total monthly income − total monthly expenses

Make two columns: income and expenses.
Convert each line to monthly and write it next to the original.
Total each column, then subtract. A negative answer is allowed and common.

Common traps: "semiannually" means twice a year, so divide by 6, not 2; skipping a small line item; dropping the negative sign.

Worked example: a full budget table

Item	Given	Per month
Salary (income)	$33,600/year	÷ 12 = $2,800
Tutoring (income)	$75/week	× 4 = $300
Rent	$1,650/month	$1,650
Groceries	$85/week	× 4 = $340
Car insurance	$480 semiannually	÷ 6 = $80
Renter's insurance	$180/year	÷ 12 = $15
Utilities	$140/month	$140
Entertainment	$450/month	$450
Car payment	$520/month	$520

Income: 2,800 + 300 = $3,100.

Expenses: 1,650 + 340 + 80 + 15 + 140 + 450 + 520 = $3,195.

Net cash flow: 3,100 − 3,195 = −$95 per month.

income $2,600/month plus $40/week from a side job

Expenses:

Rent $900/month
Food $70/week
Car insurance $600 semiannually
Gym $240/year
Gas $95/month
Textbooks $400/year
Other $300/month

Income: 2,600 + 160 = $2,760. Expenses: 900 + 280 + 100 + 20 + 95 + 33.33 + 300 ≈ $1,728.33. Net ≈ $1,032 per month.

Hardest #8 · Budgeting

Is this spending above, at, or below the national average?

57% · A little over half right on the first try

The test gives a chart of the percent of spending by category and age group. You turn the person's spending into a percent and compare.

Percent spent = category spending ÷ income × 100%. Use the same time period for both (monthly ÷ monthly).
Find the person's age group column and the right category row on the chart.
Higher than the chart → above average. Lower → below. Within about a point → at.

Common traps: reading the wrong age group; comparing a dollar amount to a percent; dividing income by spending (upside down).

Worked example: A 45-year-old earns $4,200/month and spends $380/month on entertainment

For practice only, suppose the chart shows 5% for entertainment in the 45–54 age group. On the test, read the real value from the chart.

380 ÷ 4,200 × 100% ≈ 9.0%

9.0% is higher than 5%, so this spending is above average.

Earns $3,800/month, spends $610 on transportation. Suppose the chart says 16%.

610 ÷ 3,800 ≈ 16.1%, essentially equal to 16%, so at the average.

Hardest #9 · Savings plans

Savings plan: find the balance, or find the monthly deposit

59% · About 3 in 5 right on the first try

Use this when money goes in every month (retirement accounts, college funds). Decide first whether you are finding the ending balance A or the monthly payment PMT.

Balance: A = PMT × (1 + APR/n)^nY − 1APR/n Deposit: PMT = A × (APR/n)(1 + APR/n)^nY − 1

"How much will it be worth?" → solve for A.
"How much should you deposit monthly?" → solve for PMT.
Check: the balance A should be more than PMT × number of months (the extra is interest), and PMT should be less than A ÷ number of months.

Common traps: using the present-value formula (that's a single deposit); forgetting the "− 1" in the numerator; dividing by APR instead of APR/12.

Worked example (balance): $250/month at 6% APR for 20 years

APR/12 = 0.005; nY = 240

(1.005)²⁴⁰ − 1 ≈ 3.31020 − 1 = 2.31020

A = 250 × 2.31020 ÷ 0.005 ≈ $115,510.22

Check: you deposited 250 × 240 = $60,000, so the rest is interest.

Excel: =FV(0.06/12, 240, -250)

Worked example (deposit): Want $120,000 in 15 years at 4.5% compounded monthly

APR/12 = 0.00375; nY = 180

PMT = 120,000 × 0.00375 ÷ ((1.00375)¹⁸⁰ − 1) ≈ 450 ÷ 0.96156 ≈ $467.99 per month

Excel: =PMT(0.045/12, 180, 0, -120000)

(a) $175/month at 5% for 30 years. (b) Monthly deposit to reach $50,000 in 10 years at 4%.

(a) A ≈ $145,645.26. (b) PMT ≈ $339.56.

Hardest #10 · Interest

Monthly interest on a credit card balance

64% · About 2 in 3 right on the first try

This one is short, but the wrong answer choices are designed around the easy mistakes.

Monthly interest = balance × APR12

Common traps: forgetting to divide by 12 (that gives a full year of interest); using 24 instead of 0.24 (answer 100 times too big); dividing the balance by 12 first and forgetting the rate.

Worked example: $2,400 average balance at 19.8% annual interest

2,400 × 0.198 ÷ 12 = $39.60 per month

The trap answer $475.20 is a whole year of interest.

$5,200 balance at 21.9% annual interest

5,200 × 0.219 ÷ 12 = $94.90

Which formula? A 10-second check

One deposit, find the future balance → compound interest A = P(1 + APR/n)^nY. Compounded annually: A = P(1 + APR)^Y.
One deposit, find what to deposit now → present value (divide).
One deposit, find how long → logarithm formula.
Monthly deposits into savings → savings plan formula.
Monthly payments on money you borrowed → loan payment formula (negative exponent).
Bought and sold → total and annual return.

More formulas: Test 2 study guide supplement · Formula sheet (PDF)