Solved Exercises on Unit Word Problems in Integrated Math 1

Master unit word problems: conversion, dimensional analysis, unit rates, and problem-solving strategies through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Unit Conversion Problem
Exercise 1
A car travels 250 miles in 4 hours. Convert the speed to feet per second. (1 mile = 5280 feet, 1 hour = 3600 seconds)
Definition:

Dimensional Analysis: A method to convert units by multiplying by conversion factors equal to 1

Unit Conversion Method:
  1. Identify the given units and desired units
  2. Set up conversion factors as fractions where numerator and denominator represent equivalent quantities
  3. Multiply the original quantity by the conversion factors
  4. Cancel out units that appear in both numerator and denominator
Given Speed
250 mi/4 hr
Convert Miles
× 5280 ft/mi
Convert Hours
× 1 hr/3600 sec
Step 1: Calculate initial speed in mph

Speed = Distance ÷ Time = 250 miles ÷ 4 hours = 62.5 miles per hour

Step 2: Set up dimensional analysis

62.5 miles/hour × 5280 feet/mile × 1 hour/3600 seconds

Step 3: Cancel units and calculate

62.5 × 5280 ÷ 3600 = 330,000 ÷ 3600 = 91.67 feet per second

Step 4: Verify the result

Units: (miles/hour) × (feet/mile) × (hour/second) = feet/second ✓

62.5 mph = 91.67 ft/sec
Final answer:

The car travels at approximately 91.67 feet per second

Applied rules:

• Dimensional Analysis: Multiply by conversion factors equal to 1

• Unit Cancellation: Units cancel when they appear in both numerator and denominator

• Order of Operations: Perform multiplication and division from left to right

2 Unit Rate Problem
Exercise 2
A recipe calls for 2 cups of flour to make 12 cookies. How many cups of flour are needed to make 45 cookies?
Definition:

Unit Rate: A rate expressed as a quantity per one unit of another quantity

Given Ratio
2 cups/12 cookies
Find Unit Rate
0.167 cups/cookie
Calculate Total
0.167 × 45 = 7.5 cups
Step 1: Find the unit rate (cups per cookie)

2 cups ÷ 12 cookies = 1/6 cup per cookie ≈ 0.167 cups per cookie

Step 2: Multiply unit rate by desired quantity

(1/6 cup per cookie) × 45 cookies = 45/6 = 7.5 cups

Step 3: Verify using proportion

2/12 = x/45 → Cross multiply: 2×45 = 12x → 90 = 12x → x = 7.5 ✓

7.5 cups of flour needed
Final answer:

7.5 cups of flour are needed to make 45 cookies

Applied rules:

• Unit Rate: Divide both quantities by the second quantity to get rate per 1

• Proportion: Two ratios are equal if their cross products are equal

• Multiplication Property: Multiply unit rate by desired quantity

3 Area Conversion Problem
Exercise 3
A rectangular garden measures 20 feet by 30 feet. Express the area in square yards. (1 yard = 3 feet)
Definition:

Area Conversion: When converting areas, conversion factors are squared because area is two-dimensional

Original Area
20×30 = 600 ft²
Conversion Factor
(1 yd/3 ft)²
Result
600/9 = 66.67 yd²
Step 1: Calculate area in original units

Area = length × width = 20 feet × 30 feet = 600 square feet

Step 2: Determine conversion factor for area

Since 1 yard = 3 feet, then 1 square yard = (3 feet)² = 9 square feet

Step 3: Apply area conversion

600 square feet × (1 square yard)/(9 square feet) = 600/9 = 66.67 square yards

Step 4: Verify the result

Units: (ft²) × (yd²/ft²) = yd² ✓

600 ft² = 66.67 yd²
Final answer:

The garden area is 66.67 square yards

Applied rules:

• Area Conversion: Linear conversion factors are squared for area conversions

• Dimensional Analysis: Conversion factor must equal 1

• Unit Squaring: When converting area, square the linear conversion factor

Unit Conversion Rules and Methods
\(\frac{\text{desired units}}{\text{given units}} = \text{conversion factor}\)
Dimensional Analysis Formula
Linear Conversion
1 mile = 5280 feet
Distance units
Time Conversion
1 hour = 3600 seconds
Time units
Area Conversion
1 yd² = 9 ft²
Area units
Key definitions:

Dimensional Analysis: Converting units by multiplying by conversion factors equal to 1

Unit Rate: A ratio comparing a quantity to one unit of another quantity

Conversion Factor: A fraction that equals 1, used to change units

Unit Conversion Methodology:
  1. Identify: Given units and desired units
  2. Plan: Find conversion factors that connect the units
  3. Arrange: Set up fractions so unwanted units cancel out
  4. Calculate: Multiply numerators and denominators
  5. Verify: Check that units match the desired outcome
Tip 1: Write conversion factors as fractions where numerator and denominator represent equivalent quantities.
Tip 2: When converting areas, square the linear conversion factor (e.g., 1 yd² = 9 ft²).
Tip 3: Always write units in your calculations to track cancellation.
Tip 4: For volume conversions, cube the linear conversion factor.
Common Mistakes: Forgetting to square/cube conversion factors for area/volume, incorrect placement of units in fractions, arithmetic errors.
Memorization Tip: Remember that larger units require smaller numbers (e.g., fewer yards than feet for the same distance).
Solution: Exercises 4 to 5
4 Speed and Distance Problem
Exercise 4
A cyclist rides at 15 km/hr for 2.5 hours. How far does she travel in meters? (1 km = 1000 m)
Definition:

Distance Formula: Distance = Speed × Time

Calculate Distance
15 × 2.5 = 37.5 km
Convert to Meters
× 1000 m/km
Result
37,500 m
Step 1: Apply distance formula

Distance = Speed × Time = 15 km/hr × 2.5 hr = 37.5 km

Step 2: Convert kilometers to meters

37.5 km × (1000 m/1 km) = 37.5 × 1000 = 37,500 meters

Step 3: Verify units cancellation

(km) × (m/km) = m ✓

37.5 km = 37,500 m
Final answer:

The cyclist travels 37,500 meters

Applied rules:

• Distance Formula: Distance = Rate × Time

• Unit Conversion: Multiply by conversion factor equal to 1

• Dimensional Analysis: Units cancel appropriately

5 Weight Conversion Problem
Exercise 5
A shipment weighs 2.5 tons. Convert this weight to ounces. (1 ton = 2000 pounds, 1 pound = 16 ounces)
Definition:

Chain Conversion: Multiple conversion factors used sequentially to convert between distant units

Starting Weight
2.5 tons
To Pounds
× 2000 lb/ton
To Ounces
× 16 oz/lb
Step 1: Convert tons to pounds

2.5 tons × (2000 pounds/1 ton) = 2.5 × 2000 = 5000 pounds

Step 2: Convert pounds to ounces

5000 pounds × (16 ounces/1 pound) = 5000 × 16 = 80,000 ounces

Step 3: Alternative chain method

2.5 tons × (2000 lb/ton) × (16 oz/lb) = 2.5 × 2000 × 16 = 80,000 oz

Step 4: Verify units cancellation

tons × (lb/ton) × (oz/lb) = oz ✓

2.5 tons = 80,000 oz
Final answer:

The shipment weighs 80,000 ounces

Applied rules:

• Chain Conversion: Connect distant units through intermediate units

• Multiplicative Property: Multiply all conversion factors together

• Sequential Verification: Check that each conversion makes sense

Unit Conversion Laws, Methods, and Formulas
\(\text{New Value} = \text{Old Value} \times \frac{\text{Desired Unit}}{\text{Given Unit}}\)
Unit Conversion Formula
Key definitions:

Conversion Factor: A ratio equal to 1 that expresses the same measurement in different units

Dimensional Analysis: A method of problem-solving that uses the units that describe matter

Unit Rate: A rate with a denominator of 1 unit

Complete Unit Conversion Methodology:
  1. Identify: Determine given units and required units
  2. Research: Find appropriate conversion factors
  3. Arrange: Set up conversion factors so unwanted units cancel
  4. Calculate: Perform the multiplication and division
  5. Verify: Check that final units match requirements
Tip 1: When converting area, square the linear conversion factor.
Tip 2: For volume, cube the linear conversion factor.
Tip 3: Always include units in your calculations to catch errors early.
Tip 4: Larger units correspond to smaller numbers (and vice versa).
Common Errors: Misplacing conversion factors, forgetting to square/cube for area/volume, calculation mistakes.
Exam Preparation: Practice multiple conversion paths, memorize common conversion factors, master dimensional analysis.
Essential Formulas to Know:

• Linear conversion: 1 mile = 5280 feet, 1 yard = 3 feet

• Area conversion: 1 yd² = 9 ft², 1 ft² = 144 in²

• Volume conversion: 1 ft³ = 1728 in³, 1 gal = 231 in³

• Weight conversion: 1 ton = 2000 lbs, 1 lb = 16 oz

• Time conversion: 1 hr = 60 min, 1 min = 60 sec

Questions & Answers

Question: I often get confused about whether to multiply or divide when doing unit conversions. How can I remember which operation to use?

Answer: Great question! The key is to use dimensional analysis with conversion factors. Instead of memorizing operations, set up fractions where the numerator and denominator represent the same quantity in different units.

For example, to convert 5 feet to inches:

  • You want to cancel "feet" and end up with "inches"
  • Use the conversion: 1 foot = 12 inches, written as 12 inches/1 foot
  • Multiply: 5 feet × (12 inches/1 foot) = 60 inches

The feet units cancel out, leaving inches. This method automatically gives you the correct operation and ensures units work out properly!

Question: Why do we square the conversion factor when converting area units, like going from square feet to square yards?

Answer: This happens because area is a two-dimensional measurement. Let's think about it:

  • If 1 yard = 3 feet, then a square that is 1 yard by 1 yard has dimensions 3 feet by 3 feet
  • So 1 square yard = (3 feet) × (3 feet) = 9 square feet
  • This means the linear relationship (1 yd = 3 ft) becomes an area relationship (1 yd² = 9 ft²)

For volume conversions, you would cube the linear factor since volume is three-dimensional (length × width × height).

Question: How can I check if my unit conversion is reasonable? Sometimes I get answers that seem too big or too small.

Answer: Here are effective verification strategies:

  • Reasonableness Check: Larger units should yield smaller numbers (e.g., fewer yards than feet for the same distance)
  • Unit Consistency: Make sure units cancel properly and your final answer has the correct units
  • Reverse Calculation: Convert your answer back to the original units to see if you get the starting value
  • Estimation: Round numbers to get a rough idea of what the answer should be

For example, if converting 100 feet to yards and getting 300 yards, you'd realize this is wrong since yards are larger units than feet.