Solved Exercises on Rate Conversion in Integrated Math 1

Master rate conversion: unit rates, dimensional analysis, complex conversions through these 5 detailed exercises with comprehensive solutions.

Solution: Exercises 1 to 3
1 Unit rate conversion
Exercise 1
Convert 60 miles per hour to feet per second. (1 mile = 5280 feet, 1 hour = 3600 seconds)
Definition:

Rate conversion: Changing a rate from one unit of measurement to another

Unit rate: A rate expressed as a quantity per single unit

Dimensional analysis: Using conversion factors to change units

Dimensional analysis method:

To convert rates:

  1. Identify the starting and ending units
  2. Find conversion factors
  3. Set up ratios so unwanted units cancel
  4. Multiply across and divide by denominators
  5. Simplify to get the final rate
Start
\(60 \frac{\text{miles}}{\text{hour}}\)
Convert miles to feet
\(60 \times \frac{5280}{1} \frac{\text{feet}}{\text{hour}}\)
Convert hours to seconds
\(60 \times \frac{5280}{3600} \frac{\text{feet}}{\text{second}}\)
Simplify
\(88 \frac{\text{feet}}{\text{second}}\)
Step 1: Write the starting rate

\(60 \frac{\text{miles}}{\text{hour}}\)

Step 2: Set up conversion factors

We need: miles → feet and hours → seconds

Given: 1 mile = 5280 feet, 1 hour = 3600 seconds

Step 3: Arrange conversion factors

For miles to feet: \(\frac{5280 \text{ feet}}{1 \text{ mile}}\) (miles cancel out)

For hours to seconds: \(\frac{1 \text{ hour}}{3600 \text{ seconds}}\) (hours cancel out)

Step 4: Multiply the conversion factors

\(60 \frac{\text{miles}}{\text{hour}} \times \frac{5280 \text{ feet}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}}\)

Step 5: Cancel units and calculate

\(60 \times \frac{5280}{3600} = \frac{316800}{3600} = 88\)

Step 6: Write the final rate

\(88 \frac{\text{feet}}{\text{second}}\)

\(88 \frac{\text{feet}}{\text{second}}\)
Final answer:

60 miles per hour equals 88 feet per second

Applied rules:

• Dimensional Analysis: Multiply by conversion factors arranged so unwanted units cancel

• Unit Cancellation: Place units in numerators/denominators to facilitate cancellation

• Conversion Factors: Equal quantities expressed as ratios (e.g., \(\frac{5280 \text{ ft}}{1 \text{ mi}}\))

Tip: Always arrange conversion factors so unwanted units cancel out!
Tip: Write units with each number to track cancellations.
2 Time conversion
Exercise 2
A printer prints 120 pages in 4 minutes. Convert this rate to pages per hour.
Definition:

Rate: A comparison of two quantities with different units

Unit conversion: Changing the time unit from minutes to hours

Conversion factor: 1 hour = 60 minutes

Start
\(120 \frac{\text{pages}}{4 \text{ minutes}}\)
Simplify
\(30 \frac{\text{pages}}{\text{minute}}\)
Convert minutes to hours
\(30 \times 60 = 1800 \frac{\text{pages}}{\text{hour}}\)
Step 1: Write the given rate

\(\frac{120 \text{ pages}}{4 \text{ minutes}}\)

Step 2: Simplify to unit rate (per minute)

\(\frac{120}{4} = 30\) pages per minute

So: \(30 \frac{\text{pages}}{\text{minute}}\)

Step 3: Convert minutes to hours

Since 1 hour = 60 minutes

Pages per hour = Pages per minute × Minutes per hour

Step 4: Calculate the final rate

\(30 \frac{\text{pages}}{\text{minute}} \times 60 \frac{\text{minutes}}{\text{hour}} = 1800 \frac{\text{pages}}{\text{hour}}\)

Step 5: Verify the units

Minutes cancel out, leaving \(\frac{\text{pages}}{\text{hour}}\)

\(1800 \frac{\text{pages}}{\text{hour}}\)
Final answer:

The printer prints 1800 pages per hour

Applied rules:

• Unit Rate: Divide to find the rate per single unit first

• Time Conversion: Use the relationship between time units

• Multiplication Factor: Multiply by the number of smaller units in the larger unit

Tip: Simplify to unit rate first, then convert to desired time unit!
Tip: Remember: 1 hour = 60 minutes = 3600 seconds.
3 Volume conversion
Exercise 3
A pump fills a tank at a rate of 5 gallons per minute. Convert this rate to quarts per hour. (1 gallon = 4 quarts)
Definition:

Volume rate: A rate involving volume measurements

Multiple conversions: Converting both volume and time units

Conversion factor: 1 gallon = 4 quarts

Start
\(5 \frac{\text{gallons}}{\text{minute}}\)
Convert gallons to quarts
\(5 \times 4 = 20 \frac{\text{quarts}}{\text{minute}}\)
Convert minutes to hours
\(20 \times 60 = 1200 \frac{\text{quarts}}{\text{hour}}\)
Step 1: Write the starting rate

\(5 \frac{\text{gallons}}{\text{minute}}\)

Step 2: Convert gallons to quarts

Since 1 gallon = 4 quarts

\(5 \frac{\text{gallons}}{\text{minute}} \times \frac{4 \text{ quarts}}{1 \text{ gallon}} = 20 \frac{\text{quarts}}{\text{minute}}\)

Step 3: Convert minutes to hours

Since 1 hour = 60 minutes

\(20 \frac{\text{quarts}}{\text{minute}} \times \frac{60 \text{ minutes}}{1 \text{ hour}} = 1200 \frac{\text{quarts}}{\text{hour}}\)

Step 4: Verify the units

Gallons cancel, minutes cancel, leaving quarts per hour

Step 5: Write the final rate

\(1200 \frac{\text{quarts}}{\text{hour}}\)

\(1200 \frac{\text{quarts}}{\text{hour}}\)
Final answer:

The pump fills at a rate of 1200 quarts per hour

Applied rules:

• Sequential Conversion: Convert one unit at a time

• Dimensional Analysis: Use conversion factors to cancel unwanted units

• Volume Relationships: Know common volume conversions (gallons to quarts, etc.)

Tip: When converting multiple units, tackle them one at a time!
Tip: Common volume conversions: 1 gallon = 4 quarts = 8 pints = 16 cups.
Rules and methods, laws,...
\(R_1 \times \frac{U_{1num}}{U_{1den}} \times \frac{U_{2num}}{U_{2den}} = R_2 \frac{U_{2num}}{U_{2den}}\)
Dimensional Analysis
\(\text{New Rate} = \text{Old Rate} \times \frac{\text{Conversion Factor 1}}{1} \times \frac{\text{Conversion Factor 2}}{1}\)
Conversion Formula
Length
1 mile = 5280 ft
1 foot = 12 inches
Time
1 hour = 60 min
1 minute = 60 sec
Volume
1 gal = 4 qt
1 qt = 2 pt
Weight
1 lb = 16 oz
1 ton = 2000 lbs
Unit Equality: Conversion factors are equal quantities expressed as ratios.
Cancellation Rule: Units in numerator and denominator cancel out.
Solution: Exercises 4 to 5
4 Complex rate conversion
Exercise 4
A truck travels at 45 miles per hour. Convert this rate to meters per second. (1 mile = 1609.34 meters, 1 hour = 3600 seconds)
Definition:

Complex rate conversion: Converting rates with metric and imperial units

International System: Using standard metric units (meters, seconds)

Decimal conversion factors: More precise conversion values

Start
\(45 \frac{\text{miles}}{\text{hour}}\)
Convert miles to meters
\(45 \times 1609.34 = 72420.3 \frac{\text{meters}}{\text{hour}}\)
Convert hours to seconds
\(\frac{72420.3}{3600} = 20.12 \frac{\text{meters}}{\text{second}}\)
Step 1: Write the starting rate

\(45 \frac{\text{miles}}{\text{hour}}\)

Step 2: Convert miles to meters

Using conversion: 1 mile = 1609.34 meters

\(45 \frac{\text{miles}}{\text{hour}} \times \frac{1609.34 \text{ meters}}{1 \text{ mile}} = 72420.3 \frac{\text{meters}}{\text{hour}}\)

Step 3: Convert hours to seconds

Using conversion: 1 hour = 3600 seconds

\(72420.3 \frac{\text{meters}}{\text{hour}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}} = \frac{72420.3}{3600} \frac{\text{meters}}{\text{second}}\)

Step 4: Calculate the final rate

\(\frac{72420.3}{3600} = 20.12\) meters per second

Step 5: Verify the units

Miles cancel, hours cancel, leaving meters per second

\(20.12 \frac{\text{meters}}{\text{second}}\)
Final answer:

The truck travels at approximately 20.12 meters per second

Applied rules:

• Precision: Use accurate conversion factors for precise results

• Sequential Conversion: Handle one unit conversion at a time

• Decimal Arithmetic: Perform calculations with decimals accurately

Tip: For metric conversions, use 1609.34 meters per mile for precision!
5 Multi-step conversion
Exercise 5
Water flows through a pipe at a rate of 2 cubic feet per second. Convert this rate to gallons per minute. (1 cubic foot = 7.48 gallons, 1 minute = 60 seconds)
Definition:

Volumetric flow rate: Rate of volume flowing per unit time

Volume conversion: Cubic feet to gallons

Time conversion: Seconds to minutes

Start
\(2 \frac{\text{ft}^3}{\text{sec}}\)
Convert cubic feet to gallons
\(2 \times 7.48 = 14.96 \frac{\text{gallons}}{\text{sec}}\)
Convert seconds to minutes
\(14.96 \times 60 = 897.6 \frac{\text{gallons}}{\text{min}}\)
Step 1: Write the starting rate

\(2 \frac{\text{cubic feet}}{\text{second}}\)

Step 2: Convert cubic feet to gallons

Using conversion: 1 cubic foot = 7.48 gallons

\(2 \frac{\text{ft}^3}{\text{sec}} \times \frac{7.48 \text{ gallons}}{1 \text{ ft}^3} = 14.96 \frac{\text{gallons}}{\text{second}}\)

Step 3: Convert seconds to minutes

Using conversion: 1 minute = 60 seconds

\(14.96 \frac{\text{gallons}}{\text{second}} \times \frac{60 \text{ seconds}}{1 \text{ minute}} = 897.6 \frac{\text{gallons}}{\text{minute}}\)

Step 4: Verify the units

Cubic feet cancel, seconds cancel, leaving gallons per minute

Step 5: Write the final rate

\(897.6 \frac{\text{gallons}}{\text{minute}}\)

\(897.6 \frac{\text{gallons}}{\text{minute}}\)
Final answer:

The water flows at a rate of 897.6 gallons per minute

Applied rules:

• Volumetric Conversion: Use appropriate volume conversion factors

• Unit Cancellation: Ensure all unwanted units cancel properly

• Sequential Processing: Handle volume and time conversions separately

Tip: For volumetric rates, convert volume first, then time!
Tip: Remember: 1 cubic foot ≈ 7.48 gallons.
Comprehensive Guide to Rate Conversion
\(\text{New Rate} = \text{Old Rate} \times \prod \left(\frac{\text{New Unit}}{\text{Old Unit}}\right)\)
General Conversion Formula
Key definitions:

Rate: A comparison of two quantities with different units (e.g., miles per hour)

Unit rate: A rate with a denominator of 1 (e.g., 60 miles per 1 hour)

Dimensional analysis: Using conversion factors to change units while preserving the value

Conversion factor: A ratio expressing equal quantities in different units

Rate conversion: Changing a rate from one set of units to another

Complete methodology:
  1. Identify starting units: Determine the units of the given rate
  2. Identify target units: Determine the units for the desired rate
  3. Find conversion factors: Locate the relationships between units
  4. Arrange factors: Position conversion factors so unwanted units cancel
  5. Multiply: Perform the multiplication and division
  6. Verify: Check that the final units match the target units
Tip 1: Always write units with your numbers to track cancellations!
Tip 2: Arrange conversion factors so the unit you want to cancel is in both numerator and denominator.
Tip 3: Common time conversions: 60 sec/min, 60 min/hr, 24 hr/day.
Tip 4: Common length conversions: 12 in/ft, 3 ft/yd, 5280 ft/mile.
Tip 5: Common volume conversions: 8 fl oz/cup, 2 cups/pint, 2 pints/qt, 4 qt/gal.
Common errors: Forgetting to write units, misplacing conversion factors, incorrect arithmetic, not cancelling units properly.
Exam preparation: Practice common conversion factors, focus on dimensional analysis, master unit cancellation.
Essential conversion facts:

• Length: 1 mile = 5280 feet = 1760 yards

• Time: 1 hour = 60 minutes = 3600 seconds

• Volume: 1 gallon = 4 quarts = 8 pints = 16 cups

• Weight: 1 pound = 16 ounces, 1 ton = 2000 pounds

• Metric: 1 meter = 100 centimeters = 1000 millimeters

Rate Conversion Workflow

📊
Conversion Process
1
Identify Units
2
Find Conversion Factors
3
Arrange Factors
4
Multiply
5
Verify Units
Common Conversion Factors
Length: 1 mile = 5280 ft
Time: 1 hr = 3600 sec
Volume: 1 gal = 4 qt
Weight: 1 lb = 16 oz
Speed: mi/hr → ft/sec
Flow: ft³/sec → gal/min
Master Unit Cancellation to Excel in Rate Conversion!

Questions & Answers

Question: How do I know which conversion factor to put in the numerator versus the denominator?

Answer: The key is to place the conversion factor so that the unit you want to cancel is in both the numerator and denominator. The unit you want to keep goes in the position that doesn't cancel.

Example: Converting 60 mph to ft/sec

  • Starting with miles in the numerator, I want to cancel miles, so I put miles in the denominator of the conversion factor: \(\frac{5280 \text{ ft}}{1 \text{ mile}}\)
  • Starting with hours in the denominator, I want to cancel hours, so I put hours in the numerator of the conversion factor: \(\frac{1 \text{ hour}}{3600 \text{ sec}}\)

The goal is to cancel out unwanted units and leave only the desired units!

Question: What should I do if I need to convert multiple units at once?

Answer: Convert one unit at a time sequentially. You can multiply all conversion factors together in a single calculation:

Example: Converting 60 mph to ft/sec

\(60 \frac{\text{miles}}{\text{hour}} \times \frac{5280 \text{ ft}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ sec}}\)

This equals: \(\frac{60 \times 5280}{3600} \frac{\text{ft}}{\text{sec}} = 88 \frac{\text{ft}}{\text{sec}}\)

The miles cancel with miles, hours cancel with hours, leaving feet per second.

Work systematically through each unit that needs conversion!

Question: How do I verify that my rate conversion is correct?

Answer: There are several verification methods:

  1. Unit check: Verify that the final units match what you wanted
  2. Magnitude check: See if the numerical value makes sense (e.g., feet per second should be smaller than miles per hour)
  3. Reverse conversion: Convert back to the original units to see if you get the original value
  4. Estimation: Use approximate values to see if your answer is reasonable

Example: 60 mph ≈ 88 ft/sec

Checking: 88 ft/sec × 3600 sec/hr ÷ 5280 ft/mile ≈ 60 mph ✓

The reverse conversion confirms our answer is correct!