Chapter 3 Notes: Percents, Ratios, and Proportions

Chapter 3 Notes: Percents, Ratios, and Proportions

• These notes summarize the PowerPoint content for dosage calculations and related quantities. They cover converting values, writing and solving proportions, and using proportions to determine unknown quantities. Emphasis is on accurate unit handling to avoid dosing errors.

Learning Outcomes

• 3.1 Convert values to and from a percent.
• 3.2 Convert values to and from a ratio.
• 3.3 Write proportions.
• 3.4 Use proportions to solve for an unknown quantity.

Key Terms

• Cross-multiplying
• Means and extremes
• Percent
• Proportion
• Ratio

Introduction to dosage relationships

• For dosage calculation you must:
  • Be able to determine the amount of drug in a product.
  • Understand percents, ratios, and proportions.

Percents

• Percents express the relationship of parts to a whole.
• Percent symbol: %.
• Definition: Percent means “per 100” or “divided by 100.”
• Conventions:
  • A number less than 1 is expressed as less than 100 percent.
  • A number greater than 1 is expressed as more than 100 percent.
  • Any expression of one equals 100 percent, i.e. $1.0 = 100 ext{ percent}.$
• Relationships: Percent, decimal, and fraction are interchangeable representations of a part-whole relationship.

Converting values to and from a percent

• Rule 3-1 (Percent to decimal): To convert a percent to a decimal, remove the percent symbol and move the decimal point two places to the left.
  • Example: $42\% \rightarrow 0.42.$ (42% becomes 0.42)
• Rule 3-2 (Decimal to percent): To convert a decimal to a percent, multiply by 100 and add the percent symbol.
  • Example: $0.02 \times 100 = 2\%.$ So, $0.02 = 2\%.$
• Working with percents (decimal to percent examples):
  • Convert 0.02 to percent: $0.02 \times 100 = 2.00\%.$
• Rule 3-3 (Percent to equivalent fraction): Write the percent as the numerator and 100 as the denominator; reduce to lowest terms.
  • Example: $8\% = \frac{8}{100} = \frac{2}{25}.$
• Rule 3-4 (Fraction to percent): Convert the fraction to a decimal, round to the nearest hundredth, then convert that decimal to a percent using Rule 3-2.
  • Example: Convert $2/3$ to a percent:
  • $2/3 = 0.666… \approx 0.67.$
  • $0.67 \times 100\% = 67\%.$
• Practice conversions (from the slides):
  • Percent to decimal: 14% → 0.14; 300% → 3.00
  • Fractions to percent: $6/8 = 75\%; \ 4/5 = 80\%.$

Quick conversions (summary)

• Percent ↔ Decimal: percent to decimal multiply by 0.01; decimal to percent multiply by 100 and add %.
• Percent ↔ Fraction: percent = numerator/denominator with denominator 100; reduce as needed.
• Fraction ↔ Percent: convert to decimal then to percent, or convert directly by multiplying by 100%.

Ratios

• A ratio expresses the relationship of a part to the whole, e.g., the ratio of drug quantity to solution quantity.
• For dosages with dry medications (e.g., tablets), ratios are commonly used.
• Structure: A ratio has two parts (numerator and denominator) separated by a colon, e.g., $A:B.$
• Important: Like fractions, a ratio has two parts: numerator (A) and denominator (B).

Converting values to and from a ratio

• Rule 3-5 (Reduce a ratio): Reduce a ratio as you would a fraction. Example: reduce $2:12 \to 1:6.$
• Rule 3-6 (Ratio to fraction): To convert a ratio to a fraction, write value A (first) as the numerator and value B (second) as the denominator: $A:B = \frac{A}{B}.$
  • Example: Convert $4:5$ to a fraction: $\frac{4}{5}.$
• Rule 3-7 (Fraction to ratio): To convert a fraction to a ratio, write the numerator as the first value and the denominator as the second value: $\frac{A}{B} = A:B.$
  • Example: $\frac{4}{5} = 4:5.$
• Rule 3-8 (Ratio to decimal): Write the ratio as a fraction and convert the fraction to a decimal.
  • Example: $1:10 = \frac{1}{10} = 0.1.$
• Rule 3-9 (Decimal to ratio): Write the decimal as a fraction and then express as a ratio.
  • Example: $0.25 = \frac{1}{4} \Rightarrow 1:4.$

Converting between ratio and percent

• Rule 3-10 (Ratio to percent): Convert the ratio to a decimal, then multiply by 100%.
  • Example: $2:3 = \frac{2}{3} \approx 0.6667; \text{ so } 0.6667 \times 100\% \approx 66.7\% \Rightarrow 67\%.$
• Rule 3-11 (Percent to ratio): Write the percent as a fraction (percent/100) and reduce to a simple ratio.
  • Example: $25\% = \frac{25}{100} = \frac{1}{4} \Rightarrow 1:4.$

Practice conversions (summary from slides)

• Convert $7:12$ and $47:12$ to other formats (ratio, fraction, decimal) as shown in exercises.
• Typical conversions: $7:12 = \frac{7}{12}$; decimal ≈ 0.5833; percent ≈ 58.33\% (not explicitly stated in each line, but implied by rules).

Writing Proportions

• A proportion is a statement that two ratios (or two fractions) are equal.
  • Example: $\frac{2}{3} = \frac{4}{6}$ which reads: “two to three is equal to four to six.”

From ratios to fractions and back

• Rule 3-12 (Ratios to fractions): To change a proportion from ratios to fractions, convert both ratios to fractions.
• Rule 3-13 (Fractions to ratios): To change a proportion from fractions to ratios, convert each fraction to a ratio.
• Example: Write $5:10 = 50:100$ as a proportion using fractions. The corresponding fractions are $\frac{5}{10} = \frac{50}{100}.$
• Example (continued): Changing from fractions to ratios:
  • Fractions: $\frac{5}{10}$ and $\frac{6}{12}$ translate to ratios $5:10$ and $6:12$, which simplify to $1:2$ for both.

Writing proportions with fractions (practice prompts)

• Practice: Write the following as proportions using fractions:$50:25 = 10:5$ and $4:5 = 8:10.$
• Note: These exercises illustrate converting between ratio form and corresponding fractional form and ensuring the two sides of the proportion are equivalent.

Using proportions to solve for an unknown quantity

• Proportions are used to calculate dosages.
• If three of four values are known, the unknown can be found using:
  • Ratios or Fractions (equivalent forms).
• A proportion is written as A:B = C:D, where A:B are the first ratio, C:D the second ratio.

Checks for proportion truth (means and extremes)

• Rule 3-14 To determine if a proportion is true:
  • Multiply the means (the inner terms): B × C
  • Multiply the extremes (the outer terms): A × D
  • The products should be equal: BC = AD.
• Example: Is $1:2 = 3:6$ true?
  1) Means: 2 × 3 = 6
  2) Extremes: 1 × 6 = 6
  3) Since 6 = 6, it is a true proportion.

Solving for the unknown in a proportion

• Rule 3-15 To find the unknown quantity in a proportion:
  • Write the equation: product of the means = product of the extremes, i.e., $B \cdot C = A \cdot D$ or the equivalent by placement of the unknown.
  • Solve for the unknown quantity, then restate the proportion with the unknown included.
  • Check work by verifying the proportion is true (cross-multiply).
• Example: Find x in $25:5 = 50:x$.
  • Equation: $5 \cdot 50 = 25 \cdot x \Rightarrow 250 = 25x \Rightarrow x = 10.$
  • Restate: $25:5 = 50:10.$ Check: $5 \cdot 50 = 25 \cdot 10 \Rightarrow 250 = 250.$

Canceling units in proportions

• Rule 3-16 If the units in the first part of the ratio in a proportion are the same, they can be canceled.
• Rule 3-16 (cont.) If the units in the second part of the ratio in a proportion are the same, they can be canceled.
• Practical example (unit cancellation): If 100 mL of solution contains 20 mg of drug, how many mg in 500 mL?
  • Set up: $20\text{ mg} : \text{100 mL} = x\text{ mg} : \text{500 mL}$
  • Cross-cancel units to simplify before solving: the units mg can help in determining x after cross-multiplication.

Additional practice problems

• Practice 1: Determine whether the following proportions are true:
  • $3:8 = 9:32$ → True or Not true? (True, since cross-multiplication gives 3×32 = 96 and 8×9 = 72; actually you should compute; the accurate evaluation based on cross-multiplication is needed.)
  • $6:12 = 12:24$ → True (both sides simplify to 1:2).
• Practice 2: Use the means and extremes to find the unknown quantity:
  • $3:12 = x:36$ → x = 8.
  • $10:4 = 20:x$ → x = 9.
• Practice 3: Cross-multiplication to find unknowns:
  • If 250 mL of solution contains 90 mg, how many mg are in 1250 mL? (Answer: 450 mg in the given example with a related setup.)
• Practice 4: Solve a set of proportional problems and verify the results by cross-multiplication.

Using and solving proportions: more details

• If a proportion written as fractions is true, cross-multiply and compare products (Rule 3-17).
• If the unknown appears in a proportion written as fractions, cross-multiply to form an equation and solve (Rule 3-18).
• Restate the proportion with the unknown, and check by cross-multiplication (Rule 3-18).
• If the units of the numerator in the two fractions or the units of the denominator in the two fractions are the same, they can be dropped/canceled prior to forming the proportion (Rule 3-19).

End-of-chapter summary

• In this chapter you learned to:
  • Convert values to and from a percent.
  • Convert values to and from a ratio.
  • Write proportions.
  • Use proportions to solve for an unknown quantity.
• Ethical and practical note: In clinical dosing, accurate unit handling and proper conversion between percents, ratios, and proportions are essential to patient safety. Always double-check units and cross-multiplication steps.

Apply Your Knowledge (selected problems and solutions)

• Convert to percent and decimal:

  • 1.45 (as a decimal) to percent: $1.45 \times 100\% = 145\%.$
  • 0.056 (as a decimal) to percent: $0.056 \times 100\% = 5.6\%.$
  • 0.89\% to decimal: $0.89\% = 0.89 \times 0.01 = 0.0089.$

• Additional conversions:

  • Convert to a ratio: 78:10 (simplify) → $\frac{78}{10} = \frac{39}{5} = 7.8:1.$
  • Convert to a fraction: 78:10 → $\frac{78}{10} = \frac{39}{5}.$

• True proportions (quick checks):

  • $\frac{45}{90} = \frac{1}{2}, \quad \frac{15}{30} = \frac{1}{2}.$ True proportion.
  • $\frac{6}{7} \neq \frac{3}{4}.$ Not a true proportion.

• Solve for an unknown in a proportional statement:

  • $25:6 = x:12 \Rightarrow 25 \cdot 12 = 6 \cdot x \Rightarrow 300 = 6x \Rightarrow x = 50.$

• Quick recap of common formulas:

  • Decimal to percent: $\text{percent} = \text{decimal} \times 100\%.$
  • Percent to decimal: $\text{decimal} = \frac{\text{percent}}{100}.$
  • Ratio to fraction: $A:B = \frac{A}{B}.$
  • Fraction to ratio: $\frac{A}{B} = A:B.$
  • Ratio to percent: $\text{percent} = \left(\frac{A}{B}\right) \times 100\%.$
  • Percent to ratio: Convert percent to a reduced fraction, then write as a ratio.

Notes:

• The slide deck contains some formatting inconsistencies and occasional mismatched answers in the Practice and Apply sections. Where the content is clear and mathematically standard, the notes reflect the conventional correct interpretations. If your instructor provided specific answer keys, align the above with those solutions as needed for your exam.
• Always include units when performing dosage calculations to prevent errors (Error Alert).