Admissions Test Preparation · Practice Set 3
ESAT Mathematics Mock Module 3 of 7
27 a full 27-question module, the same length and shape as one sitting of the real test. Mathematics 1 is compulsory for every ESAT candidate. Number, ratio, algebra, geometry, statistics and probability, drawn from across the whole specification. Every question states the specification sub-topic it was written against.
How to use this set
Every question carries a target time, and no question needs a calculator. Attempt each one against the clock with the solution collapsed. If you are still setting up algebra when the target time passes, stop and open the solution: the shortcut is what the question is testing, and the arithmetic is not.
Read the Key Idea before the full breakdown. If the key idea alone lets you redo the question in under a minute, the method has landed and you can move on. If it does not, work the step-by-step breakdown and then return to the question a day later.
The Common Mistake section on each question is the slower route that still reaches the right answer. Under exam conditions that route is usually the difference between finishing the module and running out of time.
Modules in this set
Select a module below to view the questions and their step-by-step worked solutions:
Mathematics
27 questions, 40 minutes in the real test. Mathematics 1 is compulsory for every ESAT candidate. Number, ratio, algebra, geometry, statistics and probability, drawn from across the whole specification. 27 worked solutions.
Shortcut index
Use this as a revision checklist. If you can state the shortcut before opening the question, you already have the mark.
Mathematics
| # | Topic | Shortcut rehearsed |
|---|---|---|
| Q1 | Roots of decimals | Rewrite the decimal as a fraction of powers of ten |
| Q2 | Compound fractions | Simplify the denominator completely before inverting |
| Q3 | Percentage profit | Profit is always measured against the cost price |
| Q4 | Interior angle sums | $(n-2)\times 180$ comes from splitting the polygon into triangles |
| Q5 | Bounds | Upper bound is half a unit above the stated value |
| Q6 | HCF and LCM | $\text{HCF}\times\text{LCM} = $ the product of the two numbers |
| Q7 | Recurring decimals | Digits over the same number of nines |
| Q8 | Direct proportion to a power | Scale by the factor raised to the power |
| Q9 | Selection without replacement | Complement again — 'at least one' means 'not none' |
| Q10 | Rectangles from perimeter | Use the semi-perimeter to halve the algebra |
| Q11 | Average speed out and back | Equal distances mean the harmonic mean, never the arithmetic mean |
| Q12 | Average speed on a climb and descent | Harmonic mean again, with a wide speed ratio |
| Q13 | Two-set counting for the overlap | Inclusion-exclusion rearranged to find the intersection |
| Q14 | Symmetric identities | Rewrite $x^{2}+y^{2}$ in terms of the sum and the product |
| Q15 | Quadratic sequences | Second differences give twice the leading coefficient |
| Q16 | Reading a pie chart | A sector angle is a fraction of 360°, applied to the total |
| Q17 | Comparing two distributions | Compare like with like: median for location, IQR for consistency |
| Q18 | Standard form products | Multiply the mantissas, add the indices, then renormalise |
| Q19 | Sum of roots | $\alpha+\beta = -b/a$ straight from the coefficients |
| Q20 | Reverse fractions | Divide by the fraction, or scale one part up |
| Q21 | Bearings | Bearings are clockwise from north, always three figures |
| Q22 | Circles from area to circumference | Recover $r$ from the area, then use it |
| Q23 | Linear sequences | Common difference gives the coefficient; the zeroth term gives the constant |
| Q24 | Powers of products | The outer index applies to every factor inside the bracket |
| Q25 | Speed unit conversion | m/s to km/h: multiply by $3.6$ |
| Q26 | Two-set problems | $n(A\cup B) = n(A)+n(B)-n(A\cap B)$ |
| Q27 | Cubes | Recover the side length, then use it everywhere |