The official Pearson VUE UCAT interface provides zero formula sheets during the 36-question, 26-minute Quantitative Reasoning subtest (43.3 seconds per question), requiring instantaneous mental retrieval of every geometric, kinematic, financial, statistical, and clinical dosing identity. This complete reference consolidates every formula tested in 2026 and 2027 UCAT Quantitative Reasoning alongside nonlinear unit conversions ($1\text{ m}^2 = 10,000\text{ cm}^2$, $1\text{ m}^3 = 1,000\text{ L}$) and fraction-to-percentage benchmarks ($1/3$ to $1/20$). Pair this reference with my Geometry, Perimeter, Area & Volume Study Note, the Mental Arithmetic & Calculator Study Note, the Geometry FSRS-6 Pulse Deck (22 Cards), and Topical QBank Chapters 162 (386 Qs) and 164 (73 Qs).
1. The Zero-Formula-Sheet Mandate in UCAT Quantitative Reasoning
The UCAT Quantitative Reasoning subtest gives you 26 minutes to solve 36 five-option multiple-choice questions across 9 data scenarios, leaving an average budget of 43.3 seconds per question. Unlike GCSE, O-Level, or A-Level mathematics examinations, the official Pearson VUE testing environment provides no on-screen formula sheet and no physical reference booklet. Every mathematical equation, geometric identity, and metric conversion factor must fire from your long-term memory in under 2 seconds.
When I engineered the BeambePrep UCAT curriculum and audited 6,190 exam-standard items, I found that candidate errors in Quantitative Reasoning rarely stem from encountering post-GCSE mathematics. Calculus, trigonometry (sin, cos, tan buttons), matrices, and quadratic equations are excluded from the UCAT specification. Instead, candidates lose marks because they hesitate for 15 seconds trying to reconstruct the volume of a cylinder, forget that the on-screen TI-108 calculator lacks a $\pi$ key and an exponent ($x^y$) button, or divide cubic centimeters by $100$ instead of $1,000,000$ when converting to cubic meters.
- Cognitive Bandwidth Conservation: In a subtest where the official cohort mean is
654out of900($N = 39,935$, overall cognitive mean1,891), spending 15 seconds deriving the surface area of a closed hemisphere from first principles consumes more than one-third of your 43.3-second question allocation before you even touch the numeric keypad. - Calculator Hardware Constraints: The Pearson VUE on-screen calculator provides only four basic arithmetic operations (
+,-,×,÷), a percentage key (%), a square root key ($\sqrt{x}$), a sign toggle (+/-), and a memory register (MRC,M+,M-). Because there is no $\pi$ button, you must enter $\pi \approx 3.14$ (or $\frac{22}{7}$) manually. Because there is no power button ($x^y$), squaring or cubing a radius requires sequential multiplication ($r \times r$ or $r \times r \times r$).
Universal Formula Standard Across UK, ANZ & AKU MBBS Admissions: Medical schools across the United Kingdom, Australia, New Zealand, and Aga Khan University (AKU) in Pakistan administer the identical computer-based Pearson VUE UCAT battery scored on the 900 to 2,700 cognitive scale. Download the Midnight Dark Edition PDF and Ink-Saving Print Edition PDF of my Geometry, Perimeter, Area & Volume Study Note and Mental Arithmetic & Calculator Ergonomics Note to keep this complete formula architecture on your revision desk, then test your recall in Topical QBank Chapter 164 (73 Geometry Questions).
2. 2D Mensuration: Polygons, Circles, Sectors, and Pythagorean Triples
Two-dimensional geometry questions in Quantitative Reasoning frequently disguise simple shapes inside practical contexts: hospital car park paving, circular running tracks, rectangular garden lawns bordered by semicircular flowerbeds, or pitched roof cross-sections.
Complete 2D Perimeter and Area Formula Table
| 2D Geometric Shape | Perimeter / Circumference Formula | Area Formula | Tactical UCAT Execution Rule |
|---|---|---|---|
| Square (side $s$) | $P = 4s$ | $A = s^2$ | Diagonal $d = s\sqrt{2} \approx 1.414s$; Area from diagonal $A = \frac{d^2}{2}$ |
| Rectangle (length $l$, width $w$) | $P = 2(l + w)$ | $A = l \times w$ | Diagonal $d = \sqrt{l^2 + w^2}$ via Pythagoras |
| Parallelogram (base $b$, slant $s$, perpendicular height $h$) | $P = 2(b + s)$ | $A = b \times h$ | Trap: Always multiply base by perpendicular height $h$, never slanted side $s$ |
| Triangle (base $b$, perpendicular height $h$) | $P = a + b + c$ | $A = \frac{1}{2} b h = 0.5 \times b \times h$ | Halve the even dimension ($b$ or $h$) mentally before multiplying |
| Equilateral Triangle (side $s$) | $P = 3s$ | $A = \frac{\sqrt{3}}{4} s^2 \approx 0.433 s^2$ | Perpendicular height $h = \frac{\sqrt{3}}{2}s \approx 0.866s$ |
| Trapezium / Trapezoid (parallel sides $a, b$; height $h$) | $P = a + b + s_1 + s_2$ | $A = \left(\frac{a + b}{2}\right) \times h$ | Average the two parallel sides mentally first, then multiply by vertical height $h$ |
| Rhombus / Kite (diagonals $d_1, d_2$) | $P = 4s$ (Rhombus) | $A = \frac{1}{2} d_1 d_2$ | Diagonals intersect at $90^\circ$, splitting into 4 right-angled triangles |
| Circle (radius $r$, diameter $d = 2r$) | $C = 2\pi r = \pi d$ | $A = \pi r^2 = \frac{\pi d^2}{4}$ | No $\pi$ button on calculator: type $3.14$ (or $\frac{22}{7}$ if $r$ is a multiple of $7$) |
| Circle Sector (central angle $\theta^\circ$) | $P_{\text{sector}} = \left(\frac{\theta}{360} \times 2\pi r\right) + 2r$ | $A_{\text{sector}} = \frac{\theta}{360} \times \pi r^2$ | Trap: Sector perimeter includes curved arc length plus two straight radii ($+2r$) |
| Annulus / Ring (outer radius $R$, inner radius $r$) | $C_{\text{total}} = 2\pi(R + r)$ | $A_{\text{ring}} = \pi(R^2 - r^2)$ | Factor out $\pi$ first: compute $(R^2 - r^2)$ mentally, then multiply by $3.14$ once |
Pythagoras' Theorem and Instant Integer Triples
For any right-angled triangle with legs $a$ and $b$ and hypotenuse $c$ (the longest side, opposite the $90^\circ$ angle):
$$a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2} \quad \Big| \quad a = \sqrt{c^2 - b^2}$$
Instead of typing squares and square roots into the on-screen calculator, memorize the four canonical Pythagorean Integer Triples and their scaled multiples to solve right-triangle lengths in 2 seconds:
- The $3 : 4 : 5$ Family: Includes $(3, 4, 5)$, $(6, 8, 10)$, $(9, 12, 15)$, $(12, 16, 20)$, and $(15, 20, 25)$.
- The $5 : 12 : 13$ Family: Includes $(5, 12, 13)$ and $(10, 24, 26)$.
- The $8 : 15 : 17$ Family: Includes $(8, 15, 17)$.
- The $7 : 24 : 25$ Family: Includes $(7, 24, 25)$.
The Annulus Difference-of-Squares Shortcut: When asked for the area of a paved circular path of width $2\text{ m}$ surrounding a circular fountain of inner radius $r = 6\text{ m}$ (so outer radius $R = 8\text{ m}$), never calculate $3.14 \times 64$ and $3.14 \times 36$ separately! Subtract the squares mentally first: $R^2 - r^2 = (8 + 6)(8 - 6) = 14 \times 2 = 28$. Then multiply $28 \times 3.14 = 87.92\text{ m}^2$ in a single calculator step, saving 15 seconds.
3. 3D Mensuration: Solids, Cylinders, Cones, Spheres, and Scaling Laws
Three-dimensional geometry stems evaluate fluid capacity (volume) and external material coverage (surface area) across storage tanks, pharmaceutical capsules, shipping containers, and hydrotherapy pools.
Complete 3D Volume and Surface Area Formula Table
| 3D Solid | Volume ($V$) | Total Surface Area ($TSA$) | Critical Exam Boundary Condition |
|---|---|---|---|
| Cube (edge $s$) | $V = s^3$ | $TSA = 6s^2$ | Open-top cube (5 faces) has surface area $5s^2$ |
| Cuboid / Rectangular Prism ($l, w, h$) | $V = l \times w \times h$ | $TSA = 2(lw + lh + wh)$ | Open-top tank (no lid) has $SA = lw + 2(lh + wh)$; 3D diagonal $d = \sqrt{l^2 + w^2 + h^2}$ |
| General Prism (cross-section $A_c$, length $L$) | $V = A_c \times L$ | $TSA = 2A_c + (P_{\text{base}} \times L)$ | Applies to triangular prisms and trapezium-floor swimming pools |
| Closed Cylinder (radius $r$, height $h$) | $V = \pi r^2 h$ | $TSA = 2\pi r h + 2\pi r^2 = 2\pi r(h + r)$ | Curved lateral wall area alone is $CSA = 2\pi r h$; open-top cylinder is $2\pi r h + \pi r^2$ |
| Right Circular Cone (radius $r$, height $h$, slant $l$) | $V = \frac{1}{3}\pi r^2 h$ | $TSA = \pi r l + \pi r^2$ | Slant height $l = \sqrt{r^2 + h^2}$; cone volume is exactly $\frac{1}{3}$ of a cylinder with equal $r$ and $h$ |
| Sphere (radius $r$) | $V = \frac{4}{3}\pi r^3$ | $TSA = 4\pi r^2$ | Enter $\frac{4}{3}\pi r^3$ as 4 3.14 r r r / 3 Enter on the NumPad |
| Solid Hemisphere (radius $r$) | $V = \frac{2}{3}\pi r^3$ | $TSA = 3\pi r^2$ (Curved $2\pi r^2$ + Base $\pi r^2$) | High-Yield Trap: Hollow dome is $2\pi r^2$, but a solid closed hemisphere is $3\pi r^2$ |
The Nonlinear Dimensional Scaling Law ($k, k^2, k^3$)
Whenever all linear dimensions of any 2D or 3D object are multiplied by a constant scale factor $k$:
- 1D Linear Measurements (side, radius, diameter, height, perimeter, circumference) scale by $k^1 = k$.
- 2D Area Measurements (cross-sectional area, base area, total surface area) scale by $k^2$.
- 3D Volume & Mass Measurements (volume, fluid capacity, weight at constant density) scale by $k^3$.
For example, if a hospital doubles the radius and height of a cylindrical oxygen tank ($k = 2$), its paintable surface area quadruples ($2^2 = 4\times$) and its gas storage volume increases eightfold ($2^3 = 8\times$). If linear dimensions increase by $20\%$ ($k = 1.20$), surface area increases by $1.20^2 = 1.44$ ($+44\%$) and volume increases by $1.20^3 = 1.728$ ($+72.8\%$).
4. Speed, Distance, Time, and Harmonic Average Speed
Motion, logistics, and multi-stage timetable problems test your mastery of the kinematic triad alongside base-60 clock conversions.
Core Kinematic and Velocity Formulas
- Fundamental Triad:
$$\text{Distance } (d) = \text{Speed } (v) \times \text{Time } (t) \quad \Big| \quad v = \frac{d}{t} \quad \Big| \quad t = \frac{d}{v}$$
- Universal Average Speed Law:
$$v_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{d_1 + d_2 + \dots + d_n}{t_1 + t_2 + \dots + t_n}$$
- Harmonic Mean for Equal-Distance Two-Way Journeys ($d_1 = d_2$):
$$v_{\text{avg}} = \frac{2 v_1 v_2}{v_1 + v_2}$$
(Never average $v_1$ and $v_2$ using $\frac{v_1 + v_2}{2}$ unless the vehicle travels for equal times $t_1 = t_2$!)- Relative Closing & Pursuit Speed:
- Opposite Directions (Head-On or Moving Apart): $v_{\text{rel}} = v_1 + v_2$, so $t_{\text{meet}} = \frac{d_{\text{initial}}}{v_1 + v_2}$.
- Same Direction (Pursuit / Catching Up): $v_{\text{rel}} = |v_1 - v_2|$, so $t_{\text{catch}} = \frac{d_{\text{headstart}}}{v_1 - v_2}$.
- Instant Velocity Multiplier ($\text{m/s} \leftrightarrow \text{km/h}$):
$$1\text{ m/s} = 3.6\text{ km/h} \implies v_{\text{km/h}} = v_{\text{m/s}} \times 3.6 \quad \Big| \quad v_{\text{m/s}} = \frac{v_{\text{km/h}}}{3.6}$$
- Base-60 Decimal Time Conversions:
$$\text{Hours} = \frac{\text{Minutes}}{60} \quad \Big| \quad \text{Minutes} = \text{Decimal Hours} \times 60$$
Memorize the six clean fractional hour anchors: $6\text{ min} = 0.10\text{ hr}$, $12\text{ min} = 0.20\text{ hr}$, $15\text{ min} = 0.25\text{ hr}$, $20\text{ min} = 0.333\text{ hr}$, $30\text{ min} = 0.50\text{ hr}$, $45\text{ min} = 0.75\text{ hr}$.
The Base-60 Decimal Clock Fallacy: Never enter hours and minutes directly into the calculator as a base-10 decimal! Treating 2 hours 45 minutes as 2.45 hours (instead of $2 + \frac{45}{60} = 2.75\text{ hours}$) or reading a calculator output of 3.3 hours as 3 hours 30 minutes (instead of $3\text{ hours}$ and $0.3 \times 60 = 18\text{ minutes}$) is one of the highest-frequency timing traps in UCAT Quantitative Reasoning.
5. Percentages, Ratios, Finance, Tax, and Interest Formulae
Percentages, proportional allocations, and financial tables account for over 60% of all calculations in Quantitative Reasoning.
| Financial / Proportional Concept | Exact Mathematical Formula | High-Speed Tactical Shortcut |
|---|---|---|
| Percentage of a Quantity | $\text{Value} = \left(\frac{r}{100}\right) \times \text{Total}$ | Use Reversible Symmetry: $X\% \text{ of } Y = Y\% \text{ of } X$ (e.g. $16\% \text{ of } 75 = 75\% \text{ of } 16 = 12$) |
| Percentage Change | $\% \Delta = \left(\frac{\text{Final} - \text{Original}}{\text{Original}}\right) \times 100\%$ | Always divide by the Original chronological baseline, never the Final value |
| Single-Step Decimal Multiplier | $\text{Final} = \text{Original} \times \left(1 \pm \frac{r}{100}\right)$ | $+18\%$ increase $\rightarrow \times 1.18$; $-15\%$ discount $\rightarrow \times 0.85$ |
| Chained Compound Multipliers | $M_{\text{net}} = M_1 \times M_2 = \left(1 \pm \frac{r_1}{100}\right)\left(1 \pm \frac{r_2}{100}\right)$ | Symmetric $+x\%$ then $-x\%$ always yields a net loss of $-\left(\frac{x^2}{100}\right)\%$ |
| Reverse Percentage (Find Original) | $\text{Original} = \frac{\text{Final}}{1 \pm \frac{r}{100}}$ | Never subtract $r\%$ from the Final value; always divide by $(1 \pm \frac{r}{100})$ |
| Removing 20% VAT (Gross to Net) | $\text{Net} = \frac{\text{Gross}}{1.20} \quad \Big\vert \quad \text{VAT Amount} = \frac{\text{Gross}}{6}$ | Dividing Gross by $6$ extracts the exact 20% VAT component in 1 step |
| Profit & Profit Margin (%) | $\text{Profit} = \text{Revenue} - \text{Cost} \quad \Big\vert \quad \text{Profit } \% = \frac{\text{Profit}}{\text{Cost}} \times 100\%$ | Verify whether the stem defines margin over Cost Price or Selling Revenue |
| Simple Interest | $I = P \times r \times t \quad \Big\vert \quad A = P(1 + rt)$ | Linear growth: interest accrues strictly on the initial principal $P$ |
| Compound Interest | $A = P\left(1 + \frac{r}{m}\right)^{m \times t}$ | No $x^y$ key on calculator: chain-multiply $P \times (1+r) \times (1+r) \dots$ |
| Part-to-Whole Ratio Partition | $\text{Share}_a = \left(\frac{a}{a + b + c}\right) \times \text{Total}$ | Find 1 unitary part first ($\frac{\text{Total}}{a+b+c}$), then scale by $a, b,$ or $c$ |
| Broker Commission Break-Even | $P^* = \frac{\text{Flat Fee } (F)}{\text{Commission Rate } (c)}$ | Below $P^*$, percentage commission is cheaper; above $P^*$, flat fee is cheaper |
6. Master Fraction-to-Decimal-to-Percentage Equivalence Table ($1/2$ to $1/20$)
Memorizing the exact decimal and percentage equivalents of every fraction from $\frac{1}{2}$ through $\frac{1}{20}$ eliminates 8 to 12 seconds of calculator entry per item. Drill these benchmarks until they become automatic:
| Fraction | Exact Decimal | Exact Percentage | High-Yield Multiples & Derived Anchors |
|---|---|---|---|
| $1/2$ | $0.50$ | $50.0\%$ | Halving benchmark |
| $1/3$ | $0.3333\dots$ | $33.33\%$ | $2/3 = 0.6667 = 66.67\%$ |
| $1/4$ | $0.25$ | $25.0\%$ | $3/4 = 0.75 = 75.0\%$ |
| $1/5$ | $0.20$ | $20.0\%$ | $2/5 = 40\%$, $3/5 = 60\%$, $4/5 = 80\%$ |
| $1/6$ | $0.1667$ | $16.67\%$ | $5/6 = 0.8333 = 83.33\%$ |
| $1/7$ | $0.1429$ | $14.29\%$ | Cyclic digits 142857: $2/7 = 28.57\%$, $3/7 = 42.86\%$, $4/7 = 57.14\%$ |
| $1/8$ | $0.125$ | $12.5\%$ | $3/8 = 37.5\%$, $5/8 = 62.5\%$, $7/8 = 87.5\%$ |
| $1/9$ | $0.1111\dots$ | $11.11\%$ | Multiply numerator by $11.11\%$: $2/9 = 22.22\%$, $4/9 = 44.44\%$, $7/9 = 77.78\%$ |
| $1/10$ | $0.10$ | $10.0\%$ | Shift decimal point 1 place left |
| $1/11$ | $0.0909\dots$ | $9.09\%$ | Multiply numerator by $9.09\%$: $2/11 = 18.18\%$, $3/11 = 27.27\%$, $5/11 = 45.45\%$ |
| $1/12$ | $0.0833$ | $8.33\%$ | $5/12 = 41.67\%$, $7/12 = 58.33\%$, $11/12 = 91.67\%$ |
| $1/15$ | $0.0667$ | $6.67\%$ | $2/15 = 13.33\%$, $4/15 = 26.67\%$ |
| $1/16$ | $0.0625$ | $6.25\%$ | Half of $1/8$ ($12.5\% \div 2 = 6.25\%$); $3/16 = 18.75\%$ |
| $1/20$ | $0.05$ | $5.0\%$ | Multiply numerator by $5\%$: $3/20 = 15\%$, $7/20 = 35\%$, $13/20 = 65\%$, $17/20 = 85\%$ |
Lock In Fractional & Geometric Reflexes With FSRS-6: Passive reading will not survive a 43.3-second exam clock. Drill these exact conversions and mensuration identities using our spaced-repetition flashcards: Percentages & Multipliers Pulse Deck (25 Cards) (Launch Instant Study), Geometry, Area & Volume Pulse Deck (22 Cards) (Launch Instant Study), or the complete Root UCAT Flashcard Suite.
7. Linear, Squared, and Cubed Unit Conversions
Unit slips destroy more marks in Quantitative Reasoning than algebra errors. While the UCAT typically provides imperial-to-metric conversion factors in a footnote (such as $1\text{ mile} = 1.61\text{ km}$ or $1\text{ UK gallon} = 4.546\text{ L}$), you are expected to know all standard metric, area, volume, and time conversions from memory:
| Measurement Dimension | Core Equivalences to Memorize | The Nonlinear / Exam Trap Warning | |||
|---|---|---|---|---|---|
| 1D Linear Length | $1\text{ km} = 1,000\text{ m}$ \ | $1\text{ m} = 100\text{ cm} = 1,000\text{ mm}$ \ | $1\text{ cm} = 10\text{ mm}$ | Approximate benchmark: $5\text{ miles} \approx 8\text{ km}$ ($1\text{ mile} \approx 1.609\text{ km}$) | |
| 2D Surface Area | $1\text{ m}^2 = 10,000\text{ cm}^2$ ($10^4\text{ cm}^2$) \ | $1\text{ cm}^2 = 100\text{ mm}^2$ \ | $1\text{ hectare} = 10,000\text{ m}^2$ | $1\text{ m}^2$ is NOT $100\text{ cm}^2$! You must square the linear factor ($100^2 = 10,000$) | |
| 3D Cubic Volume | $1\text{ m}^3 = 1,000,000\text{ cm}^3$ ($10^6\text{ cm}^3$) \ | $1\text{ cm}^3 = 1,000\text{ mm}^3$ | $1\text{ m}^3$ is NOT $1,000\text{ cm}^3$! You must cube the linear factor ($100^3 = 1,000,000$) | ||
| Liquid Fluid Capacity | $1\text{ mL} = 1\text{ cm}^3$ \ | $1\text{ L} = 1,000\text{ mL} = 1,000\text{ cm}^3$ \ | $1\text{ m}^3 = 1,000\text{ L}$ | A $1\text{ m} \times 1\text{ m} \times 1\text{ m}$ tank holds exactly $1,000\text{ Liters}$ of liquid | |
| Mass & Clinical Weight | $1\text{ tonne} = 1,000\text{ kg}$ \ | $1\text{ kg} = 1,000\text{ g}$ \ | $1\text{ g} = 1,000\text{ mg}$ \ | $1\text{ mg} = 1,000\text{ mcg } (\mu\text{g})$ | Shifting from $\text{g} \rightarrow \text{mg} \rightarrow \text{mcg}$ scales by $1,000$ at each step |
| Time & Calendar | $1\text{ hr} = 60\text{ min} = 3,600\text{ s}$ \ | $1\text{ day} = 24\text{ hr} = 1,440\text{ min}$ \ | $1\text{ yr} = 52\text{ wks} = 365\text{ days}$ | Always check whether a rate is per minute, per hour, per quarter ($4/\text{yr}$), or per year |
8. Clinical Drug Dosing, Dilutions, and IV Infusion Formulas
In recent UCAT cycles, applied clinical drug-dosing scenarios have surged in frequency because they directly mirror the numerical safety assessments medical students sit during MBBS training. Memorize these five clinical formulas and always harmonize units ($\text{mg}$ vs $\text{mcg}$, $\text{L}$ vs $\text{mL}$) before dividing:
- Weight-Based Patient Dose ($\text{mg/kg}$ Scaling):
$$\text{Single Dose Required} = \text{Prescribed Dose Rate } (\text{mg/kg}) \times \text{Patient Body Mass } (\text{kg})$$
$$\text{Total Daily Dose} = \text{Single Dose} \times \text{Number of Doses per Day}$$
(Note: If the regimen specifies $\text{mg/kg/day}$ divided into $N$ equal doses, divide the total daily dose by $N$ to find the individual administration dose!)- Liquid Suspension / Injection Volume to Administer ("What You Want Over What You Have"):
$$\text{Volume to Give } (\text{mL}) = \left(\frac{\text{Dose Prescribed } (\text{mg})}{\text{Stock Strength Available } (\text{mg})}\right) \times \text{Stock Suspension Volume } (\text{mL})$$
- Tablet / Capsule Count:
$$\text{Number of Tablets} = \frac{\text{Total Prescribed Dose } (\text{mg})}{\text{Active Drug Mass per Tablet } (\text{mg})}$$
- Intravenous (IV) Infusion Flow Rate ($\text{mL/hr}$):
$$\text{Infusion Rate } (\text{mL/hr}) = \frac{\text{Total Fluid Volume } (\text{mL})}{\text{Infusion Duration } (\text{hours})}$$
- Percentage Weight/Volume ($\text{w/v}\%$) Solution Concentration:
$$X\%\text{ w/v Solution} = X\text{ grams of solute per } 100\text{ mL of solution} = (10 \times X)\text{ mg/mL}$$
For example, a $1\%\text{ w/v}$ lidocaine solution contains $1\text{ g}$ ($1,000\text{ mg}$) per $100\text{ mL}$, which equals $10\text{ mg/mL}$. A $0.9\%\text{ w/v}$ normal saline bag contains $0.9\text{ g}$ ($900\text{ mg}$) of $\text{NaCl}$ per $100\text{ mL}$, or $9\text{ g}$ per $1,000\text{ mL}$ ($1\text{ Liter}$).
The NHS & AKU Medical Prescribing Safety Assessment (PSA): Every clinical dosing formula tested in UCAT Quantitative Reasoning is identical to the calculations you will perform daily on pediatric and intensive-care wards and in your final-year Prescribing Safety Assessment. Converting micrograms ($\text{mcg}$) to milligrams ($\div 1,000$) before applying $\frac{\text{Dose Prescribed}}{\text{Stock Strength}} \times \text{Stock Volume}$ prevents thousandfold dosing errors in neonatal care.
9. Central Tendency, Spread, and Probability Reference
Finally, consolidate the statistical and probabilistic formulas tested across QR data tables:
- Arithmetic Mean & The "Think in Totals" Identity:
$$\bar{x} = \frac{\sum x}{n} \implies \text{Total Sum } \left(\sum x\right) = \bar{x} \times n$$
- Weighted Mean Across Unequal Groups:
$$\bar{x}_{\text{weighted}} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2 + \dots + n_k \bar{x}_k}{n_1 + n_2 + \dots + n_k}$$
- Missing Value to Hit a Target Mean:
$$x_{\text{missing}} = (\bar{x}_{\text{target}} \times n_{\text{total}}) - \sum x_{\text{known}}$$
- Median Position in Ordered Data ($n$ items):
$$\text{Median Index} = \frac{n + 1}{2}$$
(If $n$ is odd, take the exact middle term; if $n$ is even, average the two central terms at $\frac{n}{2}$ and $\frac{n}{2} + 1$.)- Range & Interquartile Range (IQR):
$$\text{Range} = \text{Max} - \text{Min} \quad \Big| \quad \text{IQR} = Q_3\text{ (75th percentile)} - Q_1\text{ (25th percentile)}$$
- Histogram Frequency Density (Unequal Class Widths):
$$\text{Frequency} = \text{Frequency Density} \times \text{Class Width}$$
- Core Probability Laws:
$$P(A) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} \quad \Big| \quad P(\text{not } A) = 1 - P(A)$$
$$P(A \text{ and } B\text{ independent}) = P(A) \times P(B) \quad \Big| \quad P(A \text{ or } B\text{ mutually exclusive}) = P(A) + P(B)$$
10. Step-by-Step Worked Multi-Formula Problems
Worked Problem 1: 3D Cylinder Volume + Cubic-Meter-to-Liter Conversion
Scenario: A hospital reserve water tank is shaped as a right circular cylinder with an internal diameter of $1.6\text{ meters}$ and a height of $2.5\text{ meters}$. The tank is currently $60\%$ full of purified water. ($\text{Use } \pi = 3.14$). Question Prompt: To the nearest 10 liters, how many liters of purified water does the tank currently hold?- Option A: 301 L
- Option B: 2,410 L
- Option C: 3,010 L
- Option D: 5,020 L
- Option E: 12,060 L
- Radius Extraction (Avoid the Diameter Trap): The given measurement $1.6\text{ m}$ is the diameter. Halve it immediately: $r = \frac{1.6}{2} = 0.8\text{ m}$. (Using $r = 1.6\text{ m}$ quadruples the volume to Option E, $12,060\text{ L}$).
- Full Cylinder Volume in $\text{m}^3$:
$$V_{\text{full}} = \pi r^2 h = 3.14 \times (0.8)^2 \times 2.5 = 3.14 \times 0.64 \times 2.5$$
Notice mentally that $0.64 \times 2.5 = 0.16 \times 10 = 1.6$. Thus $V_{\text{full}} = 3.14 \times 1.6 = 5.024\text{ m}^3$.
- Apply the $60\%$ Fill Factor:
$$V_{\text{current}} = 5.024 \times 0.60 = 3.0144\text{ m}^3$$
(Option D, $5,020\text{ L}$, traps candidates who forget the $60\%$ multiplier).
- Convert Cubic Meters to Liters ($1\text{ m}^3 = 1,000\text{ L}$):
$$\text{Liters} = 3.0144 \times 1,000 = 3,014.4\text{ L} \approx 3,010\text{ L (Option C)}$$
(Option A, $301\text{ L}$, traps candidates who multiply by $100$ instead of $1,000$).
Worked Problem 2: Pediatric $\text{mg/kg}$ Dosing + Suspension Volume Formula
Scenario: A pediatric patient weighing $24\text{ kg}$ is prescribed an oral antibiotic at a total daily regimen of $15\text{ mg/kg/day}$, to be administered in 3 equally divided doses every 8 hours. The hospital pharmacy dispenses the antibiotic as a liquid suspension labeled $200\text{ mg per } 5\text{ mL}$. Question Prompt: How many milliliters ($\text{mL}$) of the suspension must be administered to the child for each individual dose?- Option A: $1.5\text{ mL}$
- Option B: $3.0\text{ mL}$
- Option C: $4.5\text{ mL}$
- Option D: $6.0\text{ mL}$
- Option E: $9.0\text{ mL}$
- Total Daily Dose ($\text{mg/day}$):
$$\text{Daily Dose} = 15\text{ mg/kg/day} \times 24\text{ kg} = 360\text{ mg/day}$$
(Mental shortcut: $24 \times 15 = 240 + 120 = 360\text{ mg}$).- Individual Single Dose ($\text{mg/dose}$): Because the daily regimen is split into 3 equal doses:
$$\text{Single Dose Prescribed} = \frac{360\text{ mg}}{3} = 120\text{ mg}$$
(Or divide $15\text{ mg/kg/day}$ by $3$ first to get $5\text{ mg/kg/dose} \times 24\text{ kg} = 120\text{ mg}$!)- Suspension Volume to Administer:
$$\text{Volume} = \left(\frac{\text{Dose Prescribed}}{\text{Stock Strength}}\right) \times \text{Stock Volume} = \left(\frac{120\text{ mg}}{200\text{ mg}}\right) \times 5\text{ mL} = 0.6 \times 5\text{ mL} = 3.0\text{ mL (Option B)}$$
(Distractor Option E, $9.0\text{ mL}$, traps candidates who calculate the full daily volume rather than a single dose).Put every formula on this page into timed action inside the UCAT Exam Hall, test your baseline with the 40-Question Diagnostic Mock, and drill all 386 Percentages & Ratios Questions and 73 Geometry & Volume Questions.
Frequently Asked Questions
Q: Does the UCAT give you a formula sheet on screen during Quantitative Reasoning?
No. The official UCAT computer interface provides zero mathematical formulas or unit conversions unless an unfamiliar imperial or currency rate is explicitly stated inside a question footnote. You must memorize all geometry, percentage, speed, average, and interest formulas before your exam.
Q: How do I calculate circle area and circumference if the UCAT calculator has no pi ($\pi$) key?
Because the Pearson VUE TI-108 on-screen calculator has no $\pi$ button, always type 3.14 (or 3.1416 if answer choices are clustered within 0.5%) into the calculator. If the radius or diameter is a clean multiple of $7$ (such as $r = 7\text{ cm}$ or $14\text{ cm}$), use the fractional approximation $\pi \approx \frac{22}{7}$ mentally so the $7$ cancels out immediately.
Q: How do I calculate powers or exponents like $(1 + r)^n$ on the UCAT calculator?
The UCAT on-screen calculator has no exponent ($x^y$) button. To raise a number to a power, multiply it by itself sequentially: for $r^2$, enter r r; for $r^3$, enter r r r; for 3 years of compound interest at $5\%$, enter Principal 1.05 1.05 1.05 Enter.
Q: What is the difference between the surface area of a hollow dome and a solid hemisphere?
A hollow hemispherical dome has only a curved outer surface equal to half of a sphere's surface area ($2\pi r^2$). However, a solid closed hemisphere also has a flat circular base of area $\pi r^2$, making its total surface area $2\pi r^2 + \pi r^2 = 3\pi r^2$. Forgetting the flat circular base is the number-one 3D geometry trap in UCAT QR.
Q: Why does $1\text{ m}^2$ equal $10,000\text{ cm}^2$ instead of $100\text{ cm}^2$?
Because a square meter measures $100\text{ cm}$ in length by $100\text{ cm}$ in width, its area in square centimeters is $100\text{ cm} \times 100\text{ cm} = 10,000\text{ cm}^2$. Similarly, a cubic meter measures $100\text{ cm} \times 100\text{ cm} \times 100\text{ cm} = 1,000,000\text{ cm}^3$ (which equals $1,000\text{ Liters}$).
Q: How do I convert between meters per second ($\text{m/s}$) and kilometers per hour ($\text{km/h}$) quickly?
Use the exact conversion multiplier $3.6$. Because $1\text{ hour} = 3,600\text{ seconds}$ and $1\text{ km} = 1,000\text{ meters}$, $1\text{ m/s} = \frac{3,600}{1,000} = 3.6\text{ km/h}$. Multiply $\text{m/s}$ by $3.6$ to get $\text{km/h}$, and divide $\text{km/h}$ by $3.6$ to get $\text{m/s}$.
Q: When should I use the harmonic mean formula for average speed in the UCAT?
Whenever a vehicle travels a fixed distance $d$ at speed $v_1$ and returns (or completes an equal-distance second leg $d$) at speed $v_2$, the average speed is the harmonic mean $v_{\text{avg}} = \frac{2v_1 v_2}{v_1 + v_2}$. Never use the simple arithmetic mean $\frac{v_1 + v_2}{2}$ unless the vehicle travels for equal durations of time ($t_1 = t_2$).
Q: What is the fastest formula for removing 20% VAT from a gross price?
To find the pre-tax Net price from a Gross price that includes $20\%$ VAT, divide by $1.20$ ($\text{Net} = \frac{\text{Gross}}{1.20}$). If the question asks strictly for the monetary VAT amount included inside the Gross price, divide the Gross price by $6$ ($\text{VAT} = \frac{\text{Gross}}{6}$). Never multiply the Gross price by $0.80$.
Q: What is the formula for liquid drug dosing in UCAT Quantitative Reasoning?
Use the universal clinical suspension formula: $\text{Volume to Give} = \left(\frac{\text{Dose Prescribed}}{\text{Stock Strength}}\right) \times \text{Stock Volume}$. Always ensure that $\text{Dose Prescribed}$ and $\text{Stock Strength}$ share identical mass units (convert $\text{grams} \rightarrow \text{mg}$ or $\text{mg} \rightarrow \text{mcg}$ by multiplying by $1,000$) before dividing.
Q: Do I need to memorize trigonometry, quadratic formulas, or standard deviation for UCAT QR?
No. Sine, cosine, tangent, the quadratic formula, calculus, logarithms, and standard deviation are outside the UCAT Quantitative Reasoning specification. If a rare question utilizes a specialized scientific or economic equation, that equation is always printed explicitly inside the stimulus text for you to substitute numbers into.
Execute Under Real Timer Pressure: Master Quantitative Reasoning
Passive reading creates the dangerous illusion of familiarity. Breaking into the 9th decile (2,270+ on the 900 to 2,700 cognitive scale) requires FSRS-6 spaced retrieval of rules and timed execution inside a true-to-life Pearson VUE simulation.