Admissions Test Preparation · Practice Set 7
ESAT Mathematics Mock Module 7 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 | Frequency density | Frequency is the area of the bar: density times class width |
| Q2 | Interpreting correlation | State direction and strength, and stop before causation |
| Q3 | Stratified sampling | Every group keeps its share of the population |
| Q4 | Choosing the right average | Match the average to the shape of the data |
| Q5 | Grouped data with gaps between the intervals | The midpoint comes from the interval as written, not from a shared boundary |
| Q6 | Ratio of mean to median | Compute both, then form the ratio — no shortcut, just order of operations |
| Q7 | When transforming and averaging commute | Averaging then transforming equals transforming then averaging — only for linear maps |
| Q8 | Choosing between the two variance formulas | Match the formula to the information you were handed |
| Q9 | Arrangements with repeated letters | Count as if distinct, then divide by the factorial of each repeat |
| Q10 | Arrangements with fixed end positions | Fill the constrained positions first, then arrange what is left |
| Q11 | Arrangements with two letters together | Glue the pair into one object, then double for their internal order |
| Q12 | Selections with a minimum requirement | Total selections minus the ones that break the requirement |
| Q13 | Counting pairs | Every handshake is a pair, so the count is $\binom{n}{2}$ |
| Q14 | Probability from combinations | Favourable selections over total selections |
| Q15 | Possibility space for two dice | Use the complement when the favourable list is long |
| Q16 | Exhaustive probabilities | Everything that can happen sums to 1 |
| Q17 | Conditional probability from a two-way table | Conditioning shrinks the denominator to the given group |
| Q18 | Adding across branches | Multiply along a branch, add across branches |
| Q19 | Relative frequency as an estimate | Relative frequency estimates probability; more trials, better estimate |
| Q20 | Expected number of outcomes | Expected frequency is probability times the number of trials |
| Q21 | The product rule for counting | Independent choices multiply |
| Q22 | Counting with repetition allowed | Repetition allowed means every position keeps its full range |
| Q23 | Arrangements with a group kept together | Glue the group, arrange the objects, then arrange within the group |
| Q24 | Selections with an exact requirement | Choose from each group separately, then multiply |
| Q25 | Expected frequencies in a tree | Work in people, not probabilities, and the tree fills itself |
| Q26 | Reading a conditional probability from a tree | Condition on the branch you are told about, not the whole group |
| Q27 | Expected frequencies with unequal branch rates | Two branches lead to the same outcome, so both must be counted |