Solved Exercises on Using Units to Solve Problems in Integrated Math 1

Master using units to solve problems: dimensional analysis, unit conversion, rate problems, and complex unit applications through these 5 detailed exercises with comprehensive solutions.

Solution: Exercises 1 to 3
1 Dimensional analysis
Exercise 1
A recipe calls for 2.5 cups of flour. How many tablespoons of flour is this? (1 cup = 16 tablespoons)
Definition:

Dimensional analysis: A method for converting units using conversion factors

Conversion factor: A ratio that expresses the same quantity in different units

Unit cancellation: The process of canceling units to obtain the desired unit

Dimensional analysis method:

To convert units using dimensional analysis:

  1. Identify the given and desired units
  2. Find the conversion factor
  3. Set up the conversion factor as a fraction
  4. Arrange so the unwanted unit cancels out
  5. Multiply and divide to get the answer
  6. Verify the final unit is correct
Given
\(2.5 \text{ cups}\)
Conversion factor
\(\frac{16 \text{ tbsp}}{1 \text{ cup}}\)
Multiply
\(2.5 \times 16 = 40\)
Final unit
tablespoons
Step 1: Identify what you have and what you want

Have: 2.5 cups

Want: tablespoons

Step 2: Find the conversion factor

Given: 1 cup = 16 tablespoons

So: \(\frac{16 \text{ tbsp}}{1 \text{ cup}}\) is the conversion factor

Step 3: Set up the multiplication

\(2.5 \text{ cups} \times \frac{16 \text{ tbsp}}{1 \text{ cup}}\)

Step 4: Cancel units

The "cups" in the numerator and denominator cancel out

\(2.5 \cancel{\text{cups}} \times \frac{16 \text{ tbsp}}{1 \cancel{\text{cup}}}\)

Step 5: Calculate the numerical result

\(2.5 \times 16 = 40\)

Step 6: Express the final answer

Result: 40 tablespoons

40 tablespoons
Final answer:

The recipe calls for 40 tablespoons of flour

Applied rules:

• Unit Conversion: Multiply by a conversion factor equal to 1

• Unit Cancellation: Place units strategically to cancel unwanted units

• Conversion Factor: Express the same quantity in different units

Tip: Always arrange conversion factors so the unit you want to cancel is in both numerator and denominator!
Tip: Write units with your numbers to track cancellations!
2 Rate and time conversion
Exercise 2
A car travels at 65 miles per hour. How many feet does it travel in 10 seconds? (1 mile = 5280 feet, 1 hour = 3600 seconds)
Definition:

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

Composite conversion: Converting both distance and time units simultaneously

Unit analysis: Tracking units through calculations to ensure correctness

Given rate
\(65 \frac{\text{miles}}{\text{hour}}\)
Convert miles to feet
\(65 \times \frac{5280}{1} = 343200 \frac{\text{feet}}{\text{hour}}\)
Convert hours to seconds
\(343200 \times \frac{1}{3600} = 95.33 \frac{\text{feet}}{\text{second}}\)
Multiply by time
\(95.33 \times 10 = 953.3 \text{ feet}\)
Step 1: Write the given rate

Speed = 65 miles per hour = \(65 \frac{\text{miles}}{\text{hour}}\)

Step 2: Convert the rate to feet per second

First, convert miles to feet: \(65 \frac{\text{miles}}{\text{hour}} \times \frac{5280 \text{ feet}}{1 \text{ mile}}\)

This gives: \(343200 \frac{\text{feet}}{\text{hour}}\)

Step 3: Continue converting to feet per second

Now convert hours to seconds: \(343200 \frac{\text{feet}}{\text{hour}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}}\)

This gives: \(95.33 \frac{\text{feet}}{\text{second}}\)

Step 4: Calculate distance traveled in 10 seconds

Distance = Rate × Time

Distance = \(95.33 \frac{\text{feet}}{\text{second}} \times 10 \text{ seconds} = 953.3 \text{ feet}\)

Step 5: Verify units cancel correctly

Hours cancel, leaving feet per second, then seconds cancel, leaving feet

Step 6: Express the final answer

The car travels approximately 953.3 feet in 10 seconds

953.3 feet
Final answer:

The car travels approximately 953.3 feet in 10 seconds

Applied rules:

• Rate Conversion: Convert one unit at a time while preserving the rate relationship

• Unit Cancellation: Arrange conversion factors so unwanted units cancel

• Distance Formula: Distance = Rate × Time

Tip: When converting rates, handle distance and time units separately!
Tip: Always verify that your final units match what the question asks for!
3 Volume and time problem
Exercise 3
A water pump fills a tank at a rate of 4 gallons per minute. How many cubic feet of water does it pump in 2 hours? (1 gallon = 0.1337 cubic feet)
Definition:

Volume rate: The amount of volume that flows per unit time

Time conversion: Converting between different time units

Composite problem: A problem requiring multiple unit conversions

Given rate
\(4 \frac{\text{gallons}}{\text{minute}}\)
Convert time
\(2 \text{ hours} = 120 \text{ minutes}\)
Calculate volume
\(4 \times 120 = 480 \text{ gallons}\)
Convert to cubic feet
\(480 \times 0.1337 = 64.176 \text{ ft}^3\)
Step 1: Identify the given information

Rate: 4 gallons per minute

Time: 2 hours

Conversion: 1 gallon = 0.1337 cubic feet

Step 2: Convert time to match the rate unit

2 hours = 2 × 60 = 120 minutes

Step 3: Calculate total volume in gallons

Volume = Rate × Time

Volume = \(4 \frac{\text{gallons}}{\text{minute}} \times 120 \text{ minutes} = 480 \text{ gallons}\)

Step 4: Convert gallons to cubic feet

480 gallons × 0.1337 cubic feet per gallon = 64.176 cubic feet

Step 5: Verify units and round appropriately

Result: 64.176 cubic feet ≈ 64.2 cubic feet

Step 6: Express the final answer

The pump delivers approximately 64.2 cubic feet of water in 2 hours

64.2 ft³
Final answer:

The pump delivers approximately 64.2 cubic feet of water in 2 hours

Applied rules:

• Rate-Time-Volume: Volume = Rate × Time

• Unit Consistency: Convert time units to match the rate units

• Sequential Conversion: Convert one unit at a time

Tip: Always convert time units to match the rate's time unit first!
Tip: Remember: 1 gallon ≈ 0.1337 cubic feet.
Rules and methods, laws,...
\(\text{Quantity}_1 \times \frac{\text{Unit}_2}{\text{Unit}_1} = \text{Quantity}_2\)
Unit Conversion Formula
\(\text{Distance} = \text{Rate} \times \text{Time}\)
Distance Formula
Length
1 mile = 5280 ft
1 foot = 12 inches
Time
1 hour = 60 min
1 minute = 60 sec
Volume
1 gal = 0.1337 ft³
1 ft³ = 7.48 gal
Weight
1 lb = 16 oz
1 ton = 2000 lbs
Unit Consistency: Units must be compatible in calculations.
Dimensional Analysis: Units must make sense in the final answer.
Solution: Exercises 4 to 5
4 Cost and quantity problem
Exercise 4
A store sells apples for $0.75 per pound. If you buy 3 kilograms of apples, how much will it cost? (1 kg = 2.2 pounds)
Definition:

Unit conversion with cost: Converting between weight units to calculate cost

Price per unit: The cost of one unit of measurement

Mass conversion: Converting between different weight units

Given
\(3 \text{ kg}, \$0.75/\text{lb}\)
Convert kg to lb
\(3 \times 2.2 = 6.6 \text{ lb}\)
Calculate cost
\(6.6 \times 0.75 = \$4.95\)
Step 1: Identify the given information

Price: $0.75 per pound

Quantity: 3 kilograms

Conversion: 1 kg = 2.2 pounds

Step 2: Convert kilograms to pounds

3 kg × 2.2 lb/kg = 6.6 lb

Step 3: Calculate the total cost

Cost = Price per pound × Number of pounds

Cost = $0.75/lb × 6.6 lb = $4.95

Step 4: Verify the units

Pounds cancel in the multiplication, leaving dollars

Step 5: Express the final answer

The cost is $4.95

$4.95
Final answer:

The cost for 3 kilograms of apples is $4.95

Applied rules:

• Unit Conversion: Convert mass units to match the price unit

• Cost Calculation: Total cost = Unit price × Quantity

• Unit Cancellation: Verify units cancel appropriately

Tip: Always convert the quantity to match the unit of the rate!
5 Complex rate problem
Exercise 5
A car uses 1 gallon of gasoline for every 25 miles it travels. How many liters of gasoline will it use to travel 400 kilometers? (1 mile = 1.609 km, 1 gallon = 3.785 liters)
Definition:

Complex rate problem: A problem requiring multiple conversions across different unit systems

Metric-imperial conversion: Converting between metric and imperial units

Chain conversion: Linking multiple conversion factors together

Given rate
\(1 \text{ gal per } 25 \text{ miles}\)
Convert km to miles
\(400 \div 1.609 = 248.6 \text{ miles}\)
Calculate gallons
\(248.6 \div 25 = 9.94 \text{ gal}\)
Convert to liters
\(9.94 \times 3.785 = 37.6 \text{ L}\)
Step 1: Identify the given information

Consumption rate: 1 gallon per 25 miles

Distance to travel: 400 kilometers

Conversions: 1 mile = 1.609 km, 1 gallon = 3.785 liters

Step 2: Convert distance from kilometers to miles

400 km ÷ 1.609 km/mile = 248.6 miles

Step 3: Calculate gasoline consumption in gallons

If 1 gallon covers 25 miles, then 248.6 miles requires:

248.6 miles ÷ 25 miles/gallon = 9.94 gallons

Step 4: Convert gallons to liters

9.94 gallons × 3.785 liters/gallon = 37.6 liters

Step 5: Verify the calculations

Units: km → miles → gallons → liters (correct sequence)

Step 6: Express the final answer

The car will use approximately 37.6 liters of gasoline

37.6 liters
Final answer:

The car will use approximately 37.6 liters of gasoline to travel 400 kilometers

Applied rules:

• Chain Conversion: Convert through intermediate units when direct conversion isn't available

• Rate Applications: Use proportional reasoning with rates

• Sequential Conversions: Handle one unit conversion at a time

Tip: For complex conversions, convert to match the units in your rate first!
Tip: Remember: 1 mile ≈ 1.609 km and 1 gallon ≈ 3.785 liters.
Comprehensive Guide to Using Units in Problem Solving
\(\text{Result} = \text{Initial Value} \times \prod \left(\frac{\text{New Unit}}{\text{Old Unit}}\right)\)
General Unit Conversion
Key definitions:

Dimensional analysis: Using units to guide calculations and verify results

Conversion factor: A ratio equal to 1 that converts between units

Unit consistency: Ensuring all units in a calculation are compatible

Unit cancellation: The process of eliminating unwanted units through division

Rate: A comparison of two quantities with different units

Complete methodology:
  1. Identify the problem: Determine what you're looking for and what you have
  2. List known conversions: Write down all relevant conversion factors
  3. Plan the path: Determine the sequence of conversions needed
  4. Set up conversions: Arrange conversion factors so units cancel appropriately
  5. Calculate: Perform the multiplication and division
  6. Verify: Check that units are correct and magnitude makes sense
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: If your final units don't match what you expect, you made a setup error!
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 = 3.785 liters = 231 cubic inches

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

• Metric: 1 meter = 100 centimeters = 1000 millimeters

Unit Conversion Workflow

📊
Problem-Solving Process
1
Identify Units
2
Find Conversion Factors
3
Arrange Factors
4
Multiply
5
Verify Units
Common Conversions
Length: mi → ft → in
Time: hr → min → sec
Volume: gal → L → mL
Weight: lb → oz → g
Distance: km → mi → m
Speed: mi/hr → ft/sec → m/s
Master Unit Cancellation to Excel in Problem Solving!

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 eliminate appears in both the numerator and denominator, allowing it to cancel out.

Example: Converting 5 miles to feet (1 mile = 5280 feet)

  • Start with: 5 miles
  • Want to eliminate miles, so place miles in the denominator: \(\frac{5280 \text{ feet}}{1 \text{ mile}}\)
  • Calculation: \(5 \text{ miles} \times \frac{5280 \text{ feet}}{1 \text{ mile}} = 26400 \text{ feet}\)

The miles cancel out, leaving feet. Always arrange conversions so unwanted units cancel!

Question: What if I need to convert multiple units in one problem?

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

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

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

Work systematically through each unit that needs conversion!

Question: How can I verify that my unit 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 should be a larger number than miles for the same distance)
  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: 1 mile = 5280 feet

If you converted 2 miles to feet and got 10560 feet, verify: 10560 ÷ 5280 = 2 miles ✓

The reverse conversion confirms your answer is correct!