AQA GCSE数学 Ratio 化简 三量比 分配 反比
比例这一块在 AQA GCSE 数学 8300 里的分值不低，基础卷占 25%，高阶卷占 20%，而且三份卷子都可能考。它不像解方程那样有固定套路，题目常把比、分数、百分比与单位换算混在一起问。下面按考纲 3.3 的条文顺序拆开讲，从化简与按比分配到反比例与复合单位，每节给出一套能直接照着算的步骤。这里引用的口径都对着 AQA 官网的 8300 specification，练习时以官方最新版本文本为准。
Ratio carries a quarter of the Foundation marks and a fifth of the Higher marks in AQA GCSE Mathematics 8300, and any of the three papers may test it. Questions rarely stay inside one idea, mixing ratios with fractions, percentages and unit conversions. What follows works through the specification's ratio section in order and gives a routine that can be used straight away.
Ratio 在 8300 中的分值与两卷差别
在 AQA 官网的 GCSE 数学 8300 specification 里，Ratio, proportion and rates of change 是一个独立的内容领域，条文代号全部以 R 开头。Ofqual 规定的权重按层级分开：基础卷这个领域占 25%，高阶卷占 20%，高阶卷把更大份额给了代数（30%）。基础卷的 Number 占 25%，高阶卷只剩 15%，同样一道比例题，两种卷子里的性价比不一样。考纲明确写着三份卷都可以考任何一部分内容，Paper 1 非计算器卷照样出比例题。很多学生以为比例只在计算器卷出现，翻开 2017 年以来的 Paper 1 就会发现比与百分比一直在。三份卷各 1 小时 30 分、各 80 分、各占三分之一，总折算分 240 分。
In the AQA GCSE Mathematics 8300 specification, ratio sits inside a content domain whose references all start with R. Ofqual fixes the weighting by tier, so ratio is worth 25 per cent at Foundation but 20 per cent at Higher, where algebra takes 30 per cent instead. Every paper may assess any part of the specification, which is why ratio questions turn up on the non-calculator paper as well. Examiners see the assumption that ratio belongs to calculator papers every June, and it costs marks on Paper 1.
化简比与三量比的合并
化简比的做法与化分数不同，两边同除同一个数，结果要写成最简整数比，不能留下小数或分数。题目给 1.2 : 0.8 时，先把两边乘 10 变成 12 : 8，再同除 4 得 3 : 2。三个量的比要先找同一个中间量的两份说法，比如 a : b = 2 : 3、b : c = 4 : 5，把 b 统一成 12，得到 a : b : c = 8 : 12 : 15。常见的错误是把两个比直接拼成 2 : 3 : 5，中间量没有统一，后面按比例分配就全错。高阶卷会出现带单位的比，例如 40 分钟 : 2 小时，必须先化成同一单位再化简，答案是 1 : 3 而不是 40 : 2。这一步最容易丢分的地方不是算错，是漏了单位。
Simplifying a ratio means dividing both sides by the same number until the integers are as small as possible, which is not the same process as simplifying a fraction. With 1.2 to 0.8, multiplying by ten and then dividing by four gives 3 to 2. Three-part ratios need the shared quantity written the same way twice before the parts can be joined, and a common slip is to paste 2 to 3 and 4 to 5 into 2 to 3 to 5. Mixed units must be converted before anything is cancelled, so 40 minutes against 2 hours is 1 to 3.
按比例分配：part 与 whole 两种问法
分配题有两类问法，一类是 part : part，一类是 part : whole，读题时先确认给的是哪一种。三人按 3 : 4 : 5 分 240 英镑，先算总份数 3 + 4 + 5 = 12，再算一份 240 除以 12 得 20，逐个乘回去得到 60、80、100。若题面写的是某人占全部的 3/8，这是 part : whole，总份数是 8，剩下的 5 份才由其他人分。学生最常把 part : whole 当成 part : part，拿 8 份又当 3 份去乘，答案直接翻倍。验算是把各部分加起来看是否回到原数，十秒钟的动作，能拦下大半低级错。
Two wordings appear in division questions, one giving part to part and the other part to whole, and the first job is deciding which one is in front of you. Dividing 240 pounds in the ratio 3 to 4 to 5 uses twelve parts, so one part is twenty and the shares are sixty, eighty and one hundred. When the question says one person takes three eighths of the total, the whole is eight parts and the other five belong to everybody else. Adding the shares back up is a ten-second check that catches most arithmetic slips.
比与分数互转时分母是谁
把比写成分数时，分母是总份数而不是另一边的份数，这一点在 part : whole 里尤其容易错。2 : 5 表示前者占全部的 2/7，后者占 5/7，而不是 2/5 与 5/2。反过来，题目给男生占 3/7，可以立刻读出男 : 女 = 3 : 4。分数与比在解题时能互换，选哪个取决于下一步做什么：要按份数分配就用比，要对整体做乘法就用分数。错得最多的是把 2 : 5 直接读成 2/5 与 5/2，接着所有计算都建在错的分母上。
Turning a ratio into a fraction puts the total number of parts in the denominator, not the number on the other side. A ratio of 2 to 5 means two sevenths and five sevenths of the whole, and reading it as two fifths and five halves ruins every line written afterwards. The conversion works both ways, so a statement that boys make up three sevenths of a class immediately gives 3 to 4. Choose the form by what comes next, ratios for sharing out and fractions for multiplying by a whole.
直接比例：判断依据与设式
直接比例指两个量同时放大或缩小，比值保持不变，写成 y = kx。判断时看单位量的值会不会变：单价固定时总价与数量成正比，速度固定时路程与时间成正比。设式时先把已知的一对数值代进去求 k，例如 4 支笔 6 英镑，k = 1.5，那么 7 支就是 10.5 英镑。用 unitary method 也行，先求一份再乘，两种写法在基础卷都给分。很多学生以为两个量一起变大就是正比，实际要看是否按同一倍数变化，2 倍价格买 3 倍数量就不成立。
Direct proportion means both quantities grow or shrink together while their quotient stays fixed, which is the same as writing y equals k x. The test is whether the value of one unit changes, since a fixed unit price keeps total cost proportional to quantity. Substituting a known pair gives k, so four pens at six pounds mean k is one and a half and seven pens cost ten and a half pounds. Two quantities rising together is not enough on its own, because tripling the quantity while doubling the cost breaks the relationship.
反比例：乘积恒定与双曲线
反比例写成 y = k/x，乘积不变是它的判据，两个量的乘积恒等于同一个常数。四个工人 6 天完工，k = 24，六个工人需要 4 天。绘图时正比是过原点的直线，反比是双曲线，两条坐标轴都是渐近线，考纲 R14 要求能读懂这两种图。难点在设式：先写 x 乘 y = k，代入已知对求出 k，再代入新值。常见的错误是把反比例写成 y = kx，然后算出工人越多工期越长，答案一眼就该知道不对。做题时先问一句一个变大另一个是不是变小，判断方向就定了。
Inverse proportion is written as y equals k over x, and its signature is a constant product rather than a constant quotient. Four workers finishing in six days give k of twenty-four, so six workers take four days. On a graph the direct case is a straight line through the origin while the inverse case is a curve approaching both axes, and the specification expects students to read either one. Writing y equals k x here leads to the absurd result that more workers need longer, which is worth a sanity check before the algebra starts.
复合单位：速度、密度与压强
考纲 R11 列出速度、单位价格、薪酬率这些复合单位，另把密度与压强放在附加基础内容里。这类题先统一单位再代公式，速度用路程除以时间，密度用质量除以体积，压强用力除以面积。单位一乱答案必然错：45 分钟要写成 0.75 小时，250 克要写成 0.25 千克，否则速度会是正确值的四倍，密度差一千倍。计算器卷也要看清题目要求的单位，写 km/h 还是 m/s 按题面最后一句。阅卷时经常看到答案数值对、单位没换，整题分就这样丢掉。
Speed, unit price and rates of pay are the compound units named in the specification, with density and pressure listed as additional foundation content. The routine is to convert units first, then substitute into the formula, dividing distance by time or mass by volume. Forty-five minutes has to become three quarters of an hour and two hundred and fifty grams a quarter of a kilogram, otherwise the answer is out by a factor of four or a thousand. Markers regularly see the right number with the wrong unit and no credit for it.
最佳购买与单价比较
best buy 题本质上是把每个包装化成同一单位的价格再比较，考纲在 R1 的注释里点名了 better value 与 best-buy。500 毫升 2.40 英镑与 750 毫升 3.30 英镑，前者每 100 毫升 0.48，后者 0.44，后者更划算。比较时保留两位或三位小数就够，写太多位反而容易在四舍五入上出错。若题面问省了多少，要答每单位省下的差额，不是总价差。学生最常把总价低的那件当成更划算，小包装的单价往往更贵，先化成单价再下结论才能避开这个坑。
A best-buy question reduces every pack to a price per unit and compares on that basis, which is the better-value case flagged in the specification notes. Five hundred millilitres at two pounds forty works out at forty-eight pence per hundred, while seven hundred and fifty at three pounds thirty is forty-four pence, so the larger pack wins. Two or three decimal places are plenty, and more digits invite rounding errors. Cheap in total is not the same as cheap per unit, and that confusion is the usual source of lost marks here.
地图比例尺与图形缩放
比例尺题把图上长度与实际长度连起来，1 : 25000 表示图上 1 厘米对应实际 25000 厘米，也就是 250 米。换算最稳的做法是先全化成厘米算完，再换回米或千米，不要在中途混用单位。图形缩放考相似，边长按同一比例放大，面积按比例的平方变化，体积按三次方变化，这一点在高阶卷经常出。很多学生以为边长放大 3 倍面积也放大 3 倍，实际是 9 倍。答题时写清换算过程，3 分以上的题里过程分通常占 1 到 2 分。
A scale such as 1 to 25000 links a measurement on the drawing to real ground, with one centimetre standing for twenty-five thousand centimetres, or two hundred and fifty metres. Converting everything to centimetres first and back to metres at the end avoids the mixed-unit errors that appear in almost every script. Enlargement follows the square and cube rules, so tripling a length multiplies area by nine and volume by twenty-seven. Writing the conversion down earns method marks that a bare answer cannot.
浓度与混合的比例算法
考纲在 R1 的实境应用里写了 mixing 与 concentrations，浓度题的表述通常是溶质与溶液的质量比或百分比。把 20 克盐溶进 80 克水，溶液总质量是 100 克，浓度是 20%，不是 25%。若浓度按体积算，就要把体积相加，不能把质量与体积混用。混合两杯不同浓度的溶液时，先分别算溶质质量，相加后再除以总质量。这一步最容易出的错是把浓度当成可加的量直接平均，20% 与 40% 各取一半并不是 30%，除非两杯质量相同。
Mixing and concentrations appear among the real contexts the specification lists for ratio. Dissolving twenty grams of salt in eighty grams of water makes one hundred grams of solution, so the concentration is twenty per cent rather than twenty-five. When concentration is defined by volume, volumes are what get added and mass must not be mixed in. Averaging two percentages only works when the two amounts are equal, which is why the mass of solute has to be calculated separately first.
Paper 1 非计算器卷的比例算术
非计算器卷的比例题靠约分与倍数关系，不靠长除法。分配 240 英镑到 3 : 4 : 5，先看 12 份与 240 的关系，240 除以 12 得 20，可以直接心算。化简 36 : 48 时两边同除 12 得 3 : 4，找最大公因数比一步步除更快。遇到 0.75 : 1.5 这类小数比，先同乘 4 变成整数比再化简。常见的错误是硬算除法，把时间耗在算术上，后面的题没时间读。三份卷各 80 分、各 1 小时 30 分，平均一分略多于一分钟，算术上的浪费直接变成丢分。
The non-calculator paper rewards cancelling and multiples rather than long division. Sharing 240 pounds across 3, 4 and 5 is quick once the twelve parts are seen, because 240 divided by 12 gives 20 and the rest is multiplication. Dividing 36 and 48 by their highest common factor of 12 is faster than repeated halving, and decimal ratios are best cleared by multiplying both sides by the same power of ten. Each paper gives eighty marks in ninety minutes, so arithmetic detours cost questions later on.
阅卷扣分点与答题收尾
比例题的分数常被扣在最后一句上，题目问还剩多少，答案却给了一方的份额。写答句时把问的对象重读一遍，单位与主体都对上。多步题要保留中间步骤，三分的题通常给一分方法、一分过程、一分答案。分数形式的比要化成最简整数比，答案写 8 : 12 而不是 3 : 2 会丢分。这一步最容易在时间紧的时候被跳过，检查顺序建议先看单位、再看是否最简、最后核对答句问的是什么。平时练习就按这三步收尾，考场上不用再想。
Ratio marks are often lost on the final line, where the question asks for what is left and the answer gives one person's share. Re-reading the demand before writing the sentence keeps the subject and the unit aligned. Multi-step questions carry method marks, so intermediate working is worth writing even when the arithmetic is mental. An answer left as 8 to 12 instead of 3 to 2 is not fully simplified and drops a mark, and checking units, then simplification, then the actual demand takes under a minute.
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