Cambridge KS3 Mathematics Revision: Core Topics from Page 95 Exercise 1 | 剑桥KS3数学复习：第95页练习1核心专题

📚 Cambridge KS3 Mathematics Revision: Core Topics from Page 95 Exercise 1 | 剑桥KS3数学复习：第95页练习1核心专题

This article revisits the key concepts typically encountered in a Cambridge KS3 mathematics practice sheet such as page 95, exercise 1. It covers whole number operations, negative numbers, fractions, decimals, percentages, and the first steps into algebra. Working through these ideas will help you tackle mixed problem sets and prepare confidently for the Checkpoint exam.

本文重新梳理在剑桥 KS3 数学练习册（如第 95 页练习 1）中经常出现的关键概念，包括整数运算、负数、分数、小数、百分数以及代数入门。通过逐一攻破这些知识点，你将能从容应对混合练习题组，并为 Checkpoint 考试做好充分准备。

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1. Whole Number Operations | 整数运算

Addition, subtraction, multiplication, and division of whole numbers form the backbone of KS3 mathematics. When adding 478 + 356, align the digits by place value: units, tens, hundreds. Begin with the units column: 8 + 6 = 14, write 4 and carry 1 to the tens column. Then 7 + 5 + 1 = 13, write 3 and carry 1 to the hundreds column. Finally, 4 + 3 + 1 = 8, giving 834. This column method eliminates confusion when numbers have different digits.

整数的加、减、乘、除是 KS3 数学的根基。计算 478 + 356 时，要按个位、十位、百位对齐数字。先从个位开始：8+6=14，写下 4，向十位进 1。然后 7+5+1=13，写下 3，向百位进 1。最后 4+3+1=8，得到 834。这种列竖式的方法避免了因数字位数不同而产生的混乱。

For multiplication, such as 24 × 36, break it into parts using the grid method. Multiply 20 × 30 = 600, 20 × 6 = 120, 4 × 30 = 120, and 4 × 6 = 24. Adding 600 + 120 + 120 + 24 gives 864. Alternatively, long multiplication with careful carrying works equally well. Always check your result by estimating: 24 × 36 is about 25 × 35 = 875, so 864 is reasonable.

对于乘法，如 24 × 36，可用网格法拆分计算：20×30=600，20×6=120，4×30=120，4×6=24。把 600+120+120+24 相加得 864。也可以使用进位长乘法，同样可靠。最后要用估算检验结果：24×36 大约等于 25×35=875，因此 864 是合理的。

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2. Negative Numbers | 负数

Working with negative numbers requires understanding the number line. Adding a negative is equivalent to moving left: 3 + (−5) = −2. Subtracting a negative means moving right: 4 − (−3) = 4 + 3 = 7. When two signs appear together, remember the rule: same signs make a positive, different signs make a negative. So −(−4) = +4, and +(−2) = −2.

处理负数需要理解数轴。加上一个负数相当于向左移动：3 + (−5) = −2。减去一个负数相当于向右移动：4 − (−3) = 4 + 3 = 7。当两个符号碰到一起时，记住“同号得正，异号得负”的规则。因此 −(−4)=+4，+(−2)=−2。

Multiplying and dividing with negatives follows a similar pattern. (−6) × (−4) = 24, while (−6) × 4 = −24. 15 ÷ (−3) = −5. These rules are vital when substituting negative values into algebraic expressions later.

负数的乘除法也遵循类似规律：（−6）×（−4）= 24，而（−6）× 4 = −24。15 ÷（−3）= −5。这些规则在后续向代数式中代入负数值时至关重要。

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3. Order of Operations (BIDMAS) | 运算顺序 (BIDMAS)

The BIDMAS rule tells us the correct sequence when a calculation involves more than one operation: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right). For example, evaluate 10 − 2 × 3². First, calculate the index: 3² = 9. Then multiply: 2 × 9 = 18. Finally, subtract: 10 − 18 = −8. If we had simply worked left to right, we would have obtained (10 − 2) × 9 = 72, which is incorrect.

当一个算式包含多种运算时，BIDMAS 规则告诉我们正确的顺序：先算括号，再算指数（乘方），接着算乘法和除法（从左到右），最后算加法和减法（从左到右）。例如计算 10 − 2 × 3²。先算指数：3² = 9，再算乘法：2×9=18，最后算减法：10−18=−8。如果只是从左往右机械计算，会得到 (10−2)×9=72，那就错了。

When brackets are present, always evaluate what is inside them first. In 24 ÷ (4 + 2) × 3, compute 4 + 2 = 6 first. Then do 24 ÷ 6 = 4, and finally 4 × 3 = 12. Inserting brackets changes the meaning of an expression, so it is important to write them clearly in your own working.

若式子中含有括号，一定要先算括号内的部分。如 24 ÷ (4 + 2) × 3，先算 4+2=6，再算 24÷6=4，最后 4×3=12。增加括号会改变整个算式的含义，因此自己写过程时也要把括号标得清清楚楚。

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4. Fractions: Simplifying and Equivalent | 分数：化简与等值分数

A fraction represents a part of a whole. To simplify a fraction such as 15/35, find the greatest common factor of the numerator and denominator. Both 15 and 35 are divisible by 5, so 15 ÷ 5 = 3 and 35 ÷ 5 = 7, giving 3/7. Simplifying makes fractions easier to compare and operate with.

分数表示整体的一部分。化简如 15/35 这样的分数，需要找到分子与分母的最大公因数。15 和 35 都能被 5 整除，因此 15÷5=3，35÷5=7，得到 3/7。化简后的分数更便于比较和计算。

Equivalent fractions can be created by multiplying or dividing the numerator and denominator by the same non-zero number. For example, 1/2, 2/4, 3/6, and 5/10 are all equivalent. You can check this by cross-multiplying: 1 × 4 equals 2 × 2, both giving 4. Equivalent fractions are useful when adding and subtracting fractions with different denominators.

将分子和分母同时乘或除以同一个非零数，就能得到等值分数。例如 1/2、2/4、3/6 和 5/10 都是等值的。你可以用交叉相乘来验证：1×4 等于 2×2，都得 4。在为异分母分数进行加减运算时，等值分数非常有用。

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5. Adding and Subtracting Fractions | 分数加减法

To add or subtract fractions, they must have the same denominator. For 3/8 + 1/4, convert 1/4 to an equivalent fraction with denominator 8: multiply the numerator and denominator by 2 to get 2/8. Then add: 3/8 + 2/8 = 5/8. The denominator stays the same while the numerators are added.

要进行分数加减，必须先化为同分母。以 3/8 + 1/4 为例，把 1/4 化成以 8 为分母的等值分数：分子分母同时乘 2，得 2/8。然后相加：3/8 + 2/8 = 5/8。分母不变，只把分子相加。

When subtracting, apply the same rule. For 7/10 − 2/5, change 2/5 to 4/10, then subtract: 7/10 − 4/10 = 3/10. If the result is an improper fraction, you may need to convert it to a mixed number. For example, 9/4 = 2 1/4. Always simplify your final answer where possible.

做减法时规则相同。计算 7/10 − 2/5，先把 2/5 化成 4/10，然后相减：7/10−4/10=3/10。如果结果为假分数，可能需要化为带分数，例如 9/4 = 2 ¼。最后一定要尽可能化简答案。

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6. Multiplying and Dividing Fractions | 分数乘除法

Multiplying fractions is straightforward: multiply the numerators together and multiply the denominators together. So 2/3 × 4/5 = (2×4)/(3×5) = 8/15. If possible, simplify before multiplying by cancelling any common factors between numerators and denominators across the fractions. For instance, 3/8 × 4/9: cancel 3 and 9 (both can be divided by 3) and 4 and 8 (divide by 4), giving 1/2 × 1/3 = 1/6.

分数乘法很直接：分子乘分子，分母乘分母。所以 2/3 × 4/5 = (2×4)/(3×5)=8/15。如果可以，在乘法运算前先约分会更简便，跨分数寻找分子分母之间的公因数。例如 3/8 × 4/9：将 3 和 9 约掉（除以 3），将 4 和 8 约掉（除以 4），得到 1/2 × 1/3 = 1/6。

Dividing fractions involves multiplying by the reciprocal of the divisor. To calculate 5/6 ÷ 2/3, flip the second fraction and multiply: 5/6 × 3/2 = 15/12. Simplify 15/12 to 5/4, which is 1 1/4. Always remember to invert only the divisor, not the first fraction.

分数除法则需乘上除数的倒数。计算 5/6 ÷ 2/3，把第二个分数翻转后相乘：5/6 × 3/2 = 15/12。化简 15/12 得 5/4，也就是 1 ¼。一定要记住，只翻转除数，不动第一个分数。

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7. Decimals and Place Value | 小数与位值

Decimals are an extension of the place value system. The columns to the right of the decimal point represent tenths, hundredths, thousandths, and so on. In the number 35.607, the digit 6 is in the tenths place (6/10), 0 in the hundredths place, and 7 in the thousandths place (7/1000). Understanding place value is essential when ordering, adding, and subtracting decimals.

小数实质上是位值体系的延伸。小数点右边的数位依次表示十分位、百分位、千分位等。在 35.607 中，数字 6 在十分位（6/10），0 在百分位，7 在千分位（7/1000）。理解位值对比较、加减小数至关重要。

When adding decimals, align the decimal points vertically. For 12.8 + 3.45, write 12.80 above 3.45, then add column by column: 0 + 5 = 5, 8 + 4 = 12 (write 2, carry 1), 2 + 3 + 1 = 6, and bring down the 1 in the tens place. The result is 16.25. Adding zeros as placeholders prevents misalignment.

做小数加法时，要把小数点上下对齐。计算 12.8 + 3.45 时，写成 12.80 对齐 3.45，再按列相加：0+5=5，8+4=12（写 2 进 1），2+3+1=6，最后把十位的 1 落下来，结果为 16.25。用零占位可以防止错位。

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8. Converting Fractions, Decimals and Percentages | 分数、小数与百分数互化

Fractions, decimals and percentages are three ways of representing the same idea. To change a fraction to a decimal, divide the numerator by the denominator: 3/8 = 3 ÷ 8 = 0.375. To convert a decimal to a percentage, multiply by 100 and add the % sign: 0.375 × 100 = 37.5%. Reversing the process, from a percentage to a decimal, divide by 100: 62% = 0.62.

分数、小数和百分数是同一概念的三种不同表示法。分数化小数，用分子除以分母：3/8 = 3÷8 = 0.375。小数化百分数，乘 100 并加上 % 号：0.375×100=37.5%。反过来，百分数化小数除以 100：62% = 0.62。

Some conversions are so common that they should be memorised. The table below lists a few key equivalents. Being able to recall these instantly makes problem-solving much faster, especially in ratio and proportion questions.

有些转换实在太常用，最好直接记住。下表列出了几个关键等价关系。能够立刻反应出这些值，会大大提高解题速度，尤其是在比例和比率问题中。

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
1/10	0.1	10%

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9. Introduction to Algebra: Substitution | 代数入门：代入求值

Algebra uses letters to represent unknown or changing numbers. Substitution means replacing a letter with a given value and calculating the result. If a = 3 and b = 4, evaluate 2a + 3b. Substitute: 2 × 3 + 3 × 4. Following BIDMAS, multiply first: 6 + 12 = 18. Always put the value in brackets when substituting if there is any chance of sign confusion, e.g. when b = −2, 3b becomes 3 × (−2) = −6.

代数用字母表示未知或可变的数。代入即用给定的数值替换字母然后计算出结果。已知 a=3，b=4，求 2a + 3b 的值：代入 2×3 + 3×4，遵照运算顺序先乘得 6+12=18。如果代入的数值是负数或易造成符号混淆，要用括号框好，例如 b=−2 时，3b 变成 3×(−2)=−6。

Another example: evaluate (x² + y) / 2 when x = 5 and y = 7. Replace x and y: (5² + 7) / 2. Calculate the index: 25 + 7 = 32, then divide by 2 to get 16. Accurate substitution is crucial before moving on to solving equations.

再如：已知 x=5，y=7，求 (x² + y) / 2。替换后得 (5² + 7) / 2，先算指数 25+7=32，再除以 2 得 16。在进一步学习解方程之前，准确的代入是基础。

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10. Solving One-step Equations | 解一步方程

An equation shows that two expressions are equal. To solve a one-step equation, perform the inverse operation to isolate the variable. For x + 7 = 15, subtract 7 from both sides: x + 7 − 7 = 15 − 7, giving x = 8. For y − 3 = 9, add 3 to both sides: y = 12. The golden rule is: whatever you do to one side, you must do to the other to keep the equation balanced.

方程表示两个表达式相等。解一步方程需要运用逆运算把变量孤立出来。以 x + 7 = 15 为例，两边同时减 7：x+7−7=15−7，得 x=8。对于 y − 3 = 9，两边同时加 3 得 y=12。黄金法则是：对一边做了什么，另一边也必须做同样的事，以保持天平平衡。

Multiplication and division equations work similarly. If 4n = 20, divide both sides by 4: n = 5. If m / 3 = 6, multiply both sides by 3: m = 18. Always check your answer by substituting it back into the original equation: for 4n = 20, 4 × 5 = 20, which is true.

乘法和除法方程同理。若 4n = 20，两边同除以 4 得 n=5。若 m / 3 = 6，两边同乘 3 得 m=18。养成好习惯，把答案代回原方程检验：4n=20 中，4×5=20，成立。

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11. Introduction to Co-ordinates | 坐标系入门

Co-ordinates describe a point’s position on a grid using an ordered pair (x, y). The x-value comes first and tells you how far to move horizontally from the origin (0,0); the y-value tells you how far to move vertically. Plot the point (2, 5) by starting at the origin, moving 2 units right and 5 units up. Negative x-values move left, negative y-values move down.

坐标用一对有序数 (x, y) 来描述网格中点的位置。x 值在前，表示从原点 (0,0) 出发沿水平方向移动的距离；y 值表示竖直移动的距离。画点 (2,5) 时从原点出发，向右移动 2 格再向上移动 5 格。x 值为负向左移，y 值为负向下移。

Being able to read and plot co-ordinates is the foundation of graphing lines and shapes. In a practice exercise like page 95, you may be asked to identify the co-ordinates of vertices of a rectangle or to complete a shape when given three vertices. Always count the grid lines carefully and double-check the order of x and y.

正确读、画坐标是后续画直线和图形的基础。在类似第 95 页的练习中，你可能会被要求写出长方形各顶点的坐标，或根据已知的三个顶点完成图形。务必仔细数网格线，并始终确认 x 和 y 的顺序。

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12. Practical Problem Solving | 实际应用问题

KS3 mathematics often combines skills in word problems. For example: “A book costs $12.50. A school buys 15 copies with a budget of $200. How much money is left?” First, find the total cost: 12.50 × 15 = $187.50. Then subtract from $200: 200 − 187.50 = $12.50 remaining. Break the problem into clear steps and decide which operations to use.

KS3 数学常通过文字题把多种技能融合在一起。比如：“一本书售价 $12.50，学校预算 $200 购买 15 本，还剩多少钱？”先求总额：12.50×15=$187.50，再拿 $200 去减：200−187.50=$12.50。把问题分解成清晰的步骤，并选择恰当的运算。

Another common type involves fractions and percentages: “In a class of 30 students, 2/5 are boys. How many girls are there?” Since 2/5 are boys, 1 − 2/5 = 3/5 are girls. 3/5 of 30 = 3/5 × 30 = 18 girls. Reading the question carefully and identifying what fraction or percentage refers to the whole is the key.

另一常见题型融合分数与百分数：“一个有 30 名学生的班级，2/5 是男生，请问有多少女生？”因为 2/5 是男生，则 1−2/5=3/5 是女生。30 的 3/5 是 3/5×30=18 名女生。仔细读题，明确分数或百分数所对应的“整体”是什么，是解题关键。

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