University Clinical Aptitude Test (UCAT) Quantitative Reasoning Syllabus

List the common metric prefixes and their factors for kilo, centi, milli, and micro.

kilo $= 10^{3}$, centi $= 10^{-2}$, milli $= 10^{-3}$, micro $= 10^{-6}$. So 1 km $= 1000$ m, 1 cm $= 0.01$ m, 1 mm $= 0.001$ m.

How many millilitres are in a litre, and how many grams in a kilogram?

$1$ litre $= 1000$ mL and $1$ kg $= 1000$ g.

How do you convert an area from $\text{m}^2$ to $\text{cm}^2$?

Since $1$ m $= 100$ cm, area scales by the square: $1\text{ m}^2 = 100^2 = 10{,}000\text{ cm}^2$. Multiply the area in $\text{m}^2$ by $10{,}000$.

In a rates-of-consumption problem, how do you find how long a supply lasts (e.g. a 600 L tank used at 25 L per hour)?

Divide the total amount by the rate of use: $\frac{600}{25} = 24$ hours.

If a machine produces 18 units per minute, how many units does it produce in 2.5 hours?

Convert time to minutes and multiply: $2.5 \times 60 = 150$ minutes, then $18 \times 150 = 2700$ units.

In a fuel-consumption problem, how do you find distance travelled given fuel used and efficiency (e.g. 40 L at 12 km per litre)?

Multiply fuel used by efficiency: $40 \times 12 = 480$ km.

When reading a value from a UCAT data table, what features must you check to avoid errors?

Check the row and column headings, the units stated, any multiplier in the title (e.g. 'figures in thousands'), and footnotes, so you interpret the figure correctly.

What does a pie chart represent, and how do you convert a sector angle into a quantity?

A pie chart shows parts of a whole, where the full circle is $360^{\circ}$ (or 100%). A sector of angle $\theta$ represents $\frac{\theta}{360} \times \text{total}$.

On a pie chart, what fraction and percentage of the total does a $90^{\circ}$ sector represent?

$\frac{90}{360} = \frac{1}{4} = 25\%$ of the total.

How do you calculate the (arithmetic) mean of a set of values?

$$\text{mean} = \frac{\text{sum of all values}}{\text{number of values}}$$

How do you find the median of a data set, including when there is an even number of values?

Arrange the values in ascending order; the median is the middle value. With an even number of values, it is the mean of the two central values.

Define the mode and the range of a data set.

The mode is the value that occurs most frequently. The range is the difference between the largest and smallest values: $\text{range} = \text{max} - \text{min}$.

If a set of 5 numbers has a mean of 12, what is their total, and how do you find a missing value?

Total $= \text{mean} \times n = 12 \times 5 = 60$. To find a missing value, subtract the sum of the known values from this total.

How do you calculate a weighted mean, e.g. averaging marks where some count more than others?

$$\text{weighted mean} = \frac{\sum (w_i x_i)}{\sum w_i}$$ Multiply each value by its weight, sum these products, and divide by the total of the weights.

How do you read a trend from a line graph and project a future value?

Identify the direction and rate of change (gradient) of the line; if the trend is roughly linear, extend it at the same rate beyond the plotted data to estimate future values, assuming the pattern continues.

What is the gradient of a straight-line graph and how is it calculated between two points?

The gradient is the rate of change of $y$ with respect to $x$: $$\text{gradient} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{change in } y}{\text{change in } x}$$

Give the formulae for the area and perimeter of a rectangle with length $l$ and width $w$.

Area $= l \times w$ and perimeter $= 2(l + w)$.

State the formula for the area of a triangle and the area of a circle.

Triangle: $\text{area} = \frac{1}{2} \times \text{base} \times \text{height}$. Circle: $\text{area} = \pi r^{2}$ where $r$ is the radius.

Give the formulae for the circumference of a circle and the area of a trapezium.

Circumference $= 2\pi r$ (or $\pi d$). Trapezium area $= \frac{1}{2}(a + b)h$, where $a$ and $b$ are the parallel sides and $h$ the perpendicular height.

What is the formula for the volume of a cuboid and the volume of a cylinder?

Cuboid: $V = l \times w \times h$. Cylinder: $V = \pi r^{2} h$, the base area times the height.

How do you find the volume of any prism?

$$V = \text{cross-sectional area} \times \text{length (height)}$$ Multiply the area of the uniform cross-section by the length of the prism.

How do you solve a simple linear equation such as $3x + 4 = 19$?

Isolate $x$ by inverse operations: subtract 4 from both sides to get $3x = 15$, then divide by 3 to get $x = 5$.

How do you rearrange the formula $A = \pi r^{2}$ to make $r$ the subject?

Divide both sides by $\pi$ to get $r^{2} = \frac{A}{\pi}$, then take the square root: $r = \sqrt{\frac{A}{\pi}}$.

How do you substitute values into a formula, e.g. find $y$ when $y = 2x^{2} - 3x$ and $x = 4$?

Replace $x$ with 4 and follow order of operations: $y = 2(4)^{2} - 3(4) = 2(16) - 12 = 32 - 12 = 20$.

When solving a UCAT applied word problem, what general process should you follow?

Read carefully to identify what is asked, extract the relevant numbers and units, choose the correct operation or formula, perform the calculation, then sense-check the answer against the options and the question's units.

In a word problem involving tax or commission, how do you find a final amount after a percentage is added (e.g. £250 plus 20% VAT)?

Multiply by the increase multiplier: $250 \times 1.20 = £300$. Equivalently, add $20\%$ of 250 (which is £50) to £250.

How do you calculate simple interest on a principal $P$ at rate $r\%$ per year for $t$ years?

$$\text{simple interest} = \frac{P \times r \times t}{100}$$ The interest is the same each year, based only on the original principal.

What is the formula for compound interest on principal $P$ at annual rate $r\%$ over $n$ years?

$$\text{final amount} = P\left(1 + \frac{r}{100}\right)^{n}$$ Interest is added to the balance each period, so it grows on the accumulated total.

How do you convert between a 12-hour and 24-hour clock when calculating elapsed time, e.g. from 09:45 to 14:20?

Use 24-hour values and subtract: from 09:45 to 14:45 is 5 hours, then back 25 minutes gives 4 hours 35 minutes. Always work in hours and minutes separately, borrowing 60 minutes from an hour when needed.

Why is it important to match units before calculating in a UCAT word problem (e.g. mixing minutes and hours)?

Formulas require consistent units; combining mismatched units (such as a speed in km/h with a time in minutes) gives wrong answers. Convert all quantities to compatible units first, e.g. minutes to hours before using $\text{distance} = \text{speed} \times \text{time}$.

What is a reliable order-of-magnitude check for a division like $\frac{4860}{18}$ in QR?

Round to friendly numbers: $\frac{4800}{20} = 240$, so the answer should be near 240 (actual is 270). This flags any answer that is off by a factor of 10 due to a decimal-point slip.