Free TMUA guide

Calculator-free arithmetic fluency for TMUA

The current TMUA specification does not permit a calculator and does not provide a formulae booklet. Arithmetic fluency therefore means choosing a reliable exact route, not trying to calculate faster by force. Use this guide with the topic map and timed practice.

About 9 minutes to read

Stay exact before you estimate

Situation	Reliable move	Example
A fraction of a quantity	Cancel before multiplying	$3/8$ of $64$ : $64/8*3=24$
A percentage change	Use a multiplier	$15$ percent increase: multiply by $1.15$
A product near a round number	Split around the round number	$196=(20-1)6=114$
A power	Use repeated structure	$2^5=22222=32$
A ratio	Use one scale factor	$2:5$ with total $21$ gives factor $3$

Worked example Cancel before you multiply

To find $5/12$ of $84$ , divide by $12$ first, then multiply by $5$ .

(5/12)*84 = 5*(84/12) = 5*7 = 35

The intermediate values stay small and exact. If the denominator does not divide immediately, factor both numbers before multiplying.

Why this works: The cancellation exposes the simplest exact calculation and reduces the chance of a place-value error.

Why the tempting route fails: Converting to a decimal first can introduce rounding before the answer has been compared with the options.

Make signs and bounds visible

Write a positive, negative or zero sign beside each factor before multiplying.
For an inequality, mark whether each operation preserves or reverses the direction.
Use a rough bound before exact work: a product of two numbers between $0$ and $1$ is between $0$ and $1$ .
Check the final answer against the bound and the units in the stem.

Worked example A sign check catches a lost negative

If $-3<x<-1$ , then multiplying by $-2$ reverses both inequality signs.

6 > -2x > 2

The order is descending because $-2x$ is positive and larger when $x$ is more negative. Write the reversal before simplifying the expression.

Why this works: The sign of the multiplier controls the direction, and the resulting interval provides a second check.

Why the tempting route fails: Multiplying the endpoints but leaving the inequality signs unchanged produces an interval in the wrong order.

Train short patterns, not random speed

Five-minute arithmetic drills
Day	Pattern	Examples to create
1	Fraction cancellation	$7/12$ of $96$ , $5/18$ of $72$
2	Squares and powers	$15^2$ , $2^7$ , numbers near a round square
3	Ratios and percentages	A total split in a ratio, then a percentage change
4	Signs and bounds	Products of signed fractions and interval estimates
5	Mixed retrieval	One example from each pattern without notes

Make the examples yourself, then check them with a written calculation. The purpose is retrieval of a route you can reproduce in a paper, not a private race against a timer.

Use fluency without abandoning structure

Arithmetic fluency should support the mathematical route. If the calculation is growing, pause and ask whether a factor, symmetry, common base, ratio or bound would make it smaller. The best calculator-free move is often a representation choice made before the arithmetic.

Keep fractions as fractions when the options are exact.
Use an estimate to reject an impossible option, then calculate exactly if two options remain.
Write one line for each place-value or sign-sensitive step.
Do not invent a memorised formula to replace a relationship you can derive safely from the given information.

Sources

TMUA Content Specification. The current calculator and formulae-booklet boundaries, plus the mathematics expected across the two papers. Read the specificationhttps://uat-wp.s3.eu-west-2.amazonaws.com/wp-content/uploads/2024/05/03165619/TMUA_Content_Specification.pdf

Notes on Mathematics for TMUA and ESAT M2. UAT-UK notes for the mathematical methods and fluency that support TMUA preparation. Open the mathematics noteshttps://uat-wp.s3.eu-west-2.amazonaws.com/wp-content/uploads/2026/06/30103537/Notes_on_Mathematics_-for_TMUA_and_ESAT_M2.pdf

TMUA preparation materials. Official preparation papers for testing arithmetic choices in context. Open the materialshttps://esat-tmua.ac.uk/tmua-preparation-materials/

Tool rules can change. Check the current official test information before sitting the test rather than relying on a study guide.

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