24 Mathematics Practice Questions & Answers
Every Mathematics practice question from the Firefighter Written Exam Practice Test, with the correct answer and a short explanation.
Start practice test →1. On an entrance exam you must evaluate: 7 + 6 x 4 - 9. What is the correct value?
- A.52
- B.43
- C.22✓ Answer
- D.31
Order of operations (PEMDAS) requires multiplication before addition or subtraction: 6 x 4 = 24, then 7 + 24 = 31, then 31 - 9 = 22. Working strictly left to right gives the trap value 43 (7 + 6 = 13, 13 x 4 = 52, 52 - 9 = 43); 31 is the result of stopping before the subtraction, and 52 stops one step earlier still.
Source: Order of operations (PEMDAS); FCTC/Cal-JAC written test, Solve Mathematical Problems section (addition, subtraction, multiplication, division)Report a problem with this question
2. A preplan drawing shows a triangular gable end wall with a base of 24 feet and a height of 9 feet. What is the area of that triangular wall?
- A.216 square feet
- B.33 square feet
- C.108 square feet✓ Answer
- D.54 square feet
The area of a triangle is one half the base times the height, because a triangle is exactly half of the rectangle that encloses it: A = 1/2 x 24 x 9 = 1/2 x 216 = 108 square feet. Answer 216 is the enclosing rectangle (forgetting the 1/2 factor), 33 comes from adding 24 + 9 instead of multiplying, and 54 halves the result twice.
Source: Area of a triangle A = 1/2 x base x height; FCTC math content area 'Area'Report a problem with this question
3. An engine's water tank is 3/8 full. The crew adds water equal to 1/4 of the tank's total capacity. What fraction of the tank is now full?
- A.1/3
- B.3/32
- C.1/8
- D.5/8✓ Answer
Fractions can only be added after they are written over a common denominator, so 1/4 must be rewritten as 2/8: 3/8 + 2/8 = 5/8. Answer 1/3 comes from the classic error of adding numerators and denominators separately (3+1 over 8+4 = 4/12), 3/32 comes from multiplying the fractions instead of adding, and 1/8 comes from subtracting.
Source: Addition of fractions with unlike denominators (common denominator rule); FCTC math content area 'Fractions/Decimals'Report a problem with this question
4. Of the 8 self-contained breathing apparatus cylinders on an apparatus, 5 have been refilled after a working fire. What percent of the cylinders have been refilled?
- A.62.5%✓ Answer
- B.58%
- C.12.5%
- D.37.5%
A fraction is converted to a percent by dividing the part by the whole and multiplying by 100: 5 / 8 = 0.625, and 0.625 x 100 = 62.5%. Answer 37.5% is the percent NOT refilled (the complement), 12.5% is the value of a single cylinder (1/8), and 58% simply misreads the digits of 5 and 8.
Source: Fraction-to-percent conversion (part / whole x 100); FCTC math content areas 'Fractions/Decimals' and 'Percentages'Report a problem with this question
5. A 750-gallon booster tank is full at the start of an incident. During the attack, 480 gallons are discharged. What percent of the tank's water REMAINS?
- A.56%
- B.27%
- C.64%
- D.36%✓ Answer
The question asks for the water remaining, so the part must be found first: 750 - 480 = 270 gallons left, and 270 / 750 = 0.36 = 36%. Answer 64% is the percent USED (480 / 750), the standard complement trap; 56% divides 270 by 480 instead of by the full tank, and 27% drops the decimal placement.
Source: Percent of a total: part / whole x 100; percent used vs. percent remaining (complement rule); FCTC math content area 'Percentages'Report a problem with this question
6. A station's monthly call volume rose from 240 calls to 300 calls. What was the percent increase?
- A.20%
- B.25%✓ Answer
- C.125%
- D.60%
Percent change is always measured against the ORIGINAL value: (new - old) / old x 100 = (300 - 240) / 240 = 60 / 240 = 0.25 = 25%. Answer 20% is the most common error, dividing the 60-call increase by the new value 300; 60 is the raw increase, not a percent; and 125% is the new total expressed as a percent of the old (300 / 240).
Source: Percent increase = (new - old) / old x 100 (denominator is the original value); FCTC math content area 'Percentages'Report a problem with this question
7. During a 24-hour shift, a company spends a total of 4 hours and 48 minutes committed to emergency calls. What percent of the shift was spent on calls?
- A.20%✓ Answer
- B.19%
- C.80%
- D.2%
Unlike units must be converted before dividing: 48 minutes is 48/60 = 0.8 of an hour, so the committed time is 4.8 hours. Then 4.8 / 24 = 0.2 = 20%. Answer 19% comes from writing the time as 4.48 hours (treating minutes as decimals), 80% is the percent of the shift NOT on calls, and 2% misplaces the decimal by a factor of ten.
Source: Unit conversion (1 hour = 60 minutes) combined with percent of a total; FCTC math content areas 'Percentages' and word-problem formatReport a problem with this question
8. On a training worksheet you are told that, for a given supply layout, each additional inch of hose diameter delivers 84 more gallons per minute (GPM). A 2-inch line is currently in use, and the incident commander needs 252 GPM MORE than that line delivers. What diameter hose is needed?
- A.A 5-inch hose✓ Answer
- B.A 4-inch hose
- C.A 3-inch hose
- D.A 6-inch hose
Set up the rate as a proportion: 252 GPM needed / 84 GPM per inch = 3 additional inches of diameter. The question asks for the resulting hose size, not the increase, so 2 + 3 = a 5-inch hose. Answer '3-inch' is the classic stop-one-step-early trap (reporting the increment instead of the final size); 4-inch and 6-inch correspond to adding 2 or 4 inches.
Source: Unit rate and proportional reasoning (given rate x quantity); Ergometrics/NTN FireTEAM math component: 'proportions'Report a problem with this question
9. A nozzle flows 150 gallons per minute. If it is opened for 12 minutes without interruption, how many gallons of water are discharged?
- A.1,800 gallons✓ Answer
- B.18,000 gallons
- C.180 gallons
- D.162 gallons
Total volume equals flow rate multiplied by time, because gallons per minute times minutes leaves gallons: 150 x 12 = 1,800 gallons. Answer 180 drops a factor of ten (150 x 12 is not 15 x 12), 18,000 adds one, and 162 comes from adding 150 + 12 instead of multiplying.
Source: Rate x time = quantity (gallons per minute x minutes = gallons); FCTC math content: multiplication word problemsReport a problem with this question
10. An engine carries a 750-gallon water tank and has a pump rated at 1,250 GPM. A crew flows a nozzle at 250 GPM using tank water only. How long will the tank water last?
- A.5 minutes
- B.3 minutes✓ Answer
- C.0.6 minutes
- D.30 minutes
Time equals volume divided by the rate actually being used: 750 gallons / 250 GPM = 3 minutes. The pump's 1,250 GPM rating is extraneous information because the nozzle, not the pump capacity, sets the flow; using it wrongly gives 750 / 1,250 = 0.6 minutes or 1,250 / 250 = 5 minutes, and 30 minutes misplaces the decimal.
Source: Volume / rate = time; identification of extraneous data in word problems (FCTC sample-item convention)Report a problem with this question
11. During a long response, an apparatus used 24 gallons of fuel to travel 180 miles. At the same rate of consumption, how many gallons would it use to travel 300 miles?
- A.2,250 gallons
- B.12.5 gallons
- C.40 gallons✓ Answer
- D.80 gallons
Set the two trips as a proportion, 24 gallons / 180 miles = x / 300 miles, and cross-multiply: x = 24 x 300 / 180 = 7,200 / 180 = 40 gallons. Answer 2,250 multiplies by the 7.5 miles-per-gallon figure instead of dividing by it, 12.5 divides 300 by 24 (the reciprocal rate), and 80 doubles the correct value.
Source: Proportion and unit rate (cross-multiplication); Ergometrics/NTN FireTEAM math component: 'proportions'Report a problem with this question
12. A ground ladder is rated for a maximum total load of 1,050 pounds. Three firefighters whose combined weight is 600 pounds will climb it at the same time, each carrying an equal amount of equipment. What is the maximum weight of equipment each firefighter may carry without exceeding the rating?
- A.200 pounds
- B.150 pounds✓ Answer
- C.450 pounds
- D.350 pounds
First find the load still available, then divide it among the climbers: 1,050 - 600 = 450 pounds available, and 450 / 3 = 150 pounds each. Answer 450 is the stop-one-step-early trap (the total remaining capacity, not the per-person share), 350 divides 1,050 by 3 without subtracting the firefighters, and 200 divides 600 by 3.
Source: Two-step linear equation: (limit - fixed load) / number of shares; FCTC math content area 'Algebra/Geometry'Report a problem with this question
13. A hose lay is built from an unknown number of 50-foot sections plus one 25-foot section. The finished lay measures 375 feet. How many 50-foot sections were used?
- A.7.5
- B.7✓ Answer
- C.6
- D.8
Translate the words into an equation and isolate the unknown with inverse operations: 50x + 25 = 375, so 50x = 350 and x = 7 sections. Answer 7.5 comes from dividing 375 by 50 without first removing the 25-foot section, 8 rounds that figure up, and 6 subtracts twice.
Source: Solving a two-step linear equation (ax + b = c); FCTC math content area 'Algebra/Geometry'Report a problem with this question
14. For a training exercise you are told that water weighs 8.34 pounds per gallon. How much does the water in a full 400-gallon tank weigh?
- A.48 pounds
- B.33,360 pounds
- C.333.6 pounds
- D.3,336 pounds✓ Answer
Substituting into the supplied relationship weight = 8.34 pounds per gallon x gallons gives 8.34 x 400 = 3,336 pounds; multiplying by 4 and then by 100 keeps the decimal in the right place. Answers 333.6 and 33,360 are decimal-placement errors by a factor of ten, and 48 comes from dividing 400 by 8.34 instead of multiplying.
Source: Substitution into a formula supplied in the stem; decimal multiplication (FCTC study guide: task-specific formulas are taught on the job, never required on the exam)Report a problem with this question
15. A warehouse is made of two attached rectangular sections. Both sections are 48 feet deep. One section has a 85-foot frontage and the other has a 115-foot frontage. Approximately how many square feet of floor area does the building have?
- A.20,000 square feet
- B.14,000 square feet
- C.6,000 square feet
- D.10,000 square feet✓ Answer
Because both parts share the same 48-foot depth, the frontages can be added first: 85 + 115 = 200 feet, and 48 x 200 = 9,600 square feet, which rounds to approximately 10,000. Answer 6,000 is one section only (48 x 115 = 5,520), 14,000 comes from using a wrong common side, and 20,000 doubles the building.
Source: Composite area by factoring a common side (A = L x W); estimation/rounding convention for 'approximately' items; FCTC math content area 'Area'Report a problem with this question
16. A preplan lists a two-story residence with a footprint of 28 feet by 45 feet. Both floors have the same dimensions. What is the total floor area of the residence?
- A.2,520 square feet✓ Answer
- B.630 square feet
- C.146 square feet
- D.1,260 square feet
The area of one floor is length times width, 28 x 45 = 1,260 square feet, and the question asks for the whole building, so both stories count: 1,260 x 2 = 2,520 square feet. Answer 1,260 is the classic one-floor trap, 146 is the perimeter (2 x 28 + 2 x 45) rather than the area, and 630 halves one floor.
Source: Rectangle area A = L x W applied per story; perimeter vs. area distinction; FCTC math content area 'Area'Report a problem with this question
17. A rectangular water storage tank measures 8 feet long, 5 feet wide, and 4 feet deep. You are told that 1 cubic foot holds 7.48 gallons. How many gallons does the tank hold when full?
- A.160 gallons
- B.1,196.8 gallons✓ Answer
- C.119.68 gallons
- D.21.4 gallons
Find the volume first, then convert with the factor given in the stem: 8 x 5 x 4 = 160 cubic feet, and 160 x 7.48 = 1,196.8 gallons. Answer 160 stops at cubic feet and never converts to the units the question asks for, 119.68 misplaces the decimal by a factor of ten, and 21.4 divides by 7.48 instead of multiplying.
Source: Rectangular volume V = L x W x H with supplied conversion 1 cubic foot = 7.48 gallons; FCTC math content area 'Volume'Report a problem with this question
18. A cylindrical storage tank is 10 feet in diameter and holds 590 gallons for every foot of water depth. The tank currently contains water to a depth of 6 feet. How many gallons are in the tank?
- A.5,900 gallons
- B.3,540 gallons✓ Answer
- C.98 gallons
- D.1,770 gallons
When a capacity per unit of depth is supplied, the volume is simply that rate times the depth: 590 x 6 = 3,540 gallons. The 10-foot diameter is extraneous information and must not be used to recompute the tank; multiplying by it gives the trap 5,900, 1,770 uses a 3-foot depth, and 98 divides 590 by 6.
Source: Capacity-per-unit-depth rate x depth; extraneous-data convention in official firefighter math sample items; FCTC math content area 'Volume'Report a problem with this question
19. A cylindrical tank has a diameter of 6 feet and a height of 10 feet. Using the formula V = pi x r^2 x h with pi = 3.14, what is the volume of the tank?
- A.188.4 cubic feet
- B.1,130.4 cubic feet
- C.282.6 cubic feet✓ Answer
- D.94.2 cubic feet
The formula uses the radius, which is half the diameter, so r = 3 feet: V = 3.14 x 3^2 x 10 = 3.14 x 9 x 10 = 282.6 cubic feet. Answer 1,130.4 is the classic diameter-for-radius error (3.14 x 36 x 10), 94.2 omits the squaring, and 188.4 is the circumference times height rather than a volume.
Source: Cylinder volume V = pi r^2 h with radius = diameter / 2; FCTC math content area 'Volume'Report a problem with this question
20. A ground ladder is raised against a vertical wall so that it forms a 75-degree angle with the level ground. What is the angle between the ladder and the wall at the point where the ladder touches the wall?
- A.105 degrees
- B.25 degrees
- C.90 degrees
- D.15 degrees✓ Answer
The ladder, the wall, and the ground form a triangle whose interior angles must total 180 degrees; the wall meets level ground at 90 degrees, so 180 - 90 - 75 = 15 degrees at the wall. Answer 105 is the supplement of 75 (the angle on the other side of the straight ground line), 90 is the ground-to-wall angle itself, and 25 has no basis in the triangle.
Source: Interior angles of a triangle sum to 180 degrees; supplementary angles on a straight line; FCTC math content area 'Angles'Report a problem with this question
21. A 25-foot ladder is placed with its base 7 feet out from the base of a wall. Using a^2 + b^2 = c^2, how far up the wall does the top of the ladder reach?
- A.18 feet
- B.26 feet
- C.32 feet
- D.24 feet✓ Answer
The ladder is the hypotenuse of a right triangle, so the vertical leg is found by rearranging the relation: 25^2 - 7^2 = 625 - 49 = 576, and the square root of 576 is 24 feet. Answer 18 subtracts the lengths directly (25 - 7), 32 adds them, and 26 treats the ladder as a leg instead of the hypotenuse.
Source: Pythagorean relation a^2 + b^2 = c^2 for ladder base-and-reach geometry; FCTC math content area 'Algebra/Geometry'Report a problem with this question
22. A unit is out of service for maintenance for 2 hours and 38 minutes. How many seconds is that?
- A.9,480 seconds✓ Answer
- B.9,600 seconds
- C.15,800 seconds
- D.158 seconds
Convert each unit before adding, since 1 hour = 3,600 seconds and 1 minute = 60 seconds: 2 x 3,600 = 7,200 and 38 x 60 = 2,280, so 7,200 + 2,280 = 9,480 seconds. Answer 158 is the time in minutes only, 15,800 multiplies those minutes by 100 instead of 60, and 9,600 rounds 38 minutes up to 40.
Source: U.S. customary time conversion: 1 hour = 60 minutes = 3,600 seconds; NWCG 'Firefighter Math' units-of-measure moduleReport a problem with this question
23. A company logs five response times, in minutes: 6.2, 4.8, 7.5, 5.0, and 6.5. What is the average (mean) response time?
- A.6.2 minutes
- B.7.5 minutes
- C.6.0 minutes✓ Answer
- D.30.0 minutes
The mean is the sum of the values divided by how many values there are: 6.2 + 4.8 + 7.5 + 5.0 + 6.5 = 30.0, and 30.0 / 5 = 6.0 minutes. Answer 30.0 is the sum without dividing, 6.2 is the median (the middle value when sorted), and 7.5 is the longest single time, not an average.
Source: Arithmetic mean = sum of values / number of values; mean vs. median distinction; FCTC math content: word problems with decimalsReport a problem with this question
24. A monthly run report lists: Station 1 - 142 EMS runs and 18 fire runs; Station 2 - 96 EMS runs and 24 fire runs; Station 3 - 130 EMS runs and 30 fire runs. How many FIRE runs did the three stations make in total?
- A.440 runs
- B.72 runs✓ Answer
- C.160 runs
- D.368 runs
Table questions are answered by selecting only the rows and column the question names, so add just the fire runs: 18 + 24 + 30 = 72. Answer 368 totals the EMS column instead (142 + 96 + 130), 440 adds every number in the table, and 160 is Station 1's combined total (142 + 18).
Source: Data interpretation: selecting only the required values from a table; Louisiana Office of State Examiner firefighter test, Math and Reasoning componentReport a problem with this question
Practice questions modeled on the common domains of firefighter written entrance exams (reading, math, mechanical reasoning, spatial orientation). Not affiliated with or endorsed by any fire department or testing vendor. Departments use different exams — check your department's announcement for its specific format. U.S. Fire Administration →