Ratio, Proportion & Basic Algebra

Where Ratio and Algebra Show Up on the FOE

Numerical reasoning shares the 25 percent General Ability allocation with Verbal and Analytical items, and ratio, proportion and one-variable algebra are the workhorses of that sub-test. They also run through the specialised areas: foam induction rates, apparatus staffing, plan scales on a pre-fire plan and the apportionment of Fire Code fees under Republic Act 9514 (the Fire Code of the Philippines of 2008) are all proportion problems in uniform. With calculators prohibited, the winning technique is to reduce every ratio to its smallest whole-number form before you multiply anything.

Ratio Notation and Simplification

A ratio compares quantities of the same kind, written a : b, and is simplified by dividing every term by the greatest common divisor. A duty roster of 36 : 24 : 24 firefighters reduces to 3 : 2 : 2. Two distinctions decide most items:

• Part-to-part compares one share against another. In 3 : 2 : 2, Shift A is three-halves the size of Shift B.
• Part-to-whole compares one share against the sum of all parts. In the same roster, Shift A is 3/7 of the station, which is 42.9 percent.

Mixing these two is the single most common ratio error on the exam. If the question says "of the station", "of the total" or "what percentage", you need part-to-whole.

Worked Example: Dividing a Duty Roster

A city fire station assigns 84 firefighters to three shifts in the ratio 3 : 2 : 2.

1. Total parts = 3 + 2 + 2 = 7.
2. One part = 84 / 7 = 12 firefighters.
3. Shifts = 3 x 12 = 36, 2 x 12 = 24, and 2 x 12 = 24. Check: 36 + 24 + 24 = 84.

Direct and Inverse Proportion

Two quantities are in direct proportion when their ratio stays constant: double one and the other doubles, so a / b = c / d and you solve by cross-multiplication (a x d = b x c). They are in inverse proportion when their product stays constant: double one and the other halves, so a x b = c x d.

Worked Example: Fire Code Fees as Direct Proportion

Under RA 9514 Section 12, one source of BFP income is one-tenth of one per centum (0.10 percent) of the verified estimated value of buildings or structures to be erected, capped at PhP 50,000.00, with at least 50 percent paid before the building permit is issued and the balance after final inspection and before the occupancy permit.

1. A warehouse with a verified estimated value of PhP 28,000,000 carries a fee of 28,000,000 x 0.001 = PhP 28,000.
2. At least half, PhP 14,000, falls due before the building permit; the remaining PhP 14,000 before the use and occupancy permit.
3. A BPO tower valued at PhP 80,000,000 would compute to PhP 80,000 - but the statutory ceiling caps it at PhP 50,000. Proportionality stops at the cap, and examiners test exactly that boundary.

Collections are then apportioned in a fixed 4 : 1 ratio - 80 percent of revenues collected by the BFP are remitted to the National Treasury for modernisation, and 20 percent is set aside and retained for the local government concerned. On PhP 4,500,000 collected in a month, that is PhP 3,600,000 and PhP 900,000 respectively.

Worked Example: Inverse Proportion on the Fireground

Six firefighters working at a constant rate clear a collapsed warehouse roof section in 8 hours. How long would four firefighters need?

1. Total work = 6 x 8 = 48 firefighter-hours (this product is the constant).
2. Time for four = 48 / 4 = 12 hours.

The direct-proportion trap answers 8 x (4/6) = 5 hours 20 minutes, which is faster with fewer people - obviously impossible. Always ask whether the second quantity should rise or fall before you compute.

Scale on Plans and Maps

A scale of 1 : 100 means one unit on the drawing equals 100 identical units in reality. Pre-fire plans, fire safety evaluation drawings and building plans submitted for a Fire Safety Evaluation Clearance (FSEC) are read this way.

• Length: a corridor measuring 96 mm on a 1 : 100 plan is 96 x 100 = 9,600 mm = 9.6 meters.
• Reverse: a 45 m frontage drawn at 1 : 500 is 45,000 mm / 500 = 90 mm on the sheet.
• Area: areas scale by the square of the linear scale. A room measuring 40 mm x 30 mm on a 1 : 100 plan is 4 m x 3 m = 12 square meters, not 1,200 times anything. Volumes scale by the cube.

Mixture and Concentration: Foam Proportioning

Foam induction rate is expressed as the percentage of concentrate in the finished foam solution, not as concentrate added to a full tank. A 3 percent concentrate means 3 parts concentrate to 97 parts water in every 100 parts of solution; 6 percent means 6 parts to 94.

Concentrate volume = finished solution volume x induction rate.

A BFP foam tender supplies a monitor discharging 950 liters per minute of finished solution for 20 minutes.

1. Finished solution = 950 x 20 = 19,000 liters.
2. At 3 percent: 19,000 x 0.03 = 570 liters of concentrate.
3. At 6 percent: 19,000 x 0.06 = 1,140 liters - exactly double.
4. Inverse view: with only 570 liters of concentrate aboard and a 6 percent system, you generate 570 / 0.06 = 9,500 liters of solution, which at 950 L/min lasts just 10 minutes. Doubling the induction rate halves the attack duration.

Translating Words Into Equations

Assign the letter to the quantity the question asks about, write one equation per stated condition, then solve.

Worked Example: One Unknown

A municipal BFP office collected PhP 186,000 in Fire Code fees in a month, from FSIC fees and FSEC fees only. FSIC fees were PhP 24,000 more than twice the FSEC fees.

• Let x = FSEC fees. Then FSIC = 2x + 24,000.
• x + (2x + 24,000) = 186,000, so 3x = 162,000 and x = PhP 54,000.
• FSIC = 2(54,000) + 24,000 = PhP 132,000. Check: 54,000 + 132,000 = 186,000.

Worked Example: Two Unknowns

A regional supply office buys 6 self-contained breathing apparatus (SCBA) cylinders and 4 fire helmets for PhP 212,000; a second order of 3 cylinders and 8 helmets costs PhP 136,000.

• 6c + 4h = 212,000 and 3c + 8h = 136,000.
• Double the second equation: 6c + 16h = 272,000. Subtract the first: 12h = 60,000, so h = PhP 5,000.
• Substitute: 6c = 212,000 - 20,000 = 192,000, so c = PhP 32,000.

Archetypes and Setups

The Traps

1. Treating an inverse relationship as direct - fewer firefighters can never mean less time.
2. Answering part-to-part when the question asks part-to-whole.
3. Forgetting a statutory ceiling, such as the PhP 50,000 cap that flattens the 0.10 percent building fee.
4. Scaling areas linearly instead of by the square of the scale.
5. Reading a foam percentage as concentrate poured into a full water tank rather than concentrate within the finished solution.
6. Cross-multiplying with mismatched units, such as liters per second on one side and liters per minute on the other.