📚 TMUA Without a Calculator: Targeted Training for Manual Calculation Efficiency | TMUA 不允许使用计算器：手算效率的针对性训练
The Test of Mathematics for University Admission (TMUA) is sat entirely without a calculator. Two papers, 75 minutes each, 20 multiple-choice questions per paper, no electronic aids of any kind. For many strong A-level students this is the single most disorienting feature of the exam: the mathematics is familiar, but the arithmetic suddenly is not. Every product, root, fraction and logarithm has to be produced by hand, in real time, under pressure.
TMUA（大学入学数学考试）全程禁止使用计算器。两张试卷，每张 75 分钟、20 道选择题，任何电子辅助工具都不允许使用。对许多 A-level 成绩优异的学生来说，这是整场考试中最令人不适应的一点：数学知识本身是熟悉的，但算术突然变得陌生。每一个乘积、根式、分数和对数，都必须在压力下实时手算得出。
The good news is that the calculator ban is not a random obstacle. It is a deliberate design choice, and it rewards a specific, trainable skill set. Students who treat manual arithmetic as a separate discipline — drilled, timed and audited like a sport — routinely gain four to eight raw marks per paper over equally knowledgeable students who do not. This article sets out exactly what the exam demands numerically and how to train for it.
好消息是，禁计算器并非随意的刁难，而是刻意的命题设计，它奖励的是一套明确且可训练的能力。把”手算”当作一门独立学科来对待——像训练运动项目那样计时、复盘、纠错——的学生，往往能在每张卷子上比知识水平相当但缺乏训练的同学多得 4 到 8 分原始分。本文将精确拆解这场考试在数值层面的要求，并给出对应的训练方法。
1. Why the Calculator Ban Changes the Game | 禁计算器如何改变游戏规则
In a calculator-permitted exam, arithmetic is a commodity: it costs almost nothing and therefore never decides the outcome. In TMUA it is a scarce resource. Each calculation consumes working memory, time and the probability of an undetected slip. A question that is conceptually trivial can still be lost because three chained multiplications each carried a 3% error chance.
在允许使用计算器的考试中，算术是”廉价商品”：几乎不耗费成本，因此从不决定成败。但在 TMUA 中，它是稀缺资源。每一次计算都会消耗工作记忆、时间，并增加未被察觉的失误概率。一道概念上很简单的题，也可能因为三步连乘各带 3% 的出错概率而丢分。
The exam’s average budget is 75 ÷ 20 = 3.75 minutes per question. That sounds generous until you realise the questions are deliberately written so that the intended route takes 60 to 120 seconds, leaving 100-plus seconds of slack that most candidates burn on arithmetic they could have avoided.
考试的平均时间预算是每道题 75 ÷ 20 = 3.75 分钟。听起来很宽裕，直到你意识到：题目的”设计解法”通常只需 60 到 120 秒，剩下的一百多秒余量，大多数考生恰恰把它烧在了本可以避免的硬算上。
The strategic consequence is simple and worth internalising before any content revision: in TMUA, avoiding a calculation is worth as much as performing one quickly.
由此得出的策略结论很简单，也值得在复习任何知识点之前就先刻进脑子里：在 TMUA 里，避开一次计算，与快速完成一次计算同等有价值。
2. What TMUA Actually Asks You to Compute | TMUA 真正要求你算的是什么
Contrary to the anxiety the ban produces, TMUA is not a test of long multiplication. The numbers are chosen to be structurally clean. You are far more likely to face 3/8 of 96 than 4 738 × 296.
与”禁计算器”引发的焦虑相反，TMUA 并不考查多位数竖式乘法。命题人刻意挑选结构干净的数字。你更可能遇到”96 的 3/8 是多少”，而不是”4738 × 296″。
It is worth being explicit about the categories that actually appear, because this determines what you drill.
明确列出实际出现的数值类型很有必要，因为这直接决定你该练什么。
| Numerical category | 数值类型 | Typical demand | 典型要求 | Calculator needed? | 需要计算器？ |
| Two-digit products | 两位数乘法 | 47 × 53, 38 × 42 | No — difference of squares | 不需要——平方差 |
| Fraction arithmetic | 分数运算 | 1/4 + 1/6, 3/5 of 2/7 | No — common denominators | 不需要——通分 |
| Surds | 根式 | √18 = 3√2, rationalising 1/(√5 − 1) | No — exact form preferred | 不需要——首选用精确形式 |
| Indices and logs | 指数与对数 | 2¹⁰ = 1024, log₁₀ 40 ≈ 1.602 | No — benchmark values | 不需要——基准值 |
| Trigonometry | 三角 | sin 150°, tan(π/3), sector area | No — exact values only | 不需要——只需精确值 |
| Estimation | 估算 | Is 3^7 bigger than 2000? Bound an integral | No — bounding is intended | 不需要——放缩就是考点 |
Notice what is absent: trigonometric function values at arbitrary angles, decimal logarithms of arbitrary arguments, roots of non-square integers expressed to five decimal places. Every one of those requires a calculator, so the exam cannot ask for them. Whenever a question appears to demand such a value, you have almost certainly chosen the wrong route.
请注意表中”缺席”的东西：任意角的三角函数值、任意真数的十进制对数、非完全平方数开方精确到五位小数。这些都必须依赖计算器，所以考试不可能要求。任何时候若一道题看起来需要这类数值，几乎可以肯定你选错了路径。
3. The Core Mental Arithmetic Toolkit | 核心心算工具箱
Every TMUA candidate should be able to produce the following instantly, without conscious effort. “Instantly” means under two seconds, with no counting on fingers and no intermediate written step.
每位 TMUA 考生都应能”瞬间”说出以下结果，无需刻意思考。”瞬间”意味着两秒以内，不掰手指，也不需要写中间步骤。
Squares from 1 to 25, and cubes from 1 to 6. These recur constantly in algebra, in sequences and in inequality questions.
1 到 25 的平方，以及 1 到 6 的立方。它们在代数、数列和不等式题中反复出现。
Powers of two up to 2¹⁰ = 1024, and the approximate equality 2¹⁰ ≈ 10³ that lets you reason about exponentials in your head.
2 的幂直到 2¹⁰ = 1024，以及近似关系 2¹⁰ ≈ 10³，它让你可以在脑中推理指数增长。
The difference of two squares is the single most valuable trick in the entire exam, because it turns a hard multiplication into an easy subtraction.
平方差公式是整场考试中最有价值的技巧，因为它把一次困难的乘法变成一次简单的减法。
(a + b)(a − b) = a² − b²
So 47 × 53 = 50² − 3² = 2500 − 9 = 2491, and 38 × 42 = 40² − 2² = 1600 − 4 = 1596. Both take about three seconds once the pattern is automatic.
于是 47 × 53 = 50² − 3² = 2500 − 9 = 2491，38 × 42 = 40² − 2² = 1600 − 4 = 1596。一旦模式自动化，两者都只需约三秒。
Other shortcuts worth automating:
其他值得自动化的捷径：
- × 11: add adjacent digits (47 × 11 = 517, since 4+7 = 11 carries). | × 11：相邻数字相加（47 × 11 = 517，因 4+7 = 11 需进位）。
- × 25: multiply by 100 then divide by 4. | × 25：乘以 100 再除以 4。
- × 5: halve then multiply by 10. | × 5：先取一半再乘 10。
- ÷ 4 and ÷ 8: halve once or twice. | ÷ 4 与 ÷ 8：连续对折一次或两次。
- Divisibility: by 3 and 9 use digit sums; by 11 alternate the digit sums. | 整除性：3 和 9 看数字和；11 看交错数字和。
- Digital-root check: reduce any integer to a single digit by summing digits, and verify sums and products modulo 9. | 数字根检验：反复求数字和直到一位数，用模 9 校验和与积。
The digital-root check deserves a special mention because it catches the most common silent error in manual arithmetic — an off-by-one in a carry — without redoing the whole calculation.
数字根检验值得单独强调，因为它能捕捉手算中最常见的”隐形错误”——进位失误——而不必重做整道计算。
4. Fractions, Decimals, Percentages and Ratios | 分数、小数、百分比与比例
TMUA heavily favours rational numbers because they stay exact. Fluency here pays for itself many times over. The following equivalences should be reflexive, not derived.
TMUA 偏爱有理数，因为它们能保持精确。这方面的熟练度会带来成倍的回报。以下等价关系应当是条件反射式的，而不是每次推导。
| Fraction | 分数 | Decimal | 小数 | Percentage | 百分比 |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 5/8 | 0.625 | 62.5% |
| 1/6 | 0.1666… | 16.67% |
| 1/7 | 0.142857… | 14.29% |
| 1/16 | 0.0625 | 6.25% |
| 1/11 | 0.0909… | 9.09% |
When adding fractions, look for the smaller denominator that divides the larger before reaching for a full product. For 1/12 + 1/18, the LCM is 36, not 216 — and choosing 36 saves both time and cancellation.
做分数加法时，先看较小的分母是否整除较大的，再去乘满。例如 1/12 + 1/18，最小公倍数是 36 而不是 216——选 36 既省时间又免去约分。
Comparing fractions is a recurring sub-task. Two reliable techniques: cross-multiply when the numbers are small, and compare to a benchmark such as 1/2 when they are not. For 7/13 versus 8/15, cross-multiplying gives 105 versus 104, so 7/13 is slightly larger.
分数比较是反复出现的子任务。两个可靠技巧：数字小时交叉相乘；数字大时与 1/2 之类的基准比较。比较 7/13 与 8/15，交叉相乘得 105 与 104，故 7/13 略大。
Percentage work should be decomposed rather than computed. 17.5% = 10% + 5% + 2.5%, and 2.5% is a quarter of 10%. Successive percentage changes multiply, they never add: a 20% rise followed by a 10% fall gives 1.2 × 0.9 = 1.08, an 8% net rise, not 10%.
百分比运算应当拆分而非硬算。17.5% = 10% + 5% + 2.5%，而 2.5% 是 10% 的四分之一。连续百分比变化是相乘而不是相加：先涨 20% 再跌 10%，得到 1.2 × 0.9 = 1.08，净涨 8%，而不是 10%。
5. Surds, Indices and Logarithm Estimates | 根式、指数与对数估算
Because calculators are banned, TMUA expects surds to stay in exact form. Simplify, combine and rationalise — but never convert to a decimal unless the answer options are decimal.
由于禁用计算器，TMUA 期望根式保持精确形式。化简、合并、有理化——但除非选项是小数，否则绝不要转成小数。
Core surd facts to have ready: √18 = 3√2, √48 = 4√3, √50 = 5√2, and the rationalisation pattern for a two-term denominator.
需备好的核心根式事实：√18 = 3√2，√48 = 4√3，√50 = 5√2，以及二项分母的有理化模式。
1/(√a + √b) = (√a − √b)/(a − b)
For order-of-magnitude comparisons you do need approximations, and these six are worth memorising: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √6 ≈ 2.449, √7 ≈ 2.646, √10 ≈ 3.162.
做数量级比较时确实需要近似值，以下六个值得背下：√2 ≈ 1.414，√3 ≈ 1.732，√5 ≈ 2.236，√6 ≈ 2.449，√7 ≈ 2.646，√10 ≈ 3.162。
Index laws must be automatic in both directions: a^(m+n) = a^m × a^n, a^(m−
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