Length Conversions

1 foot (ft) = 12 inches (in)

1 yard (yd) = 3 feet = 36 inches

1 mile (mi) = 5,280 feet = 1,760 yards

Weight Conversions

1 pound (lb) = 16 ounces (oz)

1 ton = 2,000 pounds

Capacity (Volume) Conversions

1 cup (c) = 8 fluid ounces (fl oz)

1 pint (pt) = 2 cups = 16 fluid ounces

1 quart (qt) = 2 pints = 4 cups

1 gallon (gal) = 4 quarts = 8 pints = 16 cups

📝 Conversion Method

• Larger to smaller: Multiply by conversion factor

• Smaller to larger: Divide by conversion factor

Metric Prefixes

kilo (k) = 1,000 base units

hecto (h) = 100 base units

deka (da) = 10 base units

BASE UNIT (meter, liter, gram)

deci (d) = 0.1 base unit

centi (c) = 0.01 base unit

milli (m) = 0.001 base unit

Length Conversions

1 kilometer (km) = 1,000 meters (m)

1 meter (m) = 100 centimeters (cm) = 1,000 millimeters (mm)

1 centimeter (cm) = 10 millimeters (mm)

Mass Conversions

1 kilogram (kg) = 1,000 grams (g)

1 gram (g) = 1,000 milligrams (mg)

1 metric ton = 1,000 kilograms

Capacity (Volume) Conversions

1 kiloliter (kL) = 1,000 liters (L)

1 liter (L) = 1,000 milliliters (mL)

⚠️ Important Rule

For AREA (squared units): Square the conversion factor

For VOLUME (cubic units): Cube the conversion factor

🔹 Square Unit Conversions (Area)

1 ft² = (12)² in² = 144 in²

1 yd² = (3)² ft² = 9 ft²

1 mi² = (5,280)² ft² = 27,878,400 ft²

1 m² = (100)² cm² = 10,000 cm²

1 cm² = (10)² mm² = 100 mm²

1 km² = (1,000)² m² = 1,000,000 m²

🔹 Cubic Unit Conversions (Volume)

1 ft³ = (12)³ in³ = 1,728 in³

1 yd³ = (3)³ ft³ = 27 ft³

1 m³ = (100)³ cm³ = 1,000,000 cm³

1 cm³ = (10)³ mm³ = 1,000 mm³

1 km³ = (1,000)³ m³ = 1,000,000,000 m³

💡 Example

Convert 5 m² to cm²:

Since 1 m = 100 cm, then 1 m² = (100)² cm² = 10,000 cm²

Therefore: 5 m² = 5 × 10,000 = 50,000 cm²

What is Precision?

Precision is the smallest unit that a measuring instrument can measure. It is determined by the place value of the last significant digit in a measurement.

🔹 How to Find Precision

Step 1: Identify the last non-zero digit to the right in the measurement

Step 2: Determine the place value of that digit

Step 3: That place value is the precision

📝 Examples

• 25.3 cm → Precision = 0.1 cm (tenths place)

• 14.56 m → Precision = 0.01 m (hundredths place)

• 120 ft → Precision = 10 ft (tens place)

• 3.005 kg → Precision = 0.001 kg (thousandths place)

What is Greatest Possible Error?

The greatest possible error (GPE) is half of the precision. It represents the maximum amount a measurement could be off.

🔹 Formula

GPE = Precision ÷ 2

Also written as: GPE = ±(Precision/2)

📝 Examples

Example 1: Measurement = 20.36 m

Precision = 0.01 m → GPE = 0.01 ÷ 2 = 0.005 m

True value: 20.36 ± 0.005 m (between 20.355 m and 20.365 m)

Example 2: Measurement = 4.2 lb

Precision = 0.1 lb → GPE = 0.1 ÷ 2 = 0.05 lb

True value: 4.2 ± 0.05 lb (between 4.15 lb and 4.25 lb)

Concept

When measurements have uncertainty (GPE), the calculated area or volume also has a range. We can find the minimum and maximum possible values.

🔹 Method

For Maximum Area/Volume:

Use the maximum possible values of each dimension (measurement + GPE)

For Minimum Area/Volume:

Use the minimum possible values of each dimension (measurement - GPE)

📝 Example

Rectangle: Length = 5.2 cm, Width = 3.4 cm (both measured to nearest 0.1 cm)

Precision = 0.1 cm → GPE = 0.05 cm

Maximum dimensions: 5.25 cm × 3.45 cm

Maximum Area = 5.25 × 3.45 = 18.1125 cm²

Minimum dimensions: 5.15 cm × 3.35 cm

Minimum Area = 5.15 × 3.35 = 17.2525 cm²

What is Percent Error?

Percent error measures how far off a measured or experimental value is from the actual or accepted value, expressed as a percentage.

🔹 Formula

Percent Error = |Measured Value - Actual Value| / Actual Value × 100%

The absolute value ensures the error is always positive

📝 Example

Actual weight of object: 50 g

Measured weight: 48 g

Percent Error = |48 - 50| / 50 × 100%

Percent Error = 2 / 50 × 100%

Percent Error = 4%

Concept

When calculating percent error for area or volume, errors in individual measurements compound. The total percent error is approximately the sum of individual percent errors.

🔹 Formulas

For Area (A = l × w):

% Error in Area ≈ % Error in length + % Error in width

For Volume (V = l × w × h):

% Error in Volume ≈ % Error in length + % Error in width + % Error in height

📝 Example

Mass measurement has 2% error

Volume measurement has 3% error

Density = Mass / Volume

% Error in Density ≈ 2% + 3% = 5%

Relationship

Density, mass, and volume are related through a fundamental formula in physics and chemistry.

🔹 Main Formula

Density (D) = Mass (m) / Volume (V)

D = m / V

🔹 Derived Formulas

Mass (m) = Density (D) × Volume (V)

Volume (V) = Mass (m) / Density (D)

🔹 Common Units

Density: g/cm³, kg/m³, g/mL

Mass: g (grams), kg (kilograms)

Volume: cm³, m³, mL, L

Note: 1 mL = 1 cm³

📝 Examples

Example 1: Find Density

Mass = 50 g, Volume = 25 cm³

Density = 50 g / 25 cm³ = 2 g/cm³

Example 2: Find Mass

Density = 0.8 g/cm³, Volume = 100 cm³

Mass = 0.8 g/cm³ × 100 cm³ = 80 g

Example 3: Find Volume

Mass = 120 g, Density = 3 g/cm³

Volume = 120 g / 3 g/cm³ = 40 cm³

🔹 Key Formulas

🔹 Conversion Rules