📚 Year 11 CIE Biology: Formula & Theorem Quick Reference Handbook | Year 11 CIE 生物：公式定理速查手册
This quick-reference handbook brings together all the essential formulas, key equations and important quantitative relationships you need to master for the CIE IGCSE Biology examination. From microscopy calculations and genetics probabilities to ecological sampling and physiological rates, each entry is presented with a concise explanation and worked context so you can apply the mathematics confidently in exam questions.
本速查手册汇集了 CIE IGCSE 生物考试中必须掌握的所有核心公式、关键方程式和重要的数量关系。从显微镜计算、遗传概率到生态取样和生理速率，每个条目都配有简明解释和实例背景，帮助你自信地将数学工具应用于考题之中。
1. Magnification Formula | 放大倍数公式
ge size ÷ Actual size. Make sure both measurements are in the same unit before calculating. The image size is often measured directly from a micrograph or drawing using a ruler, while the actual size may be given or calculated from a scale bar.
放大倍数表示图像比实物放大了多少倍。公式为：放大倍数 = 图像大小 ÷ 实际大小。计算之前务必确保两者单位一致。图像大小通常用直尺从显微照片或绘图上直接量取，实际大小则可能直接给出或通过比例尺求出。
Magnification = Image size / Actual size
放大倍数 = 图像大小 / 实际大小
e the length of a labelled structure (in mm) and divide by its given real length (often in µm). Convert mm to µm by multiplying by 1000 before dividing, or remember that 1 mm = 1000 µm.
当你在印好的显微照片上看到标注时，量出标记结构的长度（以 mm 计），再除以其给定的实际长度（通常为 µm）。记得在相除之前将 mm 换算为 µm（1 mm = 1000 µm），或者直接用换算后的值进行计算。
2. Actual Size Calculation | 实际大小计算
ze ÷ Magnification. This is the most common calculation in the alternative-to-practical paper. Always write the answer in appropriate units such as micrometres (µm) or millimetres (mm).
若已知放大倍数和图像大小，可通过公式变形求出标本的实际大小：实际大小 = 图像大小 ÷ 放大倍数。这是实验替代试卷中最常见的计算题。答案一定要用合适的单位，如微米 (µm) 或毫米 (mm)。
Actual size = Image size / Magnification
实际大小 = 图像大小 / 放大倍数
der a magnification of ×400, the actual length is 50 mm / 400 = 0.125 mm, which equals 125 µm. Always show your working clearly; partial marks are awarded for correct substitution.
例如，若一个细胞图示在 ×400 的放大倍数下测量的直径是 50 mm，实际长度则为 50 mm / 400 = 0.125 mm，即 125 µm。解题时务必清晰展示步骤，代入正确数值即可获得部分分数。
3. Photosynthesis Equation | 光合作用方程式
Photosynthesis is the process by which green plants manufacture glucose using carbon dioxide and water, with light energy trapped by chlorophyll. The overall word equation summarises the reactants and products. The balanced chemical equation shows that six molecules of carbon dioxide and six molecules of water produce one molecule of glucose and six molecules of oxygen.
光合作用是绿色植物利用叶绿素捕获光能，将二氧化碳和水转化为葡萄糖的过程。总文字方程式概括了反应物和生成物。配平的化学方程式则表明 6 个二氧化碳分子和 6 个水分子生成 1 个葡萄糖分子和 6 个氧气分子。
Word equation:
carbon dioxide + water → glucose + oxygen
文字方程式：
二氧化碳 + 水 → 葡萄糖 + 氧气
Balanced chemical equation:
6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂
化学方程式：
6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂
Chlorophyll and light energy are necessary conditions; they are written above the arrow in detailed equations but are not reactants. The oxygen released as a by-product comes from the splitting of water molecules during the light-dependent stage.
叶绿素和光能是必要条件，在详细的方程式书写时常写在箭头上方，但它们本身并非反应物。释放出的氧气副产品来自光反应阶段水分子的裂解。
4. Aerobic Respiration Equation | 需氧呼吸方程式
Aerobic respiration is the complete breakdown of glucose in the presence of oxygen to release a relatively large amount of energy. The word equation and the balanced symbol equation are exact opposites of photosynthesis, reflecting the cycling of gases in nature. This reaction occurs continuously in most cells, especially in muscle cells during prolonged exercise.
需氧呼吸是在有氧条件下将葡萄糖彻底分解并释放较多能量的过程。其文字方程式和配平的化学方程式恰好与光合作用相反，反映了自然界气体的循环。此反应在多数细胞中持续进行，尤其在长时间运动的肌肉细胞中更活跃。
Word equation:
glucose + oxygen → carbon dioxide + water (+ energy)
文字方程式：
葡萄糖 + 氧气 → 二氧化碳 + 水（+ 能量）
Balanced chemical equation:
C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O
化学方程式：
C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O
Remember that energy released is captured in the form of ATP, not as a separate chemical product. The equation shows the overall fate of the atoms: carbon and oxygen end up in carbon dioxide, while hydrogen combines with oxygen to form water.
请记住，释放的能量以 ATP 形式储存，并非作为一个独立的化学产物出现。总方程式反映了原子的最终去向：碳和部分氧进入二氧化碳，氢与氧结合生成水。
5. Monohybrid Crosses & Mendel's Law of Segregation | 单基因杂交与孟德尔分离定律
Mendel' ormation so that each gamete carries only one allele. A monohybrid cross between two heterozygous parents (e.g. Aa × Aa) produces offspring with a predictable genotypic and phenotypic ratio.
孟德尔分离定律指出，每个个体针对某一性状具有一对等位基因，在配子形成时这对等位基因彼此分离，每个配子只携带其中一个等位基因。两个杂合亲本（如 Aa × Aa）进行单基因杂交，其后代具有可预测的基因型比例和表现型比例。
The Punnett square below shows the cross for a gene where allele A is dominant and allele a is recessive.
下面的庞纳特方格展示了基因 A 为显性、a 为隐性时的杂交结果。
| A | a | |
| A | AA | Aa |
| a | Aa | aa |
Genotypic ratio: 1 AA : 2 Aa : 1 aa. Phenotypic ratio: 3 dominant : 1 recessive, provided that A is completely dominant over a.
基因型比例：1 AA : 2 Aa : 1 aa。表现型比例：3 显性 : 1 隐性，前提是 A 对 a 完全显性。
A test cross between a homozygous recessive (aa) and a heterozygous (Aa) individual gives a 1:1 phenotypic ratio, which is used to determine an unknown genotype.
隐性纯合子 (aa) 与杂合子 (Aa) 的测交后代表现型比例为 1:1，常用于鉴定未知基因型。
6. Probability in Genetics | 遗传学中的概率计算
Genetic outcomes can be predicted using simple probability rules. When two events are independent, the probability that both occur together is the product of their individual probabilities. This rule is extremely useful for dihybrid traits or when predicting the chance of a child inheriting a combination of sex-linked or autosomal conditions.
遗传结果可以通过简单的概率法则来预测。当两个事件相互独立时，两者同时发生的概率等于各自发生概率的乘积。该规则在双性状杂交或预测子女同时遗传某种组合的伴性或常染色体疾病时非常有用。
e AaBb from independent assortment is ½ × ½ = ¼. This multiplication rule underlies the 9:3:3:1 dihybrid ratio.
例如，若从父方获得等位基因 A 的概率为 ½，从母方获得等位基因 B 的概率也为 ½，则在自由组合下孩子同时具有 AaBb 基因型的概率为 ½ × ½ = ¼。这一乘积法则正是 9:3:3:1 双性状杂交比的基础。
two girls in a row is ½ × ½ = ¼. Always state the assumption that events are independent.
预测子代性别时，生男生女的概率各约 ½，因此连续生两个女儿的概率为 ½ × ½ = ¼。解题时须明确指出假设事件相互独立的前提。
7. Estimating Population Size – Lincoln Index | 种群大小估算——林肯指数
The Lincoln Index (also called the capture-mark-recapture method) is used to estimate the total population of motile organisms in a given area. The formula assumes that the marked individuals mix completely with the unmarked population between capture sessions, and that no significant births, deaths, immigration or emigration occur.
林肯指数（又称标记重捕法）用于估算某一区域内活动性生物的总种群数量。该公式假设标记个体在两次捕捉之间与未标记个体充分混合，且没有明显的出生、死亡、迁入或迁出现象发生。
N = (M × C) / R
N = (M × C) / R
n the second sample, and R = number of marked individuals recaptured in the second sample.
其中，N = 估算的种群总数，M = 第一次捕获并标记的个体数，C = 第二次捕获的总个体数，R = 第二次捕获中的已标记个体数。
If the calculated value is known as N, the examiner may ask you to evaluate the reliability of the estimate. A small number of recaptures (R) leads to a less precise estimate. In an exam, always check that the capture method does not harm the organisms or make them more conspicuous to predators.
如果题目要求求出 N 值，考官可能会进一步询问估算的可靠性。重捕数 (R) 过小会导致估算不准。作答时还应注意捕捉方法是否伤害生物或使其更容易被捕食者发现。
8. Cardiac Output | 心输出量
Cardiac output is the volume of blood pumped out by one ventricle per minute. It is the product of heart rate (number of beats per minute) and stroke volume (volume of blood ejected per beat). This concept helps to explain how exercise affects blood supply to muscles.
心输出量指的是一个心室每分钟泵出的血液总量，等于心率（每分钟搏动次数）与每搏输出量（每次搏动射出的血量）的乘积。这一概念有助于解释运动如何影响肌肉的血液供应。
Cardiac output (ml/min) = Heart rate (beats/min) × Stroke volume (ml/beat)
心输出量 (ml/min) = 心率 (次/min) × 每搏输出量 (ml/次)
During strenuous exercise, heart rate can more than double and stroke volume may also increase, so cardiac output rises significantly. Well-trained athletes often have a larger stroke volume and a lower resting heart rate, allowing them to achieve the same cardiac output with less strain on the heart.
剧烈运动时，心率可增长一倍以上，每搏输出量也可能升高，从而使心输出量显著增加。训练有素的运动员通常每搏输出量更大、静息心率更低，从而能在心脏负担较小的情况下达到同样的心输出量。
9. Body Mass Index (BMI) | 身体质量指数
BMI is a simple index used to classify underweight, normal weight, overweight and obesity in adults. It is calculated from a person's mass in kilograms and height in metres. Note that the height is squared, so the formula produces a unit of kg/m².
BMI 是一种用于划分成年人体重过轻、正常、超重和肥胖的简易指数，通过体重（千克）与身高（米）算出。请注意，计算时身高值需平方，因此该指数的单位为 kg/m²。
BMI = body mass (kg) ÷ [height (m)]²
BMI = 体重 (kg) ÷ [身高 (m)]²
2.9, which falls in the normal range. Although BMI is widely used, it does not distinguish between fat mass and muscle mass, so very muscular individuals may be incorrectly categorised as overweight.
一个体重 70 kg、身高 1.75 m 的人，其 BMI 为 70 / (1.75 × 1.75) = 22.9，属于正常范围。尽管 BMI 使用广泛，但它不能区分脂肪和肌肉质量，因此肌肉发达者可能被错误地归类为超重。
10. Respiratory Quotient (RQ) | 呼吸商
The respiratory quotient indicates the type of substrate being respired. It is defined as the ratio of the volume of carbon dioxide produced to the volume of oxygen consumed per unit time. The RQ value changes depending on whether carbohydrates, lipids or proteins are being oxidised.
呼吸商反映了呼吸底物的类型。其定义为每单位时间内产生的二氧化碳体积与消耗的氧气体积之比。RQ 值随被氧化的底物种类（碳水化合物、脂肪或蛋白质）而变化。
RQ = Volume of CO₂ produced / Volume of O₂ consumed
RQ = 产生的 CO₂ 体积 / 消耗的 O₂ 体积
For pure carbohydrate respiration, RQ is typically 1.0 because equal volumes of CO₂ and O₂ are involved. For lipids, RQ is about 0.7, reflecting the lower proportion of oxygen already present in fat molecules. This measurement is often studied using a respirometer, and it helps to interpret metabolic fuel use in organisms.
纯碳水化合物呼吸时，因消耗与产生的气体体积相等，RQ 通常为 1.0；脂肪呼吸时 RQ 约为 0.7，这反映了脂肪分子中本身含氧比例较低。此测量常借助呼吸计进行，有助于分析生物体内代谢燃料的利用情况。
11. Surface Area to Volume Ratio | 表面积与体积比
The surface area to volume ratio (SA:V) is a fundamental concept that explains why cells are small and why larger organisms need transport systems. It is calculated by dividing the total surface area of an organism or cell by its volume. As size increases, volume grows much faster than surface area, so the SA:V ratio decreases.
表面积与体积比 (SA:V) 是解释细胞为何微小、大型生物为何需要运输系统的基本原理。它的计算方法是用生物体或细胞的总表面积除以体积。随着尺寸增大，体积的增长远快于表面积，因此 SA:V 比值下降。
SA:V ratio = Total surface area / Volume
SA:V 比值 = 总表面积 / 体积
diffusion of oxygen into the cell and waste products out, which is essential for unicellular organisms. Multi-cellular organisms develop specialised surfaces such as alveoli, villi and root hairs to increase the effective surface area for exchange.
边长 1 cm 的立方体，SA:V 为 6:1；边长 2 cm 的立方体则为 24:8 = 3:1。高 SA:V 比值使得氧气能迅速扩散进入细胞、废物能快速排出，这对单细胞生物至关重要。多细胞生物则进化出肺泡、绒毛、根毛等特化结构，以增大物质交换的有效表面积。
12. Rate of Enzyme Reaction | 酶促反应速率
The rate of an enzyme-controlled reaction can be determined by measuring how quickly the substrate disappears or the product appears over a set period. This provides a quantitative way to investigate the effect of factors such as temperature, pH and substrate concentration.
酶控制反应的速率可通过测定单位时间内底物的减少量或产物的生成量来确定。这为研究温度、pH 值和底物浓度等因素的影响提供了量化手段。
Rate = Change in amount (substrate or product) / Time taken
速率 = 变化量（底物或产物）/ 所需时间
When plotting a graph of volume of oxygen produced against time during the catalase-hydrogen peroxide reaction, the initial rate of reaction is obtained by drawing a tangent at time zero and calculating its gradient. Understanding the initial rate helps to avoid complications from product inhibition or substrate depletion later in the reaction.
在过氧化氢酶催化过氧化氢分解实验中，绘制氧气生成体积随时间变化的曲线时，可通过在零时刻处画切线并求斜率来得到初始反应速率。掌握初始速率可避免后期因产物抑制或底物耗尽而带来的复杂影响。
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