Calculator Formulas Quick Reference

Financial Calculations

Loan & Mortgage Payments

Monthly Payment Formula: $M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}$

Where:

• $M$ = Monthly payment
• $P$ = Principal loan amount
• $r$ = Monthly interest rate (annual rate ÷ 12)
• $n$ = Total number of payments (years × 12)

Example: $200,000 loan at 5% APR for 30 years

• $P = 200,000$
• $r = 0.05 ÷ 12 = 0.004167$
• $n = 30 × 12 = 360$
• M = 200,000 × \frac{0.004167(1.004167)^{360}}{(1.004167)^{360} - 1} = \1,073.64$

Compound Interest

Formula: $A = P(1 + \frac{r}{n})^{nt}$

Where:

• $A$ = Final amount
• $P$ = Principal
• $r$ = Annual interest rate
• $n$ = Compounding frequency per year
• $t$ = Time in years

Example: 1,000 at 6% compounded monthly for 5 years $$A = 1000(1 + \frac{0.06}{12})^{12 \times 5} = \1,348.85$$

Simple Interest

Formula: $I = P \times r \times t$ $A = P + I = P(1 + rt)$

Example: 500 at 4% for 3 years $$I = 500 × 0.04 × 3 = \60$$

Percentage Calculations

Basic Percentage Formulas

• Percentage of a number: $\frac{P}{100} \times N$
• What percent is X of Y: $\frac{X}{Y} \times 100$
• Percentage change: $\frac{\text{New} - \text{Old}}{\text{Old}} \times 100$

Examples:

• 25% of 80: $\frac{25}{100} \times 80 = 20$
• 15 is what % of 60: $\frac{15}{60} \times 100 = 25\%$
• Change from 50 to 65: $\frac{65-50}{50} \times 100 = 30\%$ increase

Tip Calculations

Tip Amount: \text{Bill} \times \frac{\text{Tip %}}{100} Total: $\text{Bill} + \text{Tip}$

Example: $45 bill with 18% tip

• Tip: $45 × 0.18 =$8.10
• Total: $45 +$8.10 = $53.10

Geometry Formulas

Area Formulas

| Shape | Formula | Example | |-------|---------|---------| | Rectangle | $A = l \times w$ | $A = 5 \times 3 = 15$ | | Square | $A = s^2$ | $A = 4^2 = 16$ | | Triangle | $A = \frac{1}{2}bh$ | $A = \frac{1}{2} \times 6 \times 4 = 12$ | | Circle | $A = \pi r^2$ | $A = \pi \times 3^2 = 28.27$ | | Trapezoid | $A = \frac{1}{2}(b_1 + b_2)h$ | $A = \frac{1}{2}(5 + 3) \times 4 = 16$ |

Perimeter/Circumference

| Shape | Formula | Example | |-------|---------|---------| | Rectangle | $P = 2(l + w)$ | $P = 2(5 + 3) = 16$ | | Square | $P = 4s$ | $P = 4 \times 4 = 16$ | | Triangle | $P = a + b + c$ | $P = 3 + 4 + 5 = 12$ | | Circle | $C = 2\pi r$ | $C = 2\pi \times 3 = 18.85$ |

Volume Formulas

| Shape | Formula | Example | |-------|---------|---------| | Cube | $V = s^3$ | $V = 3^3 = 27$ | | Rectangular Prism | $V = l \times w \times h$ | $V = 4 \times 3 \times 2 = 24$ | | Cylinder | $V = \pi r^2 h$ | $V = \pi \times 2^2 \times 5 = 62.83$ | | Sphere | $V = \frac{4}{3}\pi r^3$ | $V = \frac{4}{3}\pi \times 3^3 = 113.10$ | | Cone | $V = \frac{1}{3}\pi r^2 h$ | $V = \frac{1}{3}\pi \times 2^2 \times 6 = 25.13$ |

Statistics & Probability

Mean, Median, Mode

• Mean (Average): $\bar{x} = \frac{\sum x_i}{n}$
• Median: Middle value when data is ordered
• Mode: Most frequently occurring value

Example: Data set: 2, 4, 4, 6, 8, 10

• Mean: $(2+4+4+6+8+10) ÷ 6 = 5.67$
• Median: $(4+6) ÷ 2 = 5$
• Mode: $4$

Standard Deviation

Population: $\sigma = \sqrt{\frac{\sum(x_i - \mu)^2}{N}}$ Sample: $s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}$

Probability

• Basic Probability: $P(A) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}}$
• Complement: $P(A') = 1 - P(A)$
• Independent Events: $P(A \text{ and } B) = P(A) \times P(B)$

BMI & Health Calculations

Body Mass Index

Formula: $\text{BMI} = \frac{\text{weight (kg)}}{\text{height (m)}^2}$

Imperial: $\text{BMI} = \frac{\text{weight (lbs)} \times 703}{\text{height (inches)}^2}$

Categories:

• Underweight: BMI < 18.5
• Normal: 18.5 ≤ BMI < 25
• Overweight: 25 ≤ BMI < 30
• Obese: BMI ≥ 30

Example: 70 kg, 1.75 m tall $\text{BMI} = \frac{70}{1.75^2} = \frac{70}{3.06} = 22.9$

Algebraic Formulas

Quadratic Formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For equation: $ax^2 + bx + c = 0$

Distance Formula

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Slope Formula

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Trigonometry

Basic Ratios

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
• $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Pythagorean Theorem

$a^2 + b^2 = c^2$

Example: Find hypotenuse with legs 3 and 4 $c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Quick Mental Math Tips

1. Multiply by 11: For 2-digit numbers, add digits and put sum in middle

   • $23 × 11 = 2(2+3)3 = 253$

2. Square numbers ending in 5:

   • $25^2 = 2 × 3 = 6$, append 25 → $625$
   • $35^2 = 3 × 4 = 12$, append 25 → $1225$

3. Percentage shortcuts:

   • 10% = move decimal left one place
   • 5% = half of 10%
   • 15% = 10% + 5%

4. Rule of 72: Years to double = $\frac{72}{\text{interest rate}}$

   • At 6% interest: $\frac{72}{6} = 12$ years to double