2.2 Ratios, Proportions, Unit Rates, and Dimensional Analysis

Ratios, Proportions, Unit Rates, and Dimensional Analysis

OpenExamPrep provides this quantitative reasoning review to prepare students for ratio and rate problems on the TSIA2 Mathematics assessment. Proportional reasoning connects arithmetic to algebra and appears across numerous college placement contexts, including nursing dosage calculations, laboratory science dilutions, engineering scale models, and business analytics.

---

1. Ratio Fundamentals: Part-to-Part vs. Part-to-Whole

A ratio is a mathematical comparison of two numbers or quantities by division. Ratios can be expressed in three interchangeable notations:

1. Word notation: a to b
2. Colon notation: a : b
3. Fraction notation: a / b

Like fractions, ratios should generally be expressed in simplest form by dividing each term by their greatest common divisor (GCD). For instance, a ratio of 24 to 36 simplifies to 2 to 3 (or 2:3 or 2/3).

The Part-to-Part vs. Part-to-Whole Distinction

The most frequent conceptual error on ratio problems is confusing a comparison between sub-groups (part-to-part) with a comparison between a sub-group and the entire population (part-to-whole).

Worked Scenario: A collegiate paramedic program accepts applicants such that 
the ratio of certified emergency medical technicians (EMTs) to non-certified 
applicants is 5 to 3. If the cohort has 96 students in total, how many 
are certified EMTs?

Step 1: Identify the ratio parts:
  Certified (Part A) = 5 parts
  Non-certified (Part B) = 3 parts
  Total whole = 5 + 3 = 8 equal parts

Step 2: Calculate the value of 1 part:
  Value per part = Total students ÷ Total parts = 96 ÷ 8 = 12 students

Step 3: Multiply the value per part by the requested group's parts:
  Certified EMTs = 5 parts × 12 students/part = 60 students
  Non-certified = 3 parts × 12 students/part = 36 students

Verification: 60 + 36 = 96 students, and 60:36 reduces to 5:3.

---

2. Proportions and Cross-Multiplication

A proportion is an equation stating that two ratios are equal:

a / b = c / d   (where b ≠ 0 and d ≠ 0)

The Cross-Multiplication Property (Means-Extremes Property)

For any proportion a/b = c/d, the product of the extremes equals the product of the means:

a × d = b × c

Cross-multiplication transforms a rational equation into a polynomial equation, eliminating denominators in a single step.

Solving Algebraic Proportions

When numerators or denominators contain binomial algebraic expressions, enclose them in parentheses before cross-multiplying to ensure proper distribution:

Problem: Solve for x in the proportion: (3x - 5) / 4 = (2x + 7) / 6

Step 1: Cross-multiply, distributing across binomials:
  6(3x - 5) = 4(2x + 7)

Step 2: Expand both sides:
  18x - 30 = 8x + 28

Step 3: Isolate x terms on one side by subtracting 8x from both sides:
  10x - 30 = 28

Step 4: Add 30 to both sides:
  10x = 58

Step 5: Divide by 10 and simplify:
  x = 58 / 10 = 29 / 5 = 5.8

Check:
  Left side:  (3(5.8) - 5) / 4 = (17.4 - 5) / 4 = 12.4 / 4 = 3.1
  Right side: (2(5.8) + 7) / 6 = (11.6 + 7) / 6 = 18.6 / 6 = 3.1
  Both sides equal 3.1; the solution x = 29/5 is verified.

---

3. Unit Rates & Economic Calculations

A rate compares two quantities measured in different units (such as miles per hour, dollars per gallon, or words per minute). A unit rate is a rate simplified so that its denominator equals 1 unit.

Unit Rate = Total Quantity A ÷ Total Quantity B

Unit Price Comparisons (Best Buy Problems)

To determine the most economical purchase among products packaged in varying sizes, compute the unit price (cost per unit of weight or volume):

Problem: A campus store sells organic peanut butter in two container sizes:
  - Size A: 16 ounces for $4.48
  - Size B: 28 ounces for $7.28
Which size provides the lower cost per ounce, and by how much?

Step 1: Compute unit price for Size A:
  Price per ounce = $4.48 ÷ 16 oz = $0.28 per ounce

Step 2: Compute unit price for Size B:
  Price per ounce = $7.28 ÷ 28 oz = $0.26 per ounce

Conclusion:
  Size B is more economical by $0.02 ($0.28 - $0.26) per ounce.

The Average Speed Trap: Equal Distance vs. Equal Time

A classic TSIA2 problem asks for the average speed of a round trip over equal distances at different rates. Because speed is non-linear with respect to time, you cannot simply calculate the arithmetic average of the two speeds.

Formula: Average Speed = Total Distance ÷ Total Time

Consider an automobile trip of 120 miles each way:

• Outbound: 120 miles driven at 40 mph → Time = 120 ÷ 40 = 3.0 hours
• Inbound: 120 miles driven at 60 mph → Time = 120 ÷ 60 = 2.0 hours
• Total Distance: 120 + 120 = 240 miles
• Total Time: 3.0 + 2.0 = 5.0 hours
• True Average Speed: 240 miles ÷ 5.0 hours = 48.0 mph

⚠️ The Arithmetic Trap: Computing (40 + 60) / 2 = 50 mph is incorrect because the vehicle spent 3 hours traveling at 40 mph and only 2 hours traveling at 60 mph. More time is spent at the slower speed, weighting the average speed downward toward 40 mph.

---

4. Multi-Step Dimensional Analysis (Factor-Label Method)

Dimensional analysis converts a measurement from one unit system to another by multiplying by a series of conversion factors (fractions equivalent to 1) so that unwanted units cancel out algebraically.

Conversion Factor Concept: Since 1 mile = 5,280 feet, 
(5,280 ft / 1 mi) = 1  and  (1 mi / 5,280 ft) = 1

Step-by-Step Conversion: Speed from Miles/Hour to Feet/Second

Convert 45 miles per hour into feet per second:

Step 1: Write initial rate as a fraction:
  (45 miles) / (1 hour)

Step 2: Set up conversion chain to eliminate miles and hours:
  Need feet in numerator: (5,280 feet / 1 mile)
  Need seconds in denominator: (1 hour / 60 minutes) and (1 minute / 60 seconds)

Step 3: Chain the fractions together:
  [45 miles / 1 hr] × [5,280 ft / 1 mile] × [1 hr / 60 min] × [1 min / 60 sec]

Step 4: Cancel matching numerator and denominator units:
  miles cancel; hours cancel; minutes cancel; leaves (feet / second)

Step 5: Compute arithmetic:
  (45 × 5,280 × 1 × 1) / (1 × 1 × 60 × 60) = 237,600 / 3,600 = 66 feet per second

Multi-Dimensional Conversions: Area and Volume

When converting area (square units) or volume (cubic units), the conversion factor must be raised to the corresponding power:

Linear:  1 yard = 3 feet
Area:    1 sq yd = (3 ft)² = 9 sq ft       (NOT 3 sq ft!)
Volume:  1 cu yd = (3 ft)³ = 27 cu ft      (NOT 3 or 9 cu ft!)

Example: How many square feet are in 15 square yards of carpeting?
  15 sq yd × (9 sq ft / 1 sq yd) = 135 sq ft

---

5. Summary of Proportions, Rates, and Conversions

---

6. TSIA2 Exam Traps & High-Frequency Pitfalls

1. Inverting Ratios: Pay strict attention to the phrasing of the question. If asked for the ratio of nurses to doctors, write N / D. If a prompt gives doctors to nurses as 2:7, the ratio of nurses to doctors is 7:2.
2. Part-to-Part vs Part-to-Whole Confusion: When given that red marbles to blue marbles is 3:4, the fraction of red marbles in the bag is 3/(3 + 4) = 3/7, not 3/4.
3. Simple Average of Rates: Never average speeds (v₁ + v₂)/2 unless the traveler spent an exactly equal amount of time at each speed. When the distance is equal, you must use total distance divided by total time.
4. Linear Scaling of Square Units: Forgetting to square conversion factors when working with square inches, square feet, or square yards leads to massive under-counting errors.