📚 GMAT Quantitative High-Frequency Formulas and Problem-Solving Applications | GMAT数学高频公式与解题应用
The GMAT Quantitative section evaluates your ability to apply core mathematical concepts to business-oriented problems. Rather than memorizing every formula you have ever learned, focusing on the high-frequency formulas that appear again and again can dramatically boost your score. This guide presents those formulas in a structured way, along with practical problem-solving strategies.
GMAT 数学部分旨在评估你将核心数学概念应用于商业场景问题的能力。与其背下所有学过的公式，不如专注那些反复出现的高频公式，这能显著提高你的分数。本指南系统性地展现这些公式，并附上实用的解题策略。
1. Arithmetic and Fractions | 算术与分数
Basic arithmetic rules are the foundation of many GMAT questions. You must be comfortable with integer properties, exponents, and radical expressions that appear throughout the exam.
基本算术规则是许多 GMAT 题目的基础。你需要熟练掌握整数性质、指数和根式表达，它们贯穿整个考试。
The rule of exponents: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, and a⁻ᵐ = 1/aᵐ.
指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁻ᵐ = 1/aᵐ。
Radicals can be written as fractional exponents: ⁿ√a = a1/n. For example, √x = x1/2.
根式可以写成指数分数形式：ⁿ√a = a1/n。例如，√x = x1/2。
Integer properties: even ± even = even, odd ± odd = even, even × odd = even; prime numbers have exactly two distinct factors.
整数性质：偶 ± 偶 = 偶，奇 ± 奇 = 偶，偶 × 奇 = 偶；质数恰好只有两个不同的因数。
aᵐ × aⁿ = aᵐ⁺ⁿ | (aᵐ)ⁿ = aᵐⁿ | ⁿ√a = a1/n
When solving, always check whether a variable might be zero or negative before applying exponent rules. This is especially important in data sufficiency questions.
解题时，应用指数法则前务必确认变量是否可能为零或负数，这在数据充分性题型中尤为重要。
2. Algebra and Equations | 代数与方程
Linear and quadratic equations appear frequently in GMAT problem solving. Factoring tools and the quadratic formula are essential.
一次方程和二次方程在 GMAT 解题中频繁出现。因式分解和求根公式是必备工具。
Quadratic solution: ax² + bx + c = 0 ⇒ x = [-b ± √b² - 4ac] / 2a. The discriminant b² - 4ac determines the number of real solutions.
二次方程求根：ax² + bx + c = 0 ⇒ x = [-b ± √b² - 4ac] / 2a。判别式 b² - 4ac 决定实数解的个数。
Important identities: (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; a² − b² = (a + b)(a − b).
常用恒等式：(a + b)² = a² + 2ab + b²；(a − b)² = a² − 2ab + b²；a² − b² = (a + b)(a − b)。
Absolute value equations: |x| = a means x = a or x = −a. Inequalities of the form |x| < a correspond to −a < x < a.
绝对值方程：|x| = a 意味着 x = a 或 x = −a。形如 |x| < a 的不等式对应 −a < x < a。
x = [-b ± √b² − 4ac] / 2a | a² − b² = (a + b)(a − b)
When you see a quadratic equation on the GMAT, try factoring first; if factors are not obvious, use the quadratic formula. For inequalities, keep track of sign changes when multiplying by negative numbers.
遇到二次方程时，先尝试因式分解；如果因式不明显，再用求根公式。对于不等式，注意乘以负数时不等号方向要改变。
3. Ratios and Proportions | 比与比例
Ratios are used to compare quantities and are often tested with mixtures, direct variation, or scale drawings. Converting ratios to multi-part relationships is a key skill.
比用于比较数量，常在混合问题、正比例或比例尺中考查。将比值转换成多项关系是关键能力。
If a/b = c/d, then ad = bc. This cross-product rule is the backbone of proportion problems.
如果 a/b = c/d，则 ad = bc。交叉相乘法则是比例问题的核心。
For a ratio a : b : c, the actual quantities are k·a, k·b, k·c for some constant k. The sum of all parts gives the total.
对于比值 a : b : c，实际量为 k·a、k·b、k·c（k 为某一常数）。所有部分之和等于总量。
Direct proportion: y = kx; inverse proportion: y = k/x.
正比例：y = kx；反比例：y = k/x。
a/b = c/d ⇒ ad = bc | y = kx (direct) | y = k/x (inverse)
In word problems, translate sentences like "three times as many apples as oranges" into a ratio equation before solving.
在应用题中，先将“苹果数量是橙子的三倍”这类句子转化为比例方程，再求解。
4. Percentages and Interest | 百分比与利息
Percentages appear in discounts, profit-loss, and compound interest questions. The formula for percent change is used repeatedly.
百分比出现在折扣、盈亏和复利问题中。变化率公式被反复使用。
Percent change = (new value − old value) / old value × 100%. Be careful with the base value.
变化率 = (新值 − 旧值) / 旧值 × 100%。注意基准值的选择。
Simple interest: I = P·r·t, where P is principal, r is the annual rate (as a decimal), and t is time in years. Total amount A = P(1 + rt).
单利：I = P·r·t，其中 P 为本金，r 为年利率（小数形式），t 为年数。本息和 A = P(1 + rt)。
Compound interest: A = P(1 + r/n)n·t, where n is the number of compounding periods per year. Continuously compounded interest: A = P·ert.
复利：A = P(1 + r/n)n·t，其中 n 为每年复利次数。连续复利：A = P·ert。
A = P(1 + r/n)n·t | Percent Change = (New − Old) / Old × 100%
For successive percentage changes (e.g., a 10% increase followed by a 10% decrease), do not simply add the percentages. Use a calculator-like step-by-step multiplication.
对于连续百分比变化（例如先增加 10%，后减少 10%），不能直接加减百分比，应按步骤逐个乘算。
5. Statistics and Averages | 统计与平均数
Descriptive statistics questions ask for averages, medians, modes, and standard deviations. Knowing how to compute weighted averages is essential for data interpretation questions.
描述统计问题会考查平均数、中位数、众数和标准差。掌握加权平均数的计算对数据解读题至关重要。
Arithmetic mean = Sum of values / Number of values. This can be rearranged to find the sum: Sum = Mean × Count.
算术平均数 = 数值之和 / 数值个数。可变形为：总和 = 平均数 × 个数。
Median: the middle value when the data are sorted. If there is an even number of data points, take the average of the two middle values.
中位数：数据排序后的中间值。若数据个数为偶数，则取中间两个数的平均值。
Weighted average = (w₁x₁ + w₂x₂ + …) / (w₁ + w₂ + …).
加权平均数 = (w₁x₁ + w₂x₂ + …) / (w₁ + w₂ + …)。
Standard deviation measures spread around the mean. For a set of N numbers, σ = √[Σ(xᵢ − μ)² / N]. On the GMAT you usually do not need to compute it from raw data, but you must understand its meaning.
标准差度量数据相对平均数的离散程度。对 N 个数，σ = √[Σ(xᵢ − μ)² / N]。GMAT 通常不需要你从原始数据计算标准差，但你需要理解其含义。
Mean = Sum / Count | σ = √[Σ(xᵢ − μ)² / N]
When the data are evenly spaced, the mean equals the median. Use this shortcut to avoid lengthy calculations.
当数据均匀间隔时，平均数等于中位数。利用这个技巧可以避免冗长计算。
6. Probability | 概率
Probability problems are based on counting favorable outcomes and applying the rules of independence and mutual exclusiveness.
概率问题基于计算有利结果和应用独立性与互斥性规则。
Basic probability: P(A) = Number of favorable outcomes / Total number of outcomes.
基本概率：P(A) = 有利结果数 / 总结果数。
Complement: P(not A) = 1 − P(A).
补集：P(非 A) = 1 − P(A)。
Multiplication rule for independent events: P(A and B) = P(A) × P(B).
独立事件的乘法规则：P(A 且 B) = P(A) × P(B)。
Addition rule for mutually exclusive events: P(A or B) = P(A) + P(B). For non-mutually exclusive events: P(A or B) = P(A) + P(B) − P(A and B).
互斥事件的加法规则：P(A 或 B) = P(A) + P(B)。非互斥事件：P(A 或 B) = P(A) + P(B) − P(A 且 B)。
P(A∪B) = P(A) + P(B) − P(A∩B) | P(A∩B) = P(A)·P(B) (independent)
When dealing with "at least one" problems, take the complement of "none": P(at least one) = 1 − P(none). This often saves time.
处理“至少一个”问题时，取“一个都没有”的补集：P(至少一个) = 1 − P(都没有)。这样通常更省时。
7. Permutations and Combinations | 排列与组合
Counting problems appear in many quantitative questions. The distinction between permutations (order matters) and combinations (order does not matter) is critical.
计数问题出现在许多数量题中。分清排列（顺序有影响）和组合（顺序无关）至关重要。
Permutation of n distinct objects taken r at a time: P(n, r) = n! / (n − r)!.
从 n 个不同对象中取 r 个的排列数：P(n, r) = n! / (n − r)!。
Combination of n distinct objects taken r at a time: C(n, r) = n! / [r!(n − r)!].
从 n 个不同对象中取 r 个的组合数：C(n, r) = n! / [r!(n − r)!]。
Circular arrangements: the number of ways to arrange n distinct objects in a circle is (n − 1)!.
圆形排列：n 个不同对象围成一圈的排列方式有 (n − 1)! 种。
P(n,r) = n!/(n−r)! | C(n,r) = n!/[r!(n−r)!]
If items are identical, divide by the factorial of repeated counts. For example, arranging n objects where a set of p and q are identical gives n! / (p!q!).
如果有相同元素，需要除以重复元素数目的阶乘。例如，排列 n 个对象，其中 p 个相同、q 个相同，则方式数为 n! / (p!q!)。
8. Geometry | 几何
Geometry questions test knowledge of lines, angles, triangles, circles, and solids. Knowing these formulas cold is essential for quick solving.
几何题目考查直线、角度、三角形、圆和立体图形的知识。熟记这些公式是快速解题的关键。
Triangle area = ½ × base × height. Sum of interior angles of an n-gon = (n − 2) × 180°.
三角形面积 = ½ × 底 × 高。n 边形内角和 = (n − 2) × 180°。
Pythagorean theorem: a² + b² = c². Special right triangles: 30-60-90 (sides: x, x√3, 2x) and 45-45-90 (sides: x, x, x√2).
勾股定理：a² + b² = c²。特殊直角三角形：30-60-90（边长 x, x√3, 2x）和 45-45-90（边长 x, x, x√2）。
Circle: area = πr²; circumference = 2πr. Arc length = (central angle / 360°) × 2πr; sector area = (central angle / 360°) × πr².
圆：面积 = πr²；周长 = 2πr。弧长 = (圆心角 / 360°) × 2πr；扇形面积 = (圆心角 / 360°) × πr²。
Rectangular box volume = lwh; surface area = 2(lw + lh + wh). Cylinder volume = πr²h; lateral surface area = 2πrh.
长方体体积 = lwh；表面积 = 2(lw + lh + wh)。圆柱体积 = πr²h；侧面积 = 2πrh。
a² + b² = c² | Area = ½·base·height | Circumference = 2πr
For data sufficiency geometry questions, redrawing the figure with new information often reveals missing relational facts. Always ask whether the figure is drawn to scale.
对于数据充分性的几何题，根据新信息重新画图常能揭示隐藏的关系。始终要问：图形是否按比例绘制？
9. Coordinate Geometry | 坐标几何
Coordinate geometry combines algebra and geometry. The key formulas involve slope, distance, and midpoints. You must also know how to graph linear inequalities.
坐标几何结合代数和几何。核心公式涉及斜率、距离和中点。你还需要知道如何绘制线性不等式的图像。
Slope of a line through (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁) / (x₂ − x₁).
过点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率：m = (y₂ − y₁) / (x₂ − x₁)。
Distance between two points: d = √[(x₂ − x₁)² + (y₂ − y₁)²].
两点间的距离：d = √[(x₂ − x₁)² + (y₂ − y₁)²]。
Midpoint formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
中点公式：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。
Line in slope-intercept form: y = mx + b. Point-slope form: y − y₁ = m(x − x₁).
直线斜截式：y = mx + b。点斜式：y − y₁ = m(x − x₁)。
Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.
平行直线斜率相等；垂直直线斜率之积为 −1。
m = (y₂ − y₁) / (x₂ − x₁) | d = √[(x₂ − x₁)² + (y₂ − y₁)²]
When finding intersections, solve the system of equations by substitution or elimination. For inequalities, shade the half-plane that satisfies the inequality.
求交点时，用代入法或消元法解方程组。对于不等式，画出满足不等式的半平面。
10. Word Problems: Speed, Work, and Mixtures | 应用题：速度、工作与混合
Word problems in the GMAT often involve rates: speed, work, and mixture. The underlying principle is that rate × time = amount, and the amount of work or distance is additive.
GMAT 应用题常涉及速率：速度、工作和混合。基本原则是 速率 × 时间 = 总量，且工作量或距离是可加的。
Speed: Distance = Rate × Time. Average speed = Total distance / Total time. Do not use the arithmetic mean of speeds.
速度：距离 = 速度 × 时间。平均速度 = 总距离 / 总时间。不能使用速度的算术平均数。
Work: If A completes a job in a days and B completes it in b days, then A's rate = 1/a job per day, B's rate = 1/b job per day, and working together their time = 1 / (1/a + 1/b).
工作：A 单独完成需要 a 天，B 单独需要 b 天，则 A 的工作效率为 1/a，B 为 1/b；合作完成的时间 = 1 / (1/a + 1/b)。
Mixtures: For two components with concentrations c₁ and c₂, the final concentration = (v₁c₁ + v₂c₂) / (v₁ + v₂).
混合：两种成分浓度分别为 c₁ 和 c₂，终浓度 = (v₁c₁ + v₂c₂) / (v₁ + v₂)。
D = R × T | Time Together = 1 / (1/a + 1/b) | Weighted Concentration = (v₁c₁ + v₂c₂)/(v₁ + v₂)
Always define variables carefully and convert units to ensure consistency. For combined work, many problems can be solved by assigning the total job a convenient value such as 1 or a common multiple.
务必明确设定变量并统一单位。对于合作问题，常可将总工作量为 1 或公倍数，以简化计算。
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