conversions

Introduction

• Lesson focuses on American units and conversions.
• Goal: Convert between American units of measure.
• The United States customary system is the measurement system used in the U.S., which is unique as it is the only industrialized nation that does not predominantly use the metric system in its commercial activities.
• The metric system is widely utilized in science, medicine, government, and various industrial sectors.

Overview of Conversions

Types of Conversions

• Length
• Capacity
• Weight
• Time

Conversion Methods

1. Multiplying and Dividing
   • Multiply when converting from a large unit to a smaller unit.
   • Divide when converting from a smaller unit to a larger unit.
2. Using Unit Fractions
   • Each unit fraction must equal one.
   • The numerator contains the desired measurement unit for the answer.
   • The denominator contains the unit to be changed.

Example Conversions

Length Conversion: Inches to Feet

1. Convert 45 inches to feet.
   • Write it in fraction form: $\frac{45\text{ inches}}{1}$
   • Multiply by a unit fraction:
   • Relationship: $1\text{ foot} = 12\text{ inches}$
   • Setup unit fraction:
     • $\frac{1\text{ foot}}{12\text{ inches}}$
   • Simplification:
     • $\text{inches}$ cancels out.
   • Calculation:
     • Numerator: $45 \times 1\text{ foot} = 45\text{ feet}$
     • Denominator: $1 \times 12 = 12$
   • Final result:
     • $\frac{45}{12} \text{ feet} = 3.75\text{ feet}$

Length Conversion: Miles to Yards

2. Convert 6 miles to yards.
   • Write: $\frac{6 \text{ miles}}{1}$
   • First conversion is from miles to feet:
     • Setup: $\frac{5280 \text{ feet}}{1 \text{ mile}}$
   • Second conversion from feet to yards:
     • Setup: $\frac{1 \text{ yard}}{3 \text{ feet}}$
   • Simplification:
     • Miles cancel out, leaving feet.
   • Final calculation:
     • $\frac{6 \times 5280 \text{ yards}}{1 \times 1 \times 3}$
     • $\frac{31680 \text{ yards}}{3} = 10560 \text{ yards}$

Weight Conversions

Weight Conversion: Tons to Pounds

3. Convert 6.3 tons to pounds.
   • Write as: $\frac{6.3 \text{ tons}}{1}$
   • Conversion: $1\text{ ton} = 2000\text{ pounds}$
   • Setup: $\frac{2000 \text{ pounds}}{1 \text{ ton}}$
   • Calculation:
     • $6.3 \times 2000 = 12600 \text{ pounds}$

Weight Conversion: Ounces to Pounds

4. Convert 86 ounces to pounds.
   • Write as: $\frac{86 \text{ ounces}}{1}$
   • Conversion: $1\text{ pound} = 16 \text{ ounces}$
   • Setup unit fraction: $\frac{1 \text{ pound}}{16 \text{ ounces}}$
   • Calculation:
     • Numerator: $86 \text{ pounds}$
     • Denominator: $16$
   • Final result:
     • $\frac{86}{16} = 5.375 \text{ pounds}$

Additional Weight Example

5. Convert 6400 ounces to tons.
   • Write as: $\frac{6400 \text{ ounces}}{1}$
   • Conversion to pounds first:
     • $\frac{1 \text{ pound}}{16 \text{ ounces}}$
   • Then to tons:
     • $\frac{1\text{ ton}}{2000 \text{ pounds}}$
   • Calculation:
     • Resulting numerators: $6400 \text{ tons}$
     • Resulting denominator: $16 \times 2000$
   • Final result:
     • $\frac{6400}{32000} = 0.2 \text{ tons}$

Capacity Conversions

Capacity Conversion: Quarts to Cups

6. Convert 3 quarts to cups.
   • Write as: $\frac{3 \text{ quarts}}{1}$
   • Conversion:
     • $1\text{ quart} = 2\text{ pints}$
   • Use unit fraction: $\frac{2 \text{ pints}}{1 \text{ quart}}$
   • Second conversion:
     • $\frac{2 \text{ cups}}{1 \text{ pint}}$
   • Simplification:
     • Result: $3 \times 2 \times 2 = 12 \text{ cups}$

Capacity Conversion: Fluid Ounces to Pints

7. Convert 72 fluid ounces to pints.
   • Write as: $\frac{72 \text{ fluid ounces}}{1}$
   • First, convert to cups:
     • Conversion: $\frac{1 \text{ cup}}{8 \text{ fluid ounces}}$
   • Then, from cups to pints:
     • Conversion: $\frac{1 \text{ pint}}{2 \text{ cups}}$
   • Simplification:
     • Result: 72 \text{ fluid ounces } \rightarrow \text{ total pints = 4.5 \text{ pints} $$

Conclusion

• The examples illustrate the process of converting various measures across different contexts, ensuring simplification throughout the calculations.
• These practical applications help solidify understanding of unit conversions within the American system of measurement.