Mathematics (AM.I.H) is the General-bank subject area candidates most often lose easy points on. Nothing in it is advanced: fractions and decimals, percentages, ratio and proportion, one-unknown algebra, powers and roots, and the area and volume formulas you use for tanks, cylinders, and sheet stock. What makes an answer wrong is almost always a unit or a direction rather than the arithmetic. Drill the setups until they are automatic and this becomes one of the most reliable scoring areas on the General test.
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Written against the FAA primary sources cited at the foot of this page.
The mathematics chapter of FAA-H-8083-30 defines the scope, and the questions stay inside it. Arithmetic comes first: adding and reducing fractions, converting between fractions, decimals, and percentages, working with signed numbers, and rounding to a stated place. Stems such as convert 5/8 to a decimal, or round 7.846 to the nearest tenth, are exactly what they look like - the points get lost by rushing, not by difficulty. Ratio and proportion carry more weight than their share of the questions suggests, because they turn up disguised. Gear and pulley problems, compression ratio, mixture and dilution ratios, and scale readings all reduce to the same setup. Expect to simplify a ratio such as 16 to 4, and to apply one, as in a 3-to-1 reduction driven at 1,500 rpm. Algebra is limited to a single unknown: solve 3x + 6 = 18, or rearrange a formula so the quantity you want stands alone. Powers, roots, and scientific notation appear on their own and inside the area formulas. Geometry and mensuration finish the area: area of a rectangle, triangle, and circle; volume of a rectangular solid and a cylinder; surface area for material and finish estimates. Tank problems are the classic case. Find the volume in cubic inches first, then convert at 231 cubic inches per gallon. Keep 144 square inches per square foot and 1,728 cubic inches per cubic foot ready too, because the distractors are built out of the wrong conversion.
ACS AM.I.H is tested on the FAA General written test, one of 1,558 ACS-tagged questions in the General bank. Every question tagged to this area carries a worked rationale and its FAA handbook reference, so you can drill the code itself rather than the whole test.
A rectangular fuel tank measures 30 inches long, 20 inches wide, and 12 inches deep. What is its volume in cubic inches?
A gear ratio is given as 3 to 1. If the driving gear turns 1,500 rpm, how fast does the driven gear turn when the ratio reduces speed?
Convert the fraction 5/8 to a decimal.
A part originally costing $80 is marked up 25 percent. What is the new price?
What is the square root of 144?
Express the ratio 16 to 4 in its simplest form.
Solve for x in the equation 3x + 6 = 18.
Round the number 7.846 to the nearest tenth.
The area of a triangle is found using which relationship?
One milliliter is equal to
If a square has an area of 9 square feet, the length of one side is
The center of gravity of a uniform triangular plate is located at the intersection of its
The center of gravity (centroid) of a triangle is located
Two forces act at 90° to each other with magnitudes of 3 lb and 4 lb. The magnitude of the resultant is
An object travels 18,000 feet in 1 minute 30 seconds. What is its average speed?
Approximately how many feet are in 100 yards?
A car moves a distance of 5 miles at a steady speed in 10 minutes. What is its speed?
A spacecraft travels a distance of 480,000 miles in 2 days. What is its average speed?
A cyclist covers a distance of 990 feet at a constant speed in 90 seconds. What is the speed?
How long will it take a car moving at 60 mph to travel 90 miles?
An aircraft travels at 300 mph for 30 minutes at a steady speed. How far does it move in that time?
A light airplane flies along a semicircular path from point A to point B. If the circle has a radius of 20 miles and the flight takes 30 minutes, what is the average speed?
How many radians are contained in a semicircle?
An angular velocity of 500 RPM is equal to how many radians per second?
One radian is defined as
A drive shaft turns at 150 pi radians per second. What is its speed in RPM?
Ten kilograms is correctly expressed numerically as
One cubic foot of water weighs 62.4 lb, and a fuel has a specific gravity of 0.81. What is the weight of 10 cubic feet of that fuel?
A car travels 24 miles in 45 minutes. What is its average speed?
Evaluate: (6 - 2) - (6 - 9) / (-3 + (-3))
Evaluate: 15.4 / 2 - 2 * (6.2 - 15.6)
A rectangular block measures 4 cm by 6 cm by 12 cm. What is its volume?
What defines a scalene triangle?
Evaluate the following expression: 4{2(5 - 1) - 3} + 8.
What is the result of 4 3/8 - 2 1/4 + 1/8?
What is the sum of 11/16 + 5/8?
The product of 3/4 multiplied by 0.82 is equal to
The fraction 7/6 can be expressed as the decimal
The ratio 6:5 is equivalent to which of the following ratios?
The expression 14^3 is equal to
The number 0.0000413 can be written in scientific notation as
What is the sum of 5/8 + 3/4?
What is the lowest common denominator for 1/6 + 1/5 + 1/17 + 1/2?
The formula for calculating the area of a triangle is
The lateral (curved) surface area of a right circular cone with base radius r and slant height l is
Determine the result of (+3) - (-4).
Express the fraction 9/20 as a percentage.
To find the area of a circle, multiply
How many centimeters are in one inch?
What is the lowest common multiple of 6, 7, and 8?
ACS code: AM.I.H
Correct answer: 7,200 cubic inches
Rationale: The volume of a rectangular solid equals length times width times height. Here 30 x 20 x 12 = 7,200 cubic inches. The 3,600 value comes from omitting one dimension, and 600 is only length times width, which gives surface area of one face, not volume.
ACS code: AM.I.H
Correct answer: 500 rpm
Rationale: A 3-to-1 reduction means the output turns one-third as fast as the input. Dividing 1,500 by 3 gives 500 rpm. Multiplying by 3 (4,500) would be a speed increase, and 1,500 would mean no ratio change at all.
ACS code: AM.I.H
Correct answer: 0.625
Rationale: A fraction is converted to a decimal by dividing the numerator by the denominator: 5 divided by 8 equals 0.625. The 0.580 value simply re-reads the digits as a decimal, and 1.600 is the result of dividing 8 by 5, which inverts the fraction.
ACS code: AM.I.H
Correct answer: $100
Rationale: A 25 percent markup is found by multiplying 80 by 0.25, giving 20, then adding it to the original: 80 + 20 = 100. The $105 value adds the wrong amount, and $85 adds only $5, which would be a much smaller percentage.
ACS code: AM.I.H
Correct answer: 12
Rationale: The square root of a number is the value that, multiplied by itself, equals that number. Since 12 x 12 = 144, the square root is 12. The value 72 is simply half of 144, and 14 does not square to 144 (14 x 14 = 196).
ACS code: AM.I.H
Correct answer: 4 to 1
Rationale: A ratio is reduced by dividing both terms by their greatest common factor, which is 4: 16/4 = 4 and 4/4 = 1, giving 4 to 1. The 8 to 2 form is correct in value but not fully reduced, and 1 to 4 reverses the order of the terms.
ACS code: AM.I.H
Correct answer: x = 4
Rationale: First subtract 6 from both sides to isolate the term with x: 3x = 12. Then divide both sides by 3 to find x = 4. The value 8 results from adding instead of subtracting the 6, and 6 results from dividing 18 by 3 without first removing the 6.
ACS code: AM.I.H
Correct answer: 7.8
Rationale: Rounding to the nearest tenth looks at the hundredths digit; since that digit is 4, the tenths digit stays unchanged, giving 7.8. The 7.9 value rounds up incorrectly, and 7.85 is rounded to the hundredth rather than the tenth.
ACS code: AM.I.H
Correct answer: One-half the base times the height
Rationale: The area of a triangle equals one-half the base multiplied by the height. Using base times height alone gives the area of a rectangle or parallelogram of the same dimensions, which is twice too large. The third option combines the terms in a way that has no geometric basis for area.
ACS code: AM.I.H
Correct answer: one one-thousandth of a liter.
Rationale: The prefix milli means one-thousandth, so one milliliter equals one-thousandth of a liter (1 mL = 0.001 L), and 1,000 milliliters make one liter. One-hundredth of a liter is a centiliter, and one-millionth is a microliter.
ACS code: AM.I.H
Correct answer: 3 feet.
Rationale: The area of a square equals the side length squared, so the side equals the square root of the area. The square root of 9 square feet is 3 feet. The values 9 feet and 4 feet do not square back to 9 square feet.
ACS code: AM.I.H
Correct answer: medians, each drawn from a vertex to the midpoint of the opposite side.
Rationale: The center of gravity (centroid) of a uniform triangular plate lies where its three medians intersect, each median running from a vertex to the midpoint of the opposite side. Angle bisectors and altitudes meet at different points and do not define the centroid.
ACS code: AM.I.H
Correct answer: one third of the median up from the base.
Rationale: The centroid of a triangle lies on each median, dividing it in a 2:1 ratio measured from the vertex. This places it one third of the median's length up from the base. The 2/3-from-base option reverses the ratio, which would put the centroid nearer the apex.
ACS code: AM.I.H
Correct answer: 5 lb.
Rationale: When two forces act at 90°, the resultant is found by the Pythagorean theorem: R = the square root of (3^2 + 4^2) = the square root of 25 = 5 lb. Simply adding (7 lb) or subtracting (1 lb) the magnitudes only applies when the vectors are parallel, not perpendicular.
ACS code: AM.I.H
Correct answer: 200 ft/sec.
Rationale: Average speed is distance divided by time. Converting 1 minute 30 seconds to 90 seconds gives a speed of 18,000 ÷ 90 = 200 ft/sec. The other figures result from using the wrong time conversion.
ACS code: AM.I.H
Correct answer: 300 feet.
Rationale: One yard equals 3 feet, so 100 yards equals 300 feet. The other values do not use the correct 3-feet-per-yard conversion.
ACS code: AM.I.H
Correct answer: 30 mph.
Rationale: Speed is distance divided by time. Covering 5 miles in 10 minutes means in one hour (six such intervals) the car covers 30 miles, so the speed is 30 mph. The 60 mph answer would require 5 miles in 5 minutes, and 15 mph would require 20 minutes.
ACS code: AM.I.H
Correct answer: 10,000 mph.
Rationale: Average speed is distance divided by time. Two days equal 48 hours, so 480,000 ÷ 48 = 10,000 mph. The 48,000 and 36,000 figures come from using days instead of hours or dividing incorrectly.
ACS code: AM.I.H
Correct answer: 11 ft/sec.
Rationale: Speed equals distance divided by time: 990 ft ÷ 90 sec = 11 ft/sec. The other figures result from using the wrong time value rather than the 90 seconds given.
ACS code: AM.I.H
Correct answer: 90 minutes.
Rationale: Time equals distance divided by speed: 90 mi ÷ 60 mph = 1.5 hours, which is 90 minutes. The 40- and 75-minute answers come from dividing the figures the wrong way or misconverting hours to minutes.
ACS code: AM.I.H
Correct answer: 150 miles.
Rationale: Distance equals speed times time. Thirty minutes is 0.5 hour, so distance = 300 mph × 0.5 h = 150 miles. The 300-mile answer assumes a full hour, and 75 miles would require only a quarter of an hour.
ACS code: AM.I.H
Correct answer: 125.7 mph.
Rationale: Average speed is path length divided by time. A semicircle of radius 20 miles has an arc length of pi x 20 = 62.83 miles, covered in 0.5 hour, giving 62.83 / 0.5 = 125.7 mph. The distractor 251.4 results from doubling, and 62.83 confuses speed with the raw arc-length figure.
ACS code: AM.I.H
Correct answer: pi radians.
Rationale: A full circle subtends 2 pi radians at its center, so a semicircle (half a circle) subtends exactly pi radians, about 3.14. The value 2 pi corresponds to a full circle, and pi/2 corresponds to a quarter circle.
ACS code: AM.I.H
Correct answer: 16.66 pi rad/s.
Rationale: To convert RPM to rad/s, multiply by 2 pi / 60. For 500 RPM this is 500 × 2 pi / 60 = 16.66 pi rad/s (about 52.4 rad/s). The conversion divides revolutions per minute by 60 to get per second and multiplies by 2 pi radians per revolution.
ACS code: AM.I.H
Correct answer: the angle subtended at the center of a circle when the arc length between two radii equals the radius.
Rationale: One radian is the angle subtended at the center of a circle when the arc length between two radii equals the radius. This is why a full circle of circumference 2 pi r contains exactly 2 pi radians. Defining it via the diameter or twice the radius gives the wrong angle.
ACS code: AM.I.H
Correct answer: 4,500 RPM.
Rationale: To convert from rad/s to RPM, multiply by 60 / (2 pi). So 150 pi × 60 / (2 pi) = 150 × 30 = 4,500 RPM. Each revolution is 2 pi radians, and there are 60 seconds in a minute.
ACS code: AM.I.H
Correct answer: 10 kg.
Rationale: The symbol for the kilogram is 'kg', so ten kilograms is written 10 kg. 'Mg' is the megagram (1,000 kg) and 'gm' is a non-standard abbreviation for grams, so neither correctly expresses ten kilograms. Correct unit notation is covered under metric measurement in the Mathematics chapter of FAA-H-8083-30.
ACS code: AM.I.H
Correct answer: 505.4 lb.
Rationale: Fuel weight equals water weight times specific gravity times volume. One cubic foot of water is 62.4 lb, so one cubic foot of fuel is 62.4 × 0.81 = 50.544 lb, and ten cubic feet weigh 505.44 lb, rounding to 505.4 lb. The 624 lb option ignores the 0.81 specific gravity (FAA-H-8083-30 Mathematics).
ACS code: AM.I.H
Correct answer: 32 mph.
Rationale: Average speed is distance divided by time. 45 minutes is 0.75 of an hour, so 24 miles divided by 0.75 hours equals 32 mph. The 18 mph answer wrongly divides by 45 directly instead of converting minutes to hours.
ACS code: AM.I.H
Correct answer: 3 1/2.
Rationale: Following the order of operations, work the parentheses first: (6 - 2) = 4 and (6 - 9) = -3, with the denominator (-3 + (-3)) = -6. Division comes before subtraction, so (-3) / (-6) = 0.5, giving 4 - 0.5 = 3.5, or 3 1/2.
ACS code: AM.I.H
Correct answer: 26.5.
Rationale: Apply the order of operations, with division and multiplication before subtraction. 15.4 / 2 = 7.7, and 6.2 - 15.6 = -9.4 so 2 * (-9.4) = -18.8. Subtracting a negative gives 7.7 - (-18.8) = 7.7 + 18.8 = 26.5.
ACS code: AM.I.H
Correct answer: 0.000288 cubic meters.
Rationale: The volume of a rectangular block is length times width times height: 4 x 6 x 12 = 288 cubic centimeters. Converting to cubic meters divides by 1,000,000 (since 1 m = 100 cm), giving 288 / 1,000,000 = 0.000288 cubic meters.
ACS code: AM.I.H
Correct answer: All three sides are unequal.
Rationale: A scalene triangle has all three sides of different lengths, and consequently all three angles differ as well. A triangle with two equal sides is isosceles, and one with all three sides equal is equilateral.
ACS code: AM.I.H
Correct answer: 28
Rationale: Work from the innermost grouping outward, following the standard order of operations. (5 - 1) = 4, then 2 x 4 = 8, and 8 - 3 = 5. Multiplying by the outer 4 gives 20, and adding 8 yields 28.
ACS code: AM.I.H
Correct answer: 2 1/4
Rationale: Convert to a common denominator of eighths: 4 3/8 = 35/8, 2 1/4 = 18/8, and 1/8 stays. So 35/8 - 18/8 + 1/8 = 18/8, which simplifies to 2 1/4.
ACS code: AM.I.H
Correct answer: 21/16
Rationale: Use a common denominator of 16: 11/16 stays and 5/8 becomes 10/16. Adding gives 11/16 + 10/16 = 21/16. The 55/128 answer wrongly multiplies the fractions instead of adding them.
ACS code: AM.I.H
Correct answer: 0.615
Rationale: The fraction 3/4 equals 0.75, and 0.75 x 0.82 = 0.615. The 1.23 result mistakenly multiplies by 1.5, and 2.46 multiplies by 3.
ACS code: AM.I.H
Correct answer: 1.166
Rationale: Dividing 7 by 6 gives 1.1666..., which rounds to 1.166. The fraction is one whole plus 1/6, and 1/6 = 0.1666, confirming the recurring decimal.
ACS code: AM.I.H
Correct answer: 24:20
Rationale: Multiplying both terms of 6:5 by the same factor keeps the ratio equal. Multiplying by 4 gives 24:20. The other options do not reduce back to 6:5.
ACS code: AM.I.H
Correct answer: 14 x 14 x 14
Rationale: An exponent, or power, of 3 means the base is multiplied by itself three times. So 14^3 means 14 x 14 x 14. Writing 14 + 14 + 14 represents multiplication by 3, not raising to the third power.
ACS code: AM.I.H
Correct answer: 413 x 10^-7
Rationale: Moving the decimal point right by seven places turns 0.0000413 into 413, so the original equals 413 x 10^-7. Checking: 413 x 10^-7 = 0.0000413, confirming the value.
ACS code: AM.I.H
Correct answer: 11/8
Rationale: With a common denominator of 8, 5/8 stays and 3/4 becomes 6/8. Adding gives 5/8 + 6/8 = 11/8. The 8/8 answer is simply wrong, and 11/4 uses the wrong denominator.
ACS code: AM.I.H
Correct answer: 510
Rationale: The lowest common denominator is the least common multiple of 6, 5, 17, and 2. Since 6 = 2 x 3, while 5 and 17 are prime, and 2 is already a factor of 6, the LCM is 2 x 3 x 5 x 17 = 510.
ACS code: AM.I.H
Correct answer: 1/2(base x height)
Rationale: The area of any triangle is one-half the base multiplied by the perpendicular height, so 1/2(base x height). In a right triangle the two legs serve directly as the base and height.
ACS code: AM.I.H
Correct answer: pi x r x l
Rationale: The lateral surface area of a right circular cone is pi times the base radius times the slant height: pi x r x l. Adding 2(pi)r^2 would give the total surface area including the base, which is not asked here.
ACS code: AM.I.H
Correct answer: +7
Rationale: Subtracting a negative is the same as adding its positive: (+3) - (-4) = 3 + 4 = +7. The two negative signs combine to make a positive.
ACS code: AM.I.H
Correct answer: 45 percent
Rationale: To express a fraction as a percentage, multiply by 100. 9/20 = 0.45, and 0.45 x 100 = 45 percent.
ACS code: AM.I.H
Correct answer: the square of the radius by pi
Rationale: The area of a circle is pi multiplied by the radius squared (pi r^2), so you multiply the square of the radius by pi. Multiplying twice the radius by pi instead gives the circumference.
ACS code: AM.I.H
Correct answer: 2.54
Rationale: One inch is defined as exactly 2.54 centimeters. The 25.4 figure is the number of millimeters in an inch, and 0.254 is off by a factor of ten.
ACS code: AM.I.H
Correct answer: 168
Rationale: The lowest common multiple is the smallest number divisible by 6, 7, and 8. Since 8 = 2^3, 6 = 2 x 3, and 7 is prime, the LCM is 2^3 x 3 x 7 = 168. As shown in FAA-H-8083-30, 84 is not divisible by 8, so it cannot be the LCM.
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