Test 1 Comprehensive Review Notes

Exam Logistics

• Date/Time/Location
– Test 1 is on the main campus, Building 1, Room 137.
– Starts exactly 12:30 p.m.; ends 2:10 p.m. ⇒ full duration $=1\text{ h }40\text{ min}=100\text{ min}$ .
– Arrive early; no extra time is added for late arrivals.
• What to bring
– 3 × 5 index card with anything you choose to write (formulas, examples, vocabulary, etc.).
– Calculator (any non-graphing scientific model is safe).
– Several sharpened pencils & eraser.
• Covered textbook sections 1.1, 1.3, 1.4, 1.5, 1.6 plus place-value and rounding.

Percent Change Fundamentals

• Two related—but different—ideas:
– Percent point change: simple subtraction of two quoted rates.
– Percentage change (a.k.a. relative change): $\dfrac{\text{new}-\text{old}}{\text{old}}\times100\%$ .
• Decrease example ("tunnel through Earth")
– Old time $=43\text{ min}$ , new $=35\text{ min}$ .
– Percent decrease $=\dfrac{43-35}{43}\times100\%=18.60\%$ (rounded).
– Interpretation: it now takes 18.60 % less time than the earlier estimate.

Unit Conversions & Dimensional Analysis

• Key equalities you may need
– $1\text{ ft}=0.3048\text{ m}$ (or $1\text{ m}\approx3.281\text{ ft}$ ).
– $1000\text{ m}=1\text{ km}$ ; $1000\text{ mL}=1\text{ L}$ .
• Multi-step strategy

Write what you’re given as a fraction over 1.
Multiply by conversion factors so units cancel diagonally.
Finish when the only remaining unit is the target unit.
• September walking example
– Steps/day $=15{,}000$ ; stride $=2\text{ ft}$ ⇒ $30{,}000\text{ ft/day}$ .
– September has $30$ days ⇒ $900{,}000\text{ ft}$ .
– $\dfrac{900{,}000\text{ ft}}{3.281\text{ ft/m}}=274{,}377.05\text{ m}=274.38\text{ km}$ (rounded).

Sales-Tax Debate: Percentage Points vs Percent

• Tax rose from $5\%$ to $5.5\%$ .
– Percent point change $=0.5\text{ points}$ ("Harry" viewpoint).
– Percentage change $=\dfrac{5.5-5}{5}\times100\%=10\%$ ("Linda" viewpoint).
• Who is right? Both—they are using different, legitimate measures. Be explicit on exams about which version you quote.

Admissions‐Table Skills

Given – or required to find – any two of: Total Applicants (TA), Number Admitted (NA), % Accepted (%A).
• Formulas
– $\%A=\dfrac{NA}{TA}\times100\%$ .
– $NA=TA\times\dfrac{\%A}{100}$ .
– $TA=\dfrac{NA}{\%A}\times100$ .
• Completed table (rounded where instructed)
– Clemson: %A $=21.93\%$ (given TA & NA).
– Florida State: $NA=14{,}065$ ; %A was given.
– UNC-Chapel Hill: $NA=331$ (10 % of 3,308).
• Reasoning questions
– Easiest (highest % accepted) ⇒ Florida State.
– Hardest (lowest %) ⇒ UNC-Chapel Hill.
– Largest absolute admits ⇒ Florida State.
– Smallest admits ⇒ UNC-Chapel Hill.

Translating Verbal Phrases to Inequalities

• "At least" ⇒ ≥ "At most" ⇒ ≤ "More than" ⇒ > "No less than" ⇒ ≥.
• Examples
– Kyle needs at least \$15 ⇒ $d\ge15$ .
– "Ratio of $x$ to 4 is not less than 9" ⇒ $\dfrac{x}{4}\ge9$ .
– "2 subtracted from $x$ is at most 3" ⇒ $x-2\le3$ .

Linear-Inequality Budget Example (Canoe Rental)

• Cost model: $\text{Total}=20+35n$ where $n=$ hrs.
• Requirement "do not exceed \$150" ⇒ $20+35n\le150$ .
• Solve
$35n\le130 \;\Rightarrow\; n\le3.714…$ ⇒ You can afford up to 3 hours (or 3 h 43 min if partial hours allowed).

Population & Social-Issue Percent Problems

Homelessness in NC

• 2022 figure: $8{,}314$ people represents an 80 % remainder after a 20 % drop from 2010.
• 2010 value $=\dfrac{8{,}314}{0.8}=10{,}393$ (rounded).
• Veterans now 10 % ⇒ $0.10\times8{,}314\approx831$ veterans.

14-Day Downward COVID Trajectory

• Jan 8 rate $=30.9\%$ ; Feb 6 rate $=23.4\%$ .
– Percent-point change $=23.4-30.9=-7.5\text{ points}$ .
– Percentage change $=\dfrac{-7.5}{30.9}\times100\%=-24.27\%$ .
• Projection (assuming equal daily decline)
– 29 days produced $-7.5$ points ⇒ $-0.2586$ points/day.
– ≈22 days to Mar 1 ⇒ $-5.69$ more points.
– Expected early-March positivity $\approx23.4-5.69=17.7\%$ .
– Caveat: real epidemiology seldom stays linear; mention assumptions if asked.

Medication Dosage via Dimensional Analysis

Problem: 130-lb patient, 2.5 mg/lb, concentration 200 mg/L.

130\,\frac{\text{lb}}{1}\times2.5\,\frac{\text{mg}}{\text{lb}}\times\frac{1\,\text{L}}{200\,\text{mg}}\times1000\,\frac{\text{mL}}{\text{L}}=1625\,\text{mL}

• Always align units so they cancel diagonally. Show every factor to earn full credit.

Rounding Rules & Practice

• If digit right of target place ≥5 ⇒ round up; else round down.
• Examples likely to appear
– $56.68\to57$
– $724.59\to725$
– $3.0752\to3.08$ (two decimals)
– $68\,345\,224\to68\,000\,000$ (nearest million)
– $5\,072\to5\,100$ (nearest hundred)
• Always keep commas every three digits for large whole numbers.

Percent Change in Enrollment

• Downward change: new $= \text{old}\times(1-\%\text{decrease})$ .
• Upward change: old $= \dfrac{\text{new}}{1+\%\text{increase}}$ .
Examples

Sixth-grade drop
$350\times(1-0.36)=224\text{ students}$ now.
Club growth
$\dfrac{182}{1.30}=140$ students last year.

Scientific Notation & World-Population Share

• Convert by moving decimal so only 1 non-zero digit left of decimal; count places ⇒ the power of 10.
– World $=7{,}200{,}000{,}000=7.2\times10^{9}$ .
– Canada $=35{,}000{,}000=3.5\times10^{7}$ .
• Percentage of world
$\dfrac{35\times10^{6}}{7.2\times10^{9}}\times100\%\approx0.5\%$ of humanity lives in Canada.

Common Mistakes to Avoid

• Mixing "percent" with "percent point".
• Forgetting to convert rate constants (e.g.\, mg/L) before computing volume.
• Rounding too early—carry 1–2 extra decimals until final step.
• Writing > when the phrase says "at least" (should be $\ge$ ).

Quick Study Checklist

☐ Know place values through the billions and thousandths.
☐ Practice translating 5–10 verbal phrases to algebra each night.
☐ Memorize baseline unit conversions (ft↔m, lb↔kg, L↔mL, %↔decimal).
☐ Write the percent-change and simple-interest formulas on your index card.
☐ Drill rounding until you can do it without a calculator.
☐ Time yourself: match the 100-minute exam window when you run practice sets.