Rational Numbers – Fractions, Decimals & Percents | Full Study Notes

• Competency & Learning Goal

  • Describe and represent any given rational number as fraction, decimal, or percentage.

  • Master the skills to convert among these three forms quickly and accurately.

  Warm-Up Problem of the Day

  • Scenario: 2 pizzas are ordered for an entire class.

    • Group A: 4 people share 1 pizza ➔ each gets $\frac{1}{4}$ of a pizza.

    • Group B: 8 people share 1 pizza ➔ each gets $\frac{1}{8}$ of a pizza.

  • Question: “Who got a bigger share?”

  • Resolution: \frac{1}{4} > \frac{1}{8}, so every person in Group A enjoyed the larger slice.

  • Take-away: Visualizes that the same whole (one pizza) can give different-sized parts depending on the number of people—an intuitive gateway to fractions.

  Rational Numbers

  • Definition: Any real number expressible as $\frac{a}{b}$ where

    • $a, b$ are integers

    • $b \neq 0$

    • $a$ and $b$ are relatively prime (their greatest common factor, GCF, is 1).

  • If a real number cannot be expressed that way, it is called irrational.

  • Relatively Prime: Two or more integers whose GCF = 1.

    • e.g. $8$ & $15$ are relatively prime (GCF = 1).

  Fractions

  • Canonical rational-number form: $\frac{\text{numerator}}{\text{denominator}}$.

    • Numerator (top) = dividend; Denominator (bottom) = divisor.

    • The fraction bar signals division.

  • Everyday examples written in the slides:

    • $\frac{3}{6},\ \frac{3}{3},\ \frac{5}{1},\ -\frac{3}{5}$ (note that negatives are also allowed).

  Decimals

  • A decimal fraction has a denominator that is a power of 10.

    • Written using a decimal point instead of a fraction bar.

  • Place-value refresher (moving left ↔ right):

  • $\dots$ hundreds ➔ tens ➔ ones ➔ tenths ➔ hundredths ➔ thousandths ➔ ten-thousandths ➔ hundred-thousandths ➔ millionths $\dots$

• Slide examples:

  • $\frac{1}{10}=0.1$

  • $\frac{31}{1000}=0.031$

  • Extended chart: $\underline{\phantom{0}}\,\underline{\phantom{0}}\,\underline{0}\,\underline{7}\,\underline{3}\,\underline{8}\,\underline{0}\,\underline{0}\,\underline{0}\,\underline{2}\,\underline{4}$

    • hundreds ( ) tens ( ) ones (0) . tenths (7) hundredths (3) thousandths (8) ten-thousandths (0) etc.

Percentages

• A percent is “per hundred”; symbol “%”.

• Because $x\% = \frac{x}{100}$, every percent is automatically a rational number.

  • Eg. $10\% = \frac{10}{100} = 0.10$ ➔ rational.

Conversion Procedures

1. Fraction ➔ Decimal

   • Perform long division: numerator ÷ denominator.

2. Fraction ➔ Percent

   • First turn into a decimal (step 1), then multiply by $100$ and append “%”.

3. Decimal ➔ Percent

   • Multiply the decimal directly by $100$, add “%”.

Worked Examples from Slides

• Example 1

  • Input: $\frac34$

  • Long-division result: $0.75$

  • Percent: $0.75 \times 100 = 75\%$

• Example 2

  • Input: $\frac5{32}$

  • Long-division (shown partially): $0.15625 \approx 0.16$ (rounded in slide)

  • Percent: $0.16 \times 100 = 16\%$ (note: exact value $15.625\%$)

• Example 3

  • Input: $\frac{25}{100}$

  • Decimal: $0.25$

  • Percent: $25\%$

• Example 4

  • Input: $\frac83 = 2\tfrac23$ (mixed number written “2 $\frac23$”)

  • Decimal: repeating $2.666\ldots \approx 2.67$

  • Percent: $2.666\ldots \times 100 \approx 267\%$

“Try This” Practice Set (with slide answers)

1. $\frac{20}{40}=\frac12$

   • Decimal: $0.5$

   • Percent: $50\%$

2. $\frac{8}{15}=0.5333\ldots \approx 0.53$

   • Percent: $53\%$

3. $\frac{8}{5}=1.6$

   • Percent: $160\%$

Visual Plot — Number Line Activity

• Points labeled A through F placed on 0 to 2 line.

  • A: $\frac34 = 0.75$

  • B: $\frac45 =0.8$? (slide lists 0.15; mapping B unclear—use slide values)

  • Reported coordinates:

    • A : $0.75$

    • B : $0.15$

    • C : $0.25$

    • D : $0.5$

    • E : $0.53$

    • F : $1.6$

• Purpose: Visually reinforce magnitude comparisons across forms.

Assessment Items (textbook p. 100)

Convert as indicated.

• $1\frac45 = 1 + \frac45 = \frac{9}{5} = 1.80$ (decimal shown)

• $5\frac34 = 5.75$

• $2\frac14 = 2.25$

• $0.161616\ldots$ (repeating) = $0.16\overline{16}=\frac{16}{99}$ (fraction form not provided on slide, but mathematical closure added).

Key Takeaways & Tips

• Any fraction, decimal, or percent that terminates or repeats is rational.

• Long division is the universal bridge from fractions to decimals.

• Multiply by $100$ to jump from decimal to percent; divide by $100$ (or move decimal 2 places left) to go back.

• Always watch rounding vs. exact form—it affects percent accuracy.

• Converting mixed numbers: change to improper fraction first.

• Repeating decimals ➔ fraction using algebra: e.g., for $x = 0.161616\ldots$,

  • $100x = 16.1616\ldots$

  • Subtract $x$: $99x = 16$ ➔ $x = \frac{16}{99}$.

Ethical & Real-World Relevance

• Fractions and percents underpin fair sharing, discounts, statistics, interest rates.

• Precision matters—financial and scientific contexts may forbid casual rounding.

Quick Reference Formula Sheet

• Fraction to Decimal: $\text{Decimal}=\dfrac{\text{Numerator}}{\text{Denominator}}$ (long division).

• Decimal to Percent: $\text{Percent}=\text{Decimal}\times 100\%$.

• Percent to Decimal: $\text{Decimal}=\dfrac{\text{Percent}}{100}$.

• Fraction to Percent: $\dfrac{a}{b} \rightarrow \left(\dfrac{a}{b}\right)\times100\%$.

• Repeating Decimal to Fraction (generic):

  • Let $x=$ repeating decimal.

  • Multiply by the power of 10 that aligns repeats, subtract, solve for $x$.

Practice Checklist Before the Exam

• ☐ Convert at least 10 random fractions to decimals (some terminating, some repeating).

• ☐ Convert the same set to percents; verify with calculator.

• ☐ Plot mixed, improper, and decimal numbers on a number line.

• ☐ Re-work textbook problems 17, 18, 24, 27 without notes.

• ☐ Explain—out loud—the difference between terminating and non-terminating repeating decimals.

• ☐ Time yourself: 30-second limit per conversion for fluency.