Mechanical Aptitude Formulas: The Complete Cheat Sheet (2026)
The mechanical aptitude formulas you actually need come down to about a dozen simple relationships — gear ratio, mechanical advantage, the lever (moment) formula, pressure (P = F/A) and Ohm’s law — and this page is a printable quick-reference for every one of them. Below you get the complete list in a single skimmable table, then each core formula paired with a plain-English explanation and one worked example. First, the honest part most study guides skip: on most mechanical reasoning tests you are rewarded for understanding the principle, not for plugging numbers into a calculator. A handful of calculation-heavy trade and electrical tests (IBEW, EEI CAST, parts of the Ramsay MAT) are the exception — there the formulas genuinely matter.
Independent prep — simulated/style questions only; not affiliated with any test owner. All practice material is original.
You rarely need to compute anything on a principle-based test like the Bennett BMCT-II™ or Wiesen WTMA™. Learn what each formula means (which way the trade-off runs) and you can answer most questions by reasoning, not arithmetic. Save the number-crunching for the trade and electrical tests below.
Do You Actually Need to Memorize Formulas?
This is the single most-searched worry about mechanical aptitude tests, and the honest answer is: it depends entirely on which test you are sitting. Mechanical aptitude tests split into two broad families, and how much math you need is completely different for each.
Principle-based tests (Bennett, Wiesen, ASVAB) — concepts over calculators
The Bennett Mechanical Comprehension Test™ (BMCT-II), the Wiesen Test of Mechanical Aptitude™ (WTMA), and the ASVAB Mechanical Comprehension subtest are built around diagrams and everyday scenarios. You look at a picture — two meshed gears, a loaded seesaw, a pulley rig — and pick the outcome. Calculators are usually not allowed, and you almost never need one. You need to know that a smaller gear spins faster, that a longer lever arm means less effort, that a movable pulley halves the force. That is understanding the formula, not memorizing it.
Calculation-based tests (IBEW, CAST, some Ramsay) — where the math counts
The IBEW / Electrical Training Alliance aptitude test (algebra and functions), the EEI CAST™ battery (graphic arithmetic and math usage), the Elevator Industry Aptitude Test (EIAT™ math section) and parts of the Ramsay MAT™ do want real calculation. Here you are given numbers — tooth counts, forces, areas, voltages — and you compute an answer against a tight clock. This is where knowing gear ratio = driven ÷ driver, P = F/A and V = IR cold pays off.
The 8 formula families worth knowing cold
Whichever test you face, these eight families cover essentially everything that appears: (1) gear ratios, (2) general mechanical advantage, (3) levers and the moment, (4) pulleys, (5) pressure and hydraulics, (6) Ohm’s law and resistance, (7) force / work / power, and (8) inclined planes and screws. Learn the eight and you have covered the mechanical, hydraulic and electrical questions on every major test.
Quick-Reference Formula Table
This is the printable cheat sheet — every formula on one screen. Each is explained in full further down the page.
Mechanical Aptitude Formulas Cheat Sheet
Every formula you need on one page — gear and pulley ratios, mechanical advantage, torque, work, power, pressure and Ohm's law.
Free. We'll email it to you and send occasional practice tips — unsubscribe anytime.
| Concept | Formula | What the symbols mean | Typical unit |
|---|---|---|---|
| Gear ratio | ratio = driven teeth ÷ driver teeth | driven = output gear, driver = input gear | ratio (no unit) |
| Mechanical advantage | MA = Fout ÷ Fin | force out ÷ force in (also load ÷ effort) | ratio (no unit) |
| Lever / moment | F1 × d1 = F2 × d2 | force × distance from pivot, each side | lb·ft or N·m |
| Pulley MA | effort = load ÷ supporting rope sections | count the rope segments holding the load | lb or N |
| Pressure | P = F ÷ A | force ÷ area it acts on | psi or Pa |
| Hydraulic press | F1÷A1 = F2÷A2 | pressure is equal on both pistons | psi or Pa |
| Ohm’s law | V = I × R | voltage = current × resistance | volts (V) |
| Series resistance | R = R1 + R2 + … | resistances add in a single loop | ohms (Ω) |
| Parallel resistance | 1÷R = 1÷R1 + 1÷R2 + … | reciprocals add across branches | ohms (Ω) |
| Force | F = m × a | mass × acceleration | newtons (N) |
| Work | W = F × d | force × distance moved | joules (J) or ft·lb |
| Power | P = W ÷ t | work ÷ time taken | watts (W) |
| Torque | T = F × r | force × lever-arm length | lb·ft or N·m |
| Inclined plane MA | MA = length ÷ height | ramp length ÷ vertical rise | ratio (no unit) |
| Screw | effort × circumference = load × pitch | one turn advances the screw one pitch | consistent units |
Simple Machines Formulas (with Worked Examples)
Gears, levers, pulleys, ramps and screws are the heart of every mechanical reasoning test. Each one trades force for distance (or speed) in a predictable way — once you see the pattern, the formula is almost automatic.
Gear ratio and gear direction
The gear ratio is simply the driven gear’s teeth divided by the driver gear’s teeth. A big gear driving a small gear spins the small one faster (but with less torque); a small gear driving a big gear does the opposite. Two meshed external gears always turn in opposite directions, and any gear in between acts as an idler that only flips direction — it does not change the speed ratio.
An old-fashioned hand drill has a 40-tooth crank gear that meshes with a 10-tooth pinion on the chuck. For each full turn of the crank, how many turns does the chuck make?
A. 1 turn B. 1/4 turn C. 4 turns
Show the answer & how to get it
Answer: C — 4 turns. One turn of the 40-tooth crank gear pushes 40 teeth past the mesh point. The 10-tooth pinion rotates once for every 10 of those teeth, so it completes 40 ÷ 10 = 4 full turns. The driver’s teeth over the driven’s teeth (40/10) gives the speed multiplication.
The two meshed gears shown have 12 teeth and 24 teeth. When they are running, which gear completes more turns each minute?
A. The 24-tooth gear B. The 12-tooth gear C. Both complete the same number of turns
Show the answer & how to get it
Answer: B — the 12-tooth gear. Meshed gears pass teeth at the same rate where they touch, so the gear with fewer teeth must go around more often to keep up. The 12-tooth gear turns twice for every single turn of the 24-tooth gear. Fewer teeth = faster spin, which is exactly what the gear-ratio formula predicts.
Levers and the moment (turning) formula
A lever balances when the moment (force × distance from the pivot) is equal on both sides: F1 × d1 = F2 × d2. The longer the effort arm, the less force you need — that is why a pry bar or a long wrench feels so powerful. The key is measuring each distance from the pivot (fulcrum).
A 60-pound child sits 6 feet to the left of a seesaw’s center pivot. How far from the pivot must a 90-pound child sit on the right to balance it?
A. 3 feet B. 6 feet C. 5 feet D. 4 feet
Show the answer & how to get it
Answer: D — 4 feet. Balance means equal moments on both sides. The left side produces 60 lb × 6 ft = 360 lb·ft, so the right side must also produce 360 lb·ft: 360 ÷ 90 lb = 4 feet. The heavier rider always sits closer to the pivot.
A 6-foot pry bar is used to lift the edge of a 200-pound crate. The fulcrum is placed 1 foot from the crate end of the bar, and the worker pushes down on the far end. About how much downward force is needed?
A. 33 pounds B. 200 pounds C. 100 pounds D. 40 pounds
Show the answer & how to get it
Answer: D — 40 pounds. The fulcrum splits the 6-foot bar into a 1-foot load arm and a 5-foot effort arm. Balance requires effort × 5 ft = 200 lb × 1 ft, so effort = 200 ÷ 5 = 40 pounds. The 5:1 arm ratio is the lever’s mechanical advantage.
Pulleys and mechanical advantage
The pulley rule is the one candidates most often get wrong. Effort = load ÷ the number of rope sections that support the moving block. A single fixed pulley supports the load with one section, so it gives no force advantage — it only redirects your pull. A single movable pulley is supported by two sections, so it halves the effort. Count the rope segments actually holding the load, not the total pulleys.
A painter hoists a 40-lb paint bucket up to a scaffold using a single fixed pulley bolted to an overhead beam, pulling straight down on the rope. Roughly how much pull is needed?
A. Less than 40 lb of pull B. More than 40 lb of pull C. About 40 lb of pull
Show the answer & how to get it
Answer: C — about 40 lb. A single fixed pulley only changes the direction of the pull; the rope carries the same tension on both sides, so raising the 40-lb bucket still takes about 40 lb. Just one rope section supports the load, so the mechanical advantage is 1. The benefit is convenience, not force savings.
In the double tackle shown, the upper fixed block and the lower movable block each have two sheaves, and four rope segments support a 200-lb compressor. Ignoring friction, how hard must the worker pull?
A. 25 lb B. 50 lb C. 100 lb D. 200 lb
Show the answer & how to get it
Answer: B — 50 lb. Four rope segments share the load, so each carries one quarter of the 200-lb compressor. The worker’s pull equals the tension in one segment: 200 ÷ 4 = 50 lb. In exchange, the worker must pull four feet of rope for every foot the load rises.
Inclined planes and screws
A ramp trades distance for force. Its mechanical advantage is MA = ramp length ÷ vertical height — a longer, gentler ramp needs less push to raise the same load. On a frictionless ramp the force to hold a load is load × (height ÷ length). A screw is just an inclined plane wrapped around a cylinder: effort × circumference = load × pitch, so finer threads (smaller pitch) multiply your force more but advance the screw more slowly per turn.
Every simple machine obeys the same trade-off: whatever you gain in force you pay back in distance. Double the mechanical advantage and you must move the effort twice as far. If an answer choice seems to give you free force and free speed, it is wrong.
Pressure and Hydraulics Formulas
Pressure = Force ÷ Area (P = F/A)
Pressure is force spread over an area: P = F ÷ A, usually in pounds per square inch (psi). The same force over a smaller area makes a higher pressure — which is why a sharp point pierces and a wide snowshoe does not sink. Rearranged, F = P × A and A = F ÷ P.
Pascal’s principle and the hydraulic press (F1/A1 = F2/A2)
In an enclosed fluid, pressure is the same everywhere (Pascal’s principle), so a small force on a small piston becomes a large force on a large piston: F1 ÷ A1 = F2 ÷ A2. The force is multiplied by the ratio of the piston areas — that is the whole idea behind a car jack, a hydraulic press and disc brakes.
Keep your area units consistent. If force is in pounds and area is in square inches, pressure comes out in psi. Mixing square inches with square feet is the most common hydraulics mistake on the test — convert first, then divide.
A hydraulic press has two pistons connected by an oil-filled chamber. The small piston has an area of 2 square inches and the large piston has an area of 10 square inches. If 40 pounds is pushed down on the small piston, how much force appears at the large piston?
A. 40 pounds B. 200 pounds C. 8 pounds
Show the answer & how to get it
Answer: B — 200 pounds. The small piston creates a pressure of 40 lb ÷ 2 sq in = 20 psi. By Pascal’s principle that 20 psi acts on every surface, so the large piston feels 20 psi × 10 sq in = 200 pounds. The area ratio (10:2 = 5×) is the force multiplication.
A shop press must lift a 900-pound engine block resting on its large piston, which has an area of 12 square inches. The small input piston has an area of 2 square inches. How much force must be applied to the input piston?
A. 75 pounds B. 450 pounds C. 150 pounds
Show the answer & how to get it
Answer: C — 150 pounds. To support the load the pressure must reach 900 lb ÷ 12 sq in = 75 psi. That same 75 psi acts on the 2-sq-in input piston, so the required force is 75 psi × 2 sq in = 150 pounds. Same pressure, both pistons — that is the whole formula.
Electrical Formulas for Mechanical & Electrical Aptitude Tests
If you are sitting an electrical aptitude test, the IBEW / Electrical Training Alliance test or the electrical portion of the EEI CAST™, three formulas cover almost everything.
Ohm’s law (V = IR)
Ohm’s law ties voltage, current and resistance together: V = I × R. Rearranged, I = V ÷ R and R = V ÷ I. Volts push current (amps) through resistance (ohms); raise the resistance and, at the same voltage, the current drops.
Series vs parallel resistance
In a series circuit (one loop) resistances simply add: R = R1 + R2 + …. In a parallel circuit (separate branches) you add the reciprocals: 1÷R = 1÷R1 + 1÷R2 + …, and the total is always less than the smallest branch. Adding bulbs in series dims them all; wiring them in parallel keeps each at full brightness.
Reading a simple circuit
Most aptitude-test circuit questions are about reasoning, not calculation: one path or many, what happens when a bulb burns out, where the current can flow. Know that series shares one current and splits the voltage, while parallel gives each branch the full voltage. See our electrical circuits guide and electronics knowledge test for more.
A single battery is connected to two identical light bulbs, A and B, wired one after the other in a single series loop. Which bulb glows brighter?
A. Bulb A B. Bulb B C. Both glow equally
Show the answer & how to get it
Answer: C — both glow equally. In a series loop there is only one path, so exactly the same current flows through every part of the circuit. The two bulbs are identical and carry the same current, so each converts the same amount of energy and they glow equally. Series = one shared current everywhere.
Force, Motion, Work & Power Formulas
These physics relationships show up most on the ASVAB Mechanical Comprehension subtest, the CAST math-usage section and other calculation-heavy tests. Keep each one to a single idea:
Force = mass × acceleration (F = ma)
Newton’s second law: the force on an object equals its mass times how fast it is speeding up. More mass or more acceleration means more force. The same push gives an empty cart more acceleration than a loaded one.
Work = force × distance (W = Fd)
Work is done only when a force moves something through a distance: W = F × d. Holding a weight still does no mechanical work — the load has to move. A simple machine can cut the force but never the total work.
Power = work ÷ time (P = W/t)
Power is how fast work is done: P = W ÷ t. Two motors can do the same job, but the more powerful one finishes sooner. Measured in watts (or horsepower).
Torque = force × lever arm (T = F × r)
Torque is turning force: T = F × r, where r is the distance from the pivot to where the force acts. A longer wrench (bigger r) produces more torque for the same pull — the rotating cousin of the lever formula.
Which Formulas Show Up on Which Test?
Not every test uses every formula. Use this map to focus your prep — and note the calculator column, because it tells you whether to drill the math or just the concept.
| Test | Formula families that appear | Calculator? | Guide |
|---|---|---|---|
| Bennett BMCT-II | Gears, levers, pulleys, pressure, force (principle-based) | No | Bennett guide |
| Wiesen WTMA | Simple machines, force & motion (everyday objects) | No | Wiesen guide |
| Ramsay MAT | Simple machines, physics, tools (mixed) | Sometimes | Ramsay guide |
| EIAT (Elevator) | Math section: ratios, arithmetic | Yes | EIAT guide |
| IBEW / ETA | Algebra & functions (no mechanical section) | Yes | IBEW guide |
| EEI CAST | Graphic arithmetic, math usage, Ohm’s law | Yes | CAST guide |
| ASVAB Mech Comp | F=ma, W=Fd, gears, levers, pulleys | No | ASVAB guide |
Common Formula Mistakes to Avoid
- Confusing gear ratio with mechanical advantage. Gear ratio compares tooth counts; mechanical advantage compares forces. A gear pair that slows the output by 4× multiplies the torque by 4× — related, but not the same number to report.
- Swapping the effort arm and load arm on a lever. Always measure each distance from the pivot, and remember the longer arm carries the smaller force.
- Miscounting the supporting ropes on a pulley. Count only the rope sections that actually hold up the moving block — not the total number of pulleys or the section you pull.
- Mixing series and parallel resistance. Series resistances add directly; parallel ones add as reciprocals and the total is smaller than any single branch.
- Inconsistent units in P = F/A. Pounds with square inches gives psi. Do not mix square inches and square feet — convert first.
Practice Applying These Formulas (Free)
Reading formulas is one thing; applying them under a clock is another. The fastest way to lock them in is to drill the question types that use them. Start with the ungated free mechanical aptitude test — no email required — then work the topic drills for gears, levers, pulleys and springs. For the full picture of what is tested, see the ultimate guide and the sample questions library.
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Frequently Asked Questions
Do you need to memorize formulas for a mechanical aptitude test?
For most mechanical aptitude tests — Bennett, Wiesen, ASVAB Mechanical Comprehension — you do not need to memorize formulas or use a calculator. They are principle-based: you reason from a diagram. Calculation-heavy trade and electrical tests (IBEW, EEI CAST, the EIAT math section and parts of the Ramsay MAT) do require you to know the formulas and compute answers, usually with a calculator allowed.
What formulas are on a mechanical aptitude test?
The core set is gear ratio (driven ÷ driver teeth), mechanical advantage (force out ÷ force in), the lever/moment formula (F₁×d₁ = F₂×d₂), pulley effort (load ÷ supporting ropes), pressure (P = F/A), Ohm’s law (V = IR), series and parallel resistance, and the physics trio F = ma, W = Fd and P = W/t, plus inclined-plane and screw ratios.
What is the gear ratio formula?
Gear ratio = number of teeth on the driven gear ÷ number of teeth on the driver gear. If a 10-tooth driver turns a 40-tooth driven gear, the ratio is 40/10 = 4:1, so the output turns one-quarter as fast but with four times the torque. Meshed external gears also turn in opposite directions.
How do you calculate mechanical advantage?
Mechanical advantage (MA) = output force ÷ input force, which also equals load ÷ effort. For a lever it is the effort-arm length ÷ the load-arm length; for a pulley system it is the number of rope sections supporting the load; for a ramp it is length ÷ height. A higher MA means less effort but more distance to move.
Is a calculator allowed on mechanical reasoning tests?
It varies by test. Principle-based mechanical reasoning tests (Bennett, Wiesen, ASVAB Mechanical Comprehension) usually do not permit a calculator and do not need one. Calculation-based tests such as the IBEW / Electrical Training Alliance test, the EEI CAST battery and the EIAT math section typically do allow a basic calculator. Always confirm with your test invitation.
What is the lever (moment) formula and how do I use it?
The lever balances when the moment on each side is equal: force × distance from the pivot on one side = force × distance on the other (F₁×d₁ = F₂×d₂). To find an unknown, multiply the known force by its distance, then divide by the distance on the other side. Example: a 60-lb child 6 ft from the pivot balances a 90-lb child at 360 ÷ 90 = 4 ft.
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