Solved Exercises on Formulas and Units in Integrated Math 1

Master formulas and units: area, volume, rate, and complex unit problems through these 5 detailed exercises with comprehensive solutions.

Solution: Exercises 1 to 3
1 Area calculation
Exercise 1
Find the area of a rectangle with length 8 cm and width 5 cm. What is the unit for area?
Definition:

Area: The amount of surface enclosed by a shape

Formula for rectangle: Area = length × width

Unit for area: Square units (length × width units)

Area calculation method:

To calculate area with correct units:

  1. Identify the formula: Match the shape to its area formula
  2. Substitute values: Replace variables with given measurements
  3. Calculate: Perform the multiplication
  4. Determine units: Multiply the units together
  5. Express result: Include both value and correct units
Formula
\(A = l \times w\)
Substitute
\(A = 8 \times 5\)
Calculate
\(A = 40\)
Units
\(cm \times cm = cm^2\)
Step 1: Write the formula

Area of rectangle: \(A = l \times w\)

Step 2: Identify given values

Length (\(l\)) = 8 cm

Width (\(w\)) = 5 cm

Step 3: Substitute values into formula

\(A = 8 \times 5\)

Step 4: Calculate the result

\(A = 40\)

Step 5: Determine the unit

When multiplying units: cm × cm = cm² (square centimeters)

Step 6: Express the final answer

Area = 40 cm²

40 cm²
Final answer:

The area is 40 square centimeters (40 cm²)

Applied rules:

• Area Formula: For rectangles, multiply length by width

• Unit Multiplication: Multiply units along with numbers

• Area Units: Area always has square units (cm², m², ft², etc.)

Tip: Area units are always square units: length × width units!
Tip: Always write units with your answer - they're part of the solution!
2 Volume calculation
Exercise 2
A rectangular prism has dimensions: length = 4 in, width = 3 in, height = 6 in. Find its volume. What is the unit for volume?
Definition:

Volume: The amount of space occupied by a three-dimensional object

Formula for rectangular prism: Volume = length × width × height

Unit for volume: Cubic units (length × width × height units)

Formula
\(V = l \times w \times h\)
Substitute
\(V = 4 \times 3 \times 6\)
Calculate
\(V = 72\)
Units
\(in \times in \times in = in^3\)
Step 1: Write the formula

Volume of rectangular prism: \(V = l \times w \times h\)

Step 2: Identify given values

Length (\(l\)) = 4 inches

Width (\(w\)) = 3 inches

Height (\(h\)) = 6 inches

Step 3: Substitute values into formula

\(V = 4 \times 3 \times 6\)

Step 4: Calculate the result

\(V = 72\)

Step 5: Determine the unit

When multiplying units: in × in × in = in³ (cubic inches)

Step 6: Express the final answer

Volume = 72 in³

72 in³
Final answer:

The volume is 72 cubic inches (72 in³)

Applied rules:

• Volume Formula: For rectangular prisms, multiply length, width, and height

• Unit Multiplication: Multiply all units together

• Volume Units: Volume always has cubic units (in³, cm³, m³, etc.)

Tip: Volume units are always cubic: length × width × height units!
Tip: Remember: 2D = square units, 3D = cubic units!
3 Rate calculation
Exercise 3
A car travels 240 miles in 4 hours. Calculate the average speed. What is the unit for speed?
Definition:

Rate: A comparison of two quantities with different units

Average speed: Total distance divided by total time

Unit for speed: Distance unit divided by time unit

Formula
\(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\)
Substitute
\(\text{Speed} = \frac{240}{4}\)
Calculate
\(\text{Speed} = 60\)
Units
\(\frac{\text{miles}}{\text{hours}} = \text{mph}\)
Step 1: Write the formula

Average speed: \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\)

Step 2: Identify given values

Distance = 240 miles

Time = 4 hours

Step 3: Substitute values into formula

\(\text{Speed} = \frac{240 \text{ miles}}{4 \text{ hours}}\)

Step 4: Calculate the result

\(\text{Speed} = 60\) miles per hour

Step 5: Determine the unit

When dividing units: \(\frac{\text{miles}}{\text{hours}} = \frac{\text{miles}}{\text{hour}}\) or mph

Step 6: Express the final answer

Speed = 60 mph

60 mph
Final answer:

The average speed is 60 miles per hour (60 mph)

Applied rules:

• Rate Formula: Rate = \(\frac{\text{Quantity 1}}{\text{Quantity 2}}\)

• Unit Division: Divide units along with numbers

• Rate Units: Rate units are always a ratio (miles/hour, dollars/hour, etc.)

Tip: Rates always have compound units like "per" or "/"
Tip: Speed = distance/time, density = mass/volume, rate = amount/time
Rules and methods, laws,...
\(\text{Area} = \text{length} \times \text{width} \Rightarrow \text{units}^2\)
Area Formula
\(\text{Volume} = \text{length} \times \text{width} \times \text{height} \Rightarrow \text{units}^3\)
Volume Formula
\(\text{Rate} = \frac{\text{quantity}}{\text{time}} \Rightarrow \frac{\text{units}}{\text{time units}}\)
Rate Formula
Area
\(A = lw\)
Units: length²
Volume
\(V = lwh\)
Units: length³
Rate
\(R = \frac{d}{t}\)
Units: \(\frac{\text{distance}}{\text{time}}\)
Density
\(\rho = \frac{m}{V}\)
Units: \(\frac{\text{mass}}{\text{volume}}\)
Unit Consistency: Units must be compatible in formulas and calculations.
Dimensional Analysis: Units must make sense in the final answer.
Solution: Exercises 4 to 5
4 Density calculation
Exercise 4
A metal block has a mass of 450 grams and a volume of 50 cubic centimeters. Calculate its density. What is the unit for density?
Definition:

Density: Mass per unit volume of a substance

Formula: Density = mass ÷ volume

Unit for density: Mass unit divided by volume unit

Formula
\(\rho = \frac{m}{V}\)
Substitute
\(\rho = \frac{450}{50}\)
Calculate
\(\rho = 9\)
Units
\(\frac{\text{g}}{\text{cm}^3}\)
Step 1: Write the formula

Density: \(\rho = \frac{m}{V}\) (where \(\rho\) = density, m = mass, V = volume)

Step 2: Identify given values

Mass (\(m\)) = 450 grams

Volume (\(V\)) = 50 cm³

Step 3: Substitute values into formula

\(\rho = \frac{450 \text{ g}}{50 \text{ cm}^3}\)

Step 4: Calculate the result

\(\rho = 9\) grams per cubic centimeter

Step 5: Determine the unit

When dividing units: \(\frac{\text{grams}}{\text{cubic centimeters}} = \frac{\text{g}}{\text{cm}^3}\)

Step 6: Express the final answer

Density = 9 g/cm³

9 g/cm³
Final answer:

The density is 9 grams per cubic centimeter (9 g/cm³)

Applied rules:

• Density Formula: Density = mass/volume

• Unit Division: Divide units along with numbers

• Density Units: Always mass per volume unit (g/cm³, kg/m³, etc.)

Tip: Density tells how much matter is packed into a given space!
5 Complex rate calculation
Exercise 5
A pump fills a 300-gallon tank in 2 hours. At this rate, how long would it take to fill a 1200-gallon tank? What are the units for the rate?
Definition:

Rate of change: How quickly one quantity changes with respect to another

Proportional reasoning: Using ratios to solve related problems

Unit rate: Rate per single unit of the second quantity

Find rate
\(\frac{300 \text{ gal}}{2 \text{ hr}} = 150 \frac{\text{gal}}{\text{hr}}\)
Use rate
\(\frac{1200}{150} = 8 \text{ hours}\)
Step 1: Find the rate of filling

Rate = \(\frac{\text{Volume filled}}{\text{Time taken}} = \frac{300 \text{ gallons}}{2 \text{ hours}} = 150 \frac{\text{gallons}}{\text{hour}}\)

Step 2: Set up the relationship for the larger tank

If the rate is 150 gallons per hour, how long to fill 1200 gallons?

Time = \(\frac{\text{Total volume}}{\text{Rate}}\)

Step 3: Calculate the time needed

Time = \(\frac{1200 \text{ gallons}}{150 \frac{\text{gallons}}{\text{hour}}} = 8 \text{ hours}\)

Step 4: Verify the calculation

In 8 hours at 150 gal/hr: \(8 \times 150 = 1200\) gallons ✓

Step 5: Express the final answer

It would take 8 hours to fill the 1200-gallon tank

8 hours
Final answer:

It would take 8 hours to fill the 1200-gallon tank

The rate is 150 gallons per hour (150 gal/hr)

Applied rules:

• Rate Calculation: Rate = Quantity ÷ Time

• Proportional Reasoning: Use the rate to solve related problems

• Unit Consistency: Ensure units match in calculations

Tip: Find the unit rate first, then use it to solve related problems!
Tip: When dividing by a rate, the time units will cancel appropriately!
Comprehensive Guide to Formulas and Units
\(\text{Quantity} = \text{Formula}(\text{Inputs}) \Rightarrow \text{Units} = \text{Function}(\text{Input Units})\)
Unit Analysis Principle
Key definitions:

Formula: A mathematical relationship between quantities

Unit: A standard measure for a physical quantity

Dimensional analysis: Using units to check the validity of calculations

Unit conversion: Changing from one unit to another while preserving the quantity

Derived units: Units formed by combining base units (m/s, kg/m³, etc.)

Complete methodology:
  1. Identify the formula: Match the situation to the appropriate formula
  2. Identify the units: Determine the units of all given quantities
  3. Substitute values: Replace variables with given measurements
  4. Perform calculations: Calculate the numerical result
  5. Handle units: Apply the same operations to units as to numbers
  6. Verify units: Check that the final units match the expected result
Tip 1: Always write units with your numbers to track them through calculations!
Tip 2: Area units are always square (length²), volume units are always cubic (length³)!
Tip 3: Rate units always have "per" or "/" (miles per hour, dollars per item, etc.)!
Tip 4: When multiplying units: cm × cm = cm², and when dividing: miles ÷ hours = miles/hour!
Tip 5: If your final units don't make sense, you likely made a calculation or formula error!
Common errors: Forgetting units, mixing incompatible units, incorrect formula application, not tracking units through calculations.
Exam preparation: Memorize common formulas, practice unit analysis, focus on dimensional consistency.
Essential formulas and units:

• Area of rectangle: \(A = lw\) → units²

• Volume of prism: \(V = lwh\) → units³

• Speed: \(s = \frac{d}{t}\) → distance/time

• Density: \(\rho = \frac{m}{V}\) → mass/volume

• Unit multiplication: length × width → area units

• Unit division: distance ÷ time → rate units

Formulas and Units Workflow

📊
Problem-Solving Process
1
Identify Formula
2
Check Units
3
Substitute Values
4
Calculate
5
Apply Units
6
Verify
Unit Relationships
Length: cm, m, km, in, ft, mi
Area: cm², m², in², ft²
Volume: cm³, m³, L, gal
Time: sec, min, hr, day
Rate: distance/time, work/time
Mass: g, kg, oz, lb
Master Unit Analysis to Excel in Mathematical Modeling!

Questions & Answers

Question: How do I know if I have the right units for my answer?

Answer: Use dimensional analysis to check your units:

For area problems: Units should be square (cm², m², ft², etc.)

For volume problems: Units should be cubic (cm³, m³, ft³, etc.)

For rate problems: Units should be a ratio (miles/hour, dollars/item, etc.)

Example: If calculating area of rectangle with length in cm and width in cm:

  • Area = length × width = cm × cm = cm²
  • If your answer has units like cm or cm³, you made an error

The units should make sense for the quantity you're calculating!

Question: What if the given measurements have different units? Should I convert them first?

Answer: Yes! You must convert all measurements to the same units before using a formula:

Example: Finding area of rectangle with length = 2 feet, width = 6 inches

  • Convert width to feet: 6 inches = 0.5 feet
  • Then: Area = 2 ft × 0.5 ft = 1 ft²

OR: Convert length to inches: 2 feet = 24 inches

  • Then: Area = 24 in × 6 in = 144 in²

Both answers are correct (1 ft² = 144 in²), but you must use consistent units!

Always convert to the unit required for your answer or to match the majority of given units.

Question: How do I handle units when doing multiple operations in one problem?

Answer: Apply the same mathematical operation to the units as you do to the numbers:

Example: Calculate volume = length × width × height

Where length = 3 m, width = 2 m, height = 4 m

  • Numerically: 3 × 2 × 4 = 24
  • Units: m × m × m = m³
  • Answer: 24 m³

Another example: Speed = distance ÷ time

Where distance = 120 miles, time = 2 hours

  • Numerically: 120 ÷ 2 = 60
  • Units: miles ÷ hours = miles/hour
  • Answer: 60 miles/hour

Treat units exactly like variables in algebra!

This technique is called "dimensional analysis" and is a powerful tool for checking your work.