Chapter 1 Notes: Ratio and Proportional Reasoning

Percents

  • Percent meaning and conversion:
    • Percent literally means “per 100.” 40% =
      40100=0.40\frac{40}{100}=0.40
    • Fractions and decimals equivalent: 80/200 and 10/25 are also 40%.
    • Example 1: 243 out of 400 like dogs →
      243400=0.6075=60.75%\frac{243}{400}=0.6075=60.75\%
    • Decimal to percent: move decimal two places to the right: 0.40 → 40%.
    • Example 2 (convert to percent):
    • a) ? → 25% (example in the text)
    • b) 0.02 → 2%
    • c) 2.35 → 235%
  • Percent of a whole:
    • If a part is a percent of a whole, then part = (percent as decimal) × whole.
    • Write the percent as a decimal by dividing by 100.
  • Calculations with percent of a tax example:
    • Tax on a purchase: if tax rate is 9.4% and price is $140, tax = 140×0.094=13.16140\times0.094=13.16
  • Relative vs absolute change (overview):
    • A change can be described as an absolute change or a relative change.
    • Absolute change uses the original units (same as the quantity).
    • Relative change is a percent change with respect to the base (starting) value.
  • Important cautions when talking about percent changes:
    • If a quantity described in percents changes (e.g., from 40% to 50%), describe the change as a percentage point change (10 percentage points) rather than a 10% change.
    • Relative change would be (50% − 40%)/40% = 25% increase.
  • Example 10 (percent points vs percent change):
    • Increasing from 40% to 50% is an increase of 10 percentage points, but a 25% relative increase.

Absolute and Relative Change

  • Absolute change:
    • Absolute change = ending quantity − starting quantity.
    • It has the same units as the original quantity.
  • Relative change:
    • Relative change = (ending − starting) / starting (often expressed as a percentage).
    • Base is the starting quantity; the base is important when interpreting the change.
  • Base sensitivity illustrated (marijuana example):
    • 80% of marijuana smokers using hard drugs vs 80% of hard drug users having smoked marijuana are not the same base; the two statements are not equivalent.
  • Example 6 (absolute vs relative with stores):
    • Absolute difference: 215 − 75 = 140 (Albertsons has 140 more stores than QFC).
    • Relative difference depends on base chosen:
    • Albertsons base: 140/215 ≈ 0.651 → QFC is about 65.1% smaller than Albertsons.
    • QFC base: 140/75 ≈ 1.867 → Albertsons is about 186.7% larger than QFC.
  • Example 7: comparing sizes with relative terms and eventually the idea of equivalent interpretation via ratios.
  • Example 8: understanding how statements like “dropout factories decreased by 17” relate to an absolute change vs a relative change, and how context matters for interpretation.
  • Example 9: reconciling two percent claims by recognizing they may refer to different bases (coalition vs combined bases).
  • Summary:
    • Absolute change gives a concrete amount.
    • Relative change gives a percent change relative to the base.
    • Choice of base dramatically affects the interpretation of the percentage change.

Averaging Percents

  • Caution: averaging percents can be misleading when bases differ.
  • Example 11 (overall field goal percentage):
    • 40% on 2-point attempts and 30% on 3-point attempts cannot be simply averaged.
    • Suppose you took 200 two-point attempts and 100 three-point attempts:
    • Made: 200(0.40)=80200(0.40)=80 two-pointers; 100(0.30)=30100(0.30)=30 three-pointers.
    • Overall FG = \frac{80+30}{200+100}=\frac{110}{300}=0.3667=36.7\%.
    • Conclusion: you must account for the number of attempts per type; simple average is not generally correct.
  • When is averaging percents appropriate?
    • If all percents are calculated relative to the same base, then a simple average can be meaningful (arithmetic mean).
    • Example: four tests with equal weight: 75%, 90%, 95%, 60% → arithmetic mean = \frac{75+90+95+60}{4}=80\%
  • Weighted averages (when weights differ, e.g., class components):
    • Final = weightHomework × Homework + weightQuizzes × Quizzes + weight_Tests × Tests
    • Given: Homework 30%, Quizzes 20%, Tests 50%; Homework 87%, Quizzes 70%, Tests 80%
    • Final = 0.3(87) + 0.2(70) + 0.5(80) = 80.1
  • General weighted average formula:
    • If scores xi have weights wi (as decimals) and sum to 1, then
      \text{Average} = \sumi wi x_i.
    • If weights do not sum to 1, normalize by dividing by the sum of weights:
      \text{Average} = \frac{\sumi wi xi}{\sumi w_i}.
  • Takeaway:
    • Arithmetic mean is appropriate when each component has equal weight and same base.
    • Weighted average is appropriate for differing weights (importance) of components.

Geometric Mean and Average Rate of Change

  • When averaging rates of change, the arithmetic mean is inappropriate.
  • Geometric mean concept:
    • For growth multipliers gi = 1 + ri (written as decimals), the average growth factor over n periods is
      \text{GM} = \left(\prod{i=1}^n gi\right)^{1/n}.
    • The corresponding average rate of increase is
      \text{Average growth rate} = \text{GM} - 1.
  • Example 13 (quarterly revenue changes):
    • Q1: +10% → g1 = 1.10;
    • Q2: +12% → g2 = 1.12;
    • Q3: +5% → g3 = 1.05;
    • Q4: −8% → g4 = 0.92;
    • GM = \left(1.10\cdot 1.12\cdot 1.05\cdot 0.92\right)^{1/4} ≈ 1.04447
    • Average rate = 1.04447 − 1 ≈ 0.04447 = 4.447\%.
  • Key takeaway:
    • The geometric mean captures the consistent proportional change across periods, even when positives and negatives occur.
    • Useful for annualized growth or composite growth rates.

1.3: Proportions and Rates

  • Rates are ratios of two quantities; a unit rate has denominator 1.
  • Proportion approach:
    • A proportion equation expresses equivalence of two rates, e.g.,
      \frac{a}{b} = \frac{c}{d}.
    • Solve by cross-multiplying: a d = b c.
  • Examples:
    • Example 15: Map scale: If ½ inch on the map corresponds to 3 real miles, then for an x-inch separation on the map, the real distance is found by setting
      \frac{1/2}{3}=\frac{x}{\text{miles}}.
      Solve to get miles.
    • Example 16: ½ inch on map ↔ 3 miles; two cities inches apart on map → real miles x. Proportion relation sets
      \frac{\text{map inches}}{\text{map unit}} = \frac{\text{real miles}}{3} and solve for real miles.
  • Proportions with rate vs unit rate:
    • If a rate is known, you can set up a proportion to scale to a new quantity (e.g., miles, gallons, etc.).

1.4: Dimensional Analysis

  • Dimensional analysis: multiply a quantity by rates to convert units; units cancel to yield the desired unit.
  • Example 1: Driving distance via rate:
    • If a car gets 20 miles per gallon and you have 40 gallons, distance = 40 × 20 = 800 miles.
    • Also you could compute gallons needed for a given miles by inverting the rate.
  • Unit conversions (reference table):
    • Length: 1 ft = 12 in; 1 yd = 3 ft; 1 mile = 5,280 ft; 1 mile = 1.609 km;
      1000 mm = 1 m; 100 cm = 1 m; 1000 m = 1 km; 2.54 cm = 1 in.
    • Weight: 1 lb = 16 oz; 1 ton = 2000 lb; 1000 mg = 1 g; 1000 g = 1 kg; 1 kg = 2.2 lb.
    • Capacity: 1 cup = 8 fl oz; 1 pint = 2 cups; 1 quart = 2 pints = 4 cups; 1 gallon = 4 quarts = 16 cups; 1000 mL = 1 L.
  • Note: 1 gallon ≈ 3.785 L; 1 fl oz is a volume measure (roughly ounces by weight for water).
  • Example 20 (dimensional analysis combined with rate): convert 40 gallons to miles using the rate 20 miles per gallon, or convert miles to gallons by inverting the rate.
  • Dimensional analysis emphasizes cancelling units rather than performing long proportional calculations.

1.5: Scientific Notation

  • Purpose: handle very large or very small numbers and compare magnitudes via exponents.
  • Form: A number in scientific notation is a\times 10^{r}withwith1\le a<10 and r an integer.
  • How to convert:
    • Move the decimal to obtain a number in [1, 10) and record the exponent as the number of places moved for the decimal point.
    • Positive exponent indicates large numbers; negative exponent indicates small numbers.
  • Keys:
    • When multiplying numbers in scientific notation, add exponents: (a\times 10^{r})(b\times 10^{s})=(ab)\times 10^{r+s}.
    • When dividing, subtract exponents: \frac{a\times 10^{r}}{b\times 10^{s}}=\frac{a}{b}\times 10^{r-s}.
  • Examples:
    • 0.000015 m → 1.5 × 10^{-5} m;
    • Jupiter: 1.899 × 10^{27} kg; use exponents to compare with Mercury (3.285 × 10^{23} kg) and note Jupiter is about 10^4 times more massive in order of magnitude; more precise comparison uses exponents and then the difference in coefficients.
  • Unemployment claims example: 1.4 million claims → 1.4 × 10^{6}; U.S. population ≈ 3.3 × 10^{8}; percentage ≈ (1.4×10^6)/(3.3×10^8) ≈ 4.2 × 10^{-3} = 0.42%.
  • Examples 27–28 demonstrate using scientific notation to compare budgets and wages and to compute large totals without long strings of zeros.

1.6: Geometry

  • Key shapes and formulas:
    • Area (Rectangle): A = L\cdot W
    • Circle: A = \pi r^2, \quad \text{Circumference } C = 2\pi r
    • Volume (Rectangular Box): V = L\cdot W\cdot H
    • Cylinder: V = \pi r^2 H
  • Example 29 (shadow method for height): similar triangles relate height to shadow lengths; if a person of height 6 ft casts a 1.5 ft shadow and a tree casts a 15 ft shadow, then h/15 = 6/1.5 → h = 60 ft.
  • Example 30 (pizzas): dough weight scales with area when pizzas have similar thickness. Areas: for 12" pizza (radius 6) A = \pi(6)^2 = 113.1\text{ in}^2;for16"pizza(radius8)A=; for 16" pizza (radius 8) A =\pi(8)^2 = 201.1\text{ in}^2; weight scales approximately with area, so weight for 16" pizza ≈ 10 oz × (201/113) ≈ 17.8 oz.
  • Example 31 (calorie scaling): volume of cylinders scales with the cube of linear dimensions; jumbo vs regular marshmallows roughly scales by a factor of 1.5 in diameter leading to about 1.5^3 ≈ 3.375 times calories.
  • Example 36 (cost estimation for deck rebuilding): uses area-based pricing to estimate material cost and includes waste factor (~10%).

1.7: Problem Solving and Estimating

  • Problem-solving approach:
    • Start at the end: identify the target question; work backward to determine needed information and procedures.
    • Build a solution pathway with a sequence of steps; estimate missing information if necessary; ignore unnecessary details.
    • Often solve approximately using real-life reasoning and estimates.
  • Examples (select highlights):
    • Example 32 Heartbeats per year: If 80 beats per minute, annual beats ≈ 80 × 60 × 24 × 365 = 42,048,000.
    • Example 33 Thickness and weight of a sheet of paper: a ream (500 sheets) ~ 2 inches thick and ~5 pounds; per-sheet measures: ~0.004 inches thick; ~0.01 pounds per sheet; ~0.16 ounces per sheet.
    • Example 34 Muffins: 12 muffins → 250 calories each; for 20 muffins, total calories = 20 × (250 × 12/12) = 3000 calories; if you eat 4 mini-muffins, total = 4 × 150 = 600 calories.
    • Example 35 Deck cost estimation: cost per square foot for decking boards; compute area of deck (e.g., 16 ft by 24 ft = 384 ft²); cost per board area, then total cost; add waste.
    • Example 36 Car choice: compare gas costs for driving vs flying, including extra costs (tickets, taxi, parking) and possible tax deductions; decide based on total cost and time trade-offs.
  • Problem-solving mindset:
    • Distinguish between exact calculations and estimates.
    • Use proportional reasoning to scale from known quantities to unknowns.
    • When data is incomplete, justify reasonable assumptions and document them.

1.8: Taxes

  • Tax types (how they are described):
    • Flat tax / proportional tax: constant percentage rate regardless of base.
    • Progressive tax: rate increases as base increases.
    • Regressive tax: rate decreases as base increases.
  • Examples:
    • Sales tax example: 9.3% of $140 → tax = $140 × 0.093 = $13.02.
    • Effective tax rate: apply to fully the base amount; example: property tax on a house valued at $215,000 with $3,200 tax → effective rate = 3200/215000 ≈ 0.01488 ≈ 1.49%.
    • U.S. federal income tax is progressive: different portions of income taxed at different rates (e.g., 10% up to $8,500; 15% on the portion above $8,500 up to $34,500 for 2011). Calculation examples show how to combine taxes across brackets and compute an effective rate.
  • Summary:
    • Flat tax: same percentage on all income or purchases.
    • Progressive tax: rate increases with base; effective rate rises with income.
    • Regressive tax: rate falls with higher base in terms of percentage of income.

1.9: Exercises (chapter-end problems)

  • A large collection of practical problems to apply the material—from percents, averages, and proportions to unit conversions, scientific notation, and real-world scenarios.
  • Typical tasks include:
    • Percent calculations, percent change and interpretation of percentage points.
    • Averaging percentages with attention to bases and weights.
    • Setting up and solving proportions and rate problems.
    • Using dimensional analysis to convert units and compute rates.
    • Applying scientific notation to large-scale or small-scale numbers and comparing magnitudes.
    • Estimating quantities in contextual problems (e.g., costs, proportions, growth).

Quick reference of key formulas and concepts

  • Percent, decimal, fraction equivalences:
    • p\% = p/100,\quad d = p/100,\quad \text{fraction} = p/100\,.
  • Part of a whole (percent of a quantity):
    • Part = (percent as decimal) × Whole, or \text{Part} = \left(\frac{p}{100}\right)\times \text{Whole}.
  • Absolute vs relative change:
    • Absolute change: \Delta = \text{Ending} - \text{Starting}.
    • Relative change: \text{Relative change} = \frac{\text{Ending} - \text{Starting}}{\text{Starting}}.
  • Percent change vs percent points:
    • Change from 40% to 50%: absolute change = 10 percentage points; relative change = \frac{50-40}{40}=0.25=25\%.
  • Averaging percents:
    • Arithmetic mean (equal weights): \bar{x} = \frac{x1+x2+\cdots+x_n}{n}.
    • Weighted average: \text{Average} = \frac{\sumi wi xi}{\sumi w_i}, where weights sum to 1 if already in decimal form.
  • Geometric mean for growth rates:
    • Growth multipliers: gi = 1 + ri.
    • Geometric mean: \text{GM} = \left(\prod{i=1}^n gi\right)^{1/n},
    • Average rate: \text{Average rate} = \text{GM} - 1.
  • Proportions and rates:
    • Proportion equation: \frac{a}{b} = \frac{c}{d}.cross−multiplication:cross-multiplication:a d = b c.
  • Dimensional analysis (unit cancellation):
    • Multiply by conversion factors so units cancel to the desired units.
  • Scientific notation:
    • Form: a\times 10^r,\quad 1\le a<10, r\in\mathbb{Z}
    • Multiplication: (a\times 10^r)(b\times 10^s) = (ab)\times 10^{r+s}
    • Division: \frac{a\times 10^r}{b\times 10^s} = \left(\frac{a}{b}\right)\times 10^{r-s}
  • Geometry (basic areas/volumes):
    • Rectangle area: A = L\cdot W
    • Circle area: A = \pi r^2\,;\quad \text{Circumference } C = 2\pi r
    • Rectangular prism volume: V = L\cdot W\cdot H
    • Cylinder volume: V = \pi r^2 H$$
  • Problem-solving process (summary):
    • Identify the question, work backward to identify needed information, build a solution pathway, fill in missing information by estimation if necessary, ignore irrelevant data, and solve.