13.2 Unit Rates, Cost per Unit & MPG

13.2 Unit Rates, Cost per Unit & MPG

Quick Answer: A unit rate is total ÷ quantity for one unit. Cost per gallon = total dollars ÷ gallons ($726 ÷ 600 = $1.21). MPG = miles driven ÷ gallons used; miles driven = ending odometer − starting odometer, then divide by gallons ((4957.6 − 4730.8) ÷ 18 = 12.6 mpg).

After time, percent, and ratio problems, the packet moves to unit rates: how much one gallon costs, and how many miles one gallon delivers. These are the same skill—divide carefully and keep the units honest—applied to apparatus fuel math you will also use on the job.

What “unit rate” means

A unit rate answers: How much per one?

Rule: Put the quantity you want “per 1 of” on the bottom. If you want dollars per gallon, gallons go on the bottom.

Method A — Cost per unit (fuel, supplies, equipment)

Step-by-step

1. Identify total cost (usually a dollar amount).
2. Identify total quantity (gallons, feet, items).
3. Compute unit cost = total cost ÷ quantity.
4. Round only if the problem or choices require it; match the choice format (often two decimal places for money).

Worked example — vacation fuel (packet Q4)

Given: Captain Robnett purchased 600 gallons of fuel for $726.

Find: Cost per gallon.

Work:

$726 ÷ 600 = $1.21 per gallon

Long division check:

600 × 1.21 = 600 × 1 + 600 × 0.20 + 600 × 0.01
           = 600 + 120 + 6
           = 726  ✓

Answer: $1.21.

Why nearby choices appear

Unit-rate method for similar fire-service costs

Trap: Do not reverse the fraction. 600 ÷ 726 ≈ 0.83 is gallons per dollar—not cost per gallon. If the answer is less than $1 when you expected about a dollar-plus fuel price (or vice versa), you likely inverted.

Money division tips under time pressure

• Move the decimal: $726 ÷ 600 = 726 ÷ 600.
• Or scale: 726 ÷ 600 = (726 ÷ 6) ÷ 100 = 121 ÷ 100 = 1.21.
• Or: 726 ÷ 600 = 1.21 exactly because 600 × 1.21 = 726.

Method B — Miles per gallon (odometer method)

Goal

Find mpg for an apparatus between two odometer readings when you know how many gallons were used (often the fill-up amount that refills the tank after the trip).

Formula (write this every time)

miles driven = ending odometer − starting odometer
mpg = miles driven ÷ gallons used

Step-by-step

1. Subtract odometer readings (ending − starting). Confirm the result is positive and sensible.
2. Identify gallons used (tank fill amount in the packet problem).
3. Divide miles by gallons.
4. Match the choice (often one decimal place).

Worked example — Engine 7 fuel economy (packet Q5)

Given:

• Odometer on June 4: 4730.8
• Odometer on June 6: 4957.6
• Tank filled with 18 gallons on June 6

Find: Engine 7’s mpg for that period.

Work:

Miles driven = 4957.6 − 4730.8 = 226.8 miles
mpg = 226.8 ÷ 18 = 12.6 mpg

Check: 18 × 12 = 216; 18 × 0.6 = 10.8; 216 + 10.8 = 226.8 ✓.

Answer: 12.6 mpg.

Distractor map

Critical MPG traps

1. Skipping the subtraction. Dividing 4957.6 by 18 (≈ 275.4) or 4730.8 by 18 (≈ 262.8) is meaningless for trip mpg—those are not miles driven.
2. Subtracting in the wrong order. Starting − ending gives a negative; if you drop the sign you still have the wrong workflow habit.
3. Using gallons in the numerator. Gallons ÷ miles would be gallons per mile (gpm fuel burn), not mpg.
4. Decimal odometers. Keep the tenth of a mile: 4957.6 − 4730.8 is 226.8, not 227 rounded early before dividing (early rounding can push you off the exact choice).

Second worked example (same method)

Odometer start 12,400.0, end 12,652.0, fuel used 20 gallons.

Miles = 12,652.0 − 12,400.0 = 252.0
mpg = 252 ÷ 20 = 12.6 mpg

Method C — General unit-rate attack for word problems

When a fire-service story feels long, strip it to a three-line setup:

WANT:   ? per 1 unit   (e.g., $ per gal, miles per gal)
HAVE:   total of A, total of B
DO:     (relevant total A) ÷ (relevant total B)

Mini examples

Overtime cost per hour: $540 overtime for 12 hours → $540 ÷ 12 = $45/hour.

Average calls per shift: 36 calls over 9 shifts → 36 ÷ 9 = 4 calls/shift.

Hose laid per minute: 300 feet in 5 minutes → 300 ÷ 5 = 60 ft/min.

Water cost per gallon (if billed): $48 for 4,000 gallons → $48 ÷ 4,000 = $0.012/gal.

Same division skill as packet Q4 and Q5; only the labels change.

Side-by-side: cost/gal vs mpg

Exam checklist for 13.2

1. Underline the two totals you will divide.
2. For MPG, box the two odometer numbers and subtract before anything else.
3. Write units on your scratch paper: $ / gal or mi / gal.
4. Multiply back (quotient × divisor) to verify you hit the original total.
5. Compare to choices; if you are between two, re-check subtraction and decimal placement—not the story wording.

Unit rates are the bridge between simple division and the next section’s volume, weight, and multi-apparatus flow totals—where you still divide and multiply, but with 7.5 gal/ft³ and 62.5 lb/ft³ constants.