1.2 Ratios, Proportions & Unit Conversions

Ratios and proportions let you scale a known relationship up or down, and unit conversions let you move between measurement systems without changing the underlying quantity. Both skills show up constantly on the NAPT — and they matter well beyond the math chapters. Navy engineering and technical manuals mix metric units (meters, liters, kilograms, Celsius) with imperial/customary units (feet, gallons, pounds, Fahrenheit), so comfortable, error-free conversion between the two systems is a study tip called out specifically for NAPT prep, and it resurfaces again in the physics and chemistry chapters ahead.

Ratios and Proportions

A ratio compares two quantities, written as a:b or a/b. A proportion is a statement that two ratios are equal, such as a/b = c/d. Proportions are solved with cross-multiplication: multiply the numerator of one side by the denominator of the other, and set the two products equal.

Worked example: In an engine room, machinist's mates and electrician's mates are staffed at a fixed ratio of 3 machinists for every 4 electricians. If a ship currently has 28 electrician's mates, how many machinist's mates does it have?

Set up the proportion: 3/4 = x/28

Cross-multiply: 4x = 3 × 28 = 84

Divide: x = 84 ÷ 4 = 21 machinist's mates.

Unit Rates

A unit rate is a ratio simplified so the second quantity is 1 — such as miles per hour or gallons per minute. Once you know a unit rate, you can scale it to any amount using multiplication.

Worked example: A bilge pump moves 45 gallons of water every 3 minutes. At that same rate, how many gallons does it move in 11 minutes?

• Find the unit rate: 45 ÷ 3 = 15 gallons per minute
• Scale to 11 minutes: 15 × 11 = 165 gallons

Part-to-Part vs. Part-to-Whole Ratios

Word problems are not always explicit about which kind of ratio they describe. A part-to-part ratio compares two categories within a group (3 machinists to 4 electricians, as above). A part-to-whole ratio compares one category to the entire group. If a work crew of 35 sailors is staffed at that same 3:4 machinist-to-electrician ratio, the group divides into 3 + 4 = 7 equal shares, so machinists make up 3/7 of the whole crew — not 3/4 of it.

Worked example: A 35-person work crew is staffed at a 3:4 machinist-to-electrician ratio. How many of the 35 sailors are machinists?

• Total ratio parts: 3 + 4 = 7
• Each part represents 35 ÷ 7 = 5 sailors
• Machinists: 3 × 5 = 15 sailors

Always check whether a problem's ratio describes two parts of a group (part-to-part) or one part against the group's total (part-to-whole) before you set up your proportion.

---

Dimensional Analysis

Dimensional analysis (also called the factor-label method) converts a measurement from one unit to another by multiplying by one or more conversion factors — fractions equal to 1, such as 5,280 ft over 1 mi. Because each conversion factor equals 1, multiplying by it changes the units without changing the actual quantity.

Worked example: Convert 65 miles per hour to feet per second.

65 mi/hr × (5,280 ft ÷ 1 mi) × (1 hr ÷ 3,600 s)

= (65 × 5,280) ÷ 3,600

= 343,200 ÷ 3,600

≈ 95.3 ft/s

Notice how the "mi" units cancel (one on top, one on bottom), and "hr" cancels the same way, leaving only ft/s.

Metric ↔ Imperial Conversions

Because Navy technical documentation uses both systems, memorize these commonly tested conversion factors:

Worked example: A technical manual lists a component's mass as 68 kilograms. What is that mass in pounds?

68 kg × 2.205 lb/kg ≈ 149.9 lb

Navy-relevant worked example: A ship's navigation system reports a speed of 18 knots. A knot is a unit of speed equal to one nautical mile per hour, and 1 knot ≈ 1.151 miles per hour. What is the ship's speed in miles per hour?

18 knots × 1.151 mph/knot ≈ 20.7 mph

This is exactly the kind of conversion-factor chain you will use throughout this guide: identify the given rate, multiply by the correct conversion factor, and confirm the units cancel correctly (knots cancel, leaving mph).

Common error: applying the conversion factor backwards. Dividing 68 by 2.205 instead of multiplying gives about 30.8 — a completely different (and wrong) quantity. Always check that your units cancel correctly: kilograms × (pounds/kilogram) leaves pounds; kilograms ÷ (pounds/kilogram) leaves kilograms², which is meaningless.

A Conversion That Is NOT a Simple Ratio

Not every metric-to-imperial conversion is a straight multiplication. Temperature is the major exception: the Fahrenheit and Celsius scales do not share a common zero point, so converting between them requires the formula F = (9/5) × C + 32, not a simple proportion. Chapter 4 (Thermodynamics & Heat Transfer) covers this formula and its worked examples in depth — the distinction matters because applying a ratio-style shortcut to temperature produces a wrong answer every time.

Test-Day Tip

When a problem gives you a ratio in words ("for every," "per," "out of"), translate it into a fraction immediately and set up the proportion before doing any arithmetic. Label every conversion factor's units and cancel them on paper — this catches backwards conversions before they become a wrong answer.