📚 Essential Maths 9C Homework Answers: Compressed Year 9 Core Maths Guide | 9年级数学核心 9C 作业答案精要指南
This guide compresses the most common Year 9 Essential Maths 9C homework topics into worked examples and answer-checking methods. Use it to review key skills and check your own working, not to copy answers without understanding.
本指南将 Year 9 Essential Maths 9C 作业中最常见的主题压缩为示例题和答案检查方法。请用它来复习关键技能并检查自己的解题过程，而不是在不理解的情况下抄写答案。
1. Number and Place Value: Rounding and Estimation | 数字与位值：四舍五入与估算
Round 0.008364 to 2 significant figures. The first two significant figures are 8 and 3, so the answer is 0.0084 because the next digit 6 is 5 or more.
将 0.008364 四舍五入到 2 位有效数字。前两位有效数字是 8 和 3，因此答案是 0.0084，因为下一位数字 6 大于等于 5。
0.008364 ≈ 0.0084 (2 s.f.)
o the estimate is 32. This is a fast way to check whether a calculator answer is sensible.
将 4.2 × 7.8 四舍五入到一位有效数字来估算：4 × 8 = 32，因此估算值为 32。这是检查计算器答案是否合理的快速方法。
Upper and lower bounds are also tested. A length given as 6.5 cm to 1 decimal place has a lower bound of 6.45 cm and an upper bound of 6.55 cm.
上界和下界也会考查。一个长度写成 6.5 cm（精确到 1 位小数）的下界是 6.45 cm，上界是 6.55 cm。
2. Fractions, Decimals and Percentages | 分数、小数和百分比
Convert 3/8 to a decimal and a percentage. Divide 3 by 8 to get 0.375, then multiply by 100 to get 37.5%.
将 3/8 转换为小数和百分比。用 3 除以 8 得到 0.375，再乘以 100 得到 37.5%。
3/8 = 0.375 = 37.5%
To increase £140 by 15%, first find 15% of 140: 0.15 × 140 = 21. Then add: 140 + 21 = 161. The final amount is £161.
要将 £140 增加 15%，先求 140 的 15%：0.15 × 140 = 21。然后相加：140 + 21 = 161。最终金额为 £161。
For a percentage decrease, subtract instead. Decreasing £80 by 10% gives 80 − 8 = 72, because 10% of 80 is 8.
对于百分比减少，则用减法。将 £80 减少 10% 得到 80 − 8 = 72，因为 80 的 10% 是 8。
3. Ratio and Proportion | 比与比例
Simplify the ratio 24:36 by dividing both sides by their highest common factor, 12. The simplified ratio is 2:3.
将比 24:36 两边同时除以它们的最大公因数 12，化简为 2:3。
24:36 = 2:3
To share £ parts: 2 + 3 = 5. One part is 300 ÷ 5 = 60. The shares are 2 × 60 = 120 and 3 × 60 = 180.
按 2:3 的比例分配 £300，先把份数相加：2 + 3 = 5。每份是 300 ÷ 5 = 60。分配结果是 2 × 60 = 120 和 3 × 60 = 180。
Proportion problems often ask for the value of one item. If 5 pens cost £3.50, then 1 pen costs 3.50 ÷ 5 = 0.70, so 8 pens cost 8 × 0.70 = 5.60.
比例问题常求单个物品的价值。如果 5 支笔花费 £3.50，那么 1 支笔花费 3.50 ÷ 5 = 0.70，因此 8 支笔花费 8 × 0.70 = 5.60。
4. Algebraic Expressions and Linear Equations | 代数表达式与线性方程
Expand 5(2x − 3) by multiplying each term inside the bracket by 5. This gives 10x − 15.
将 5(2x − 3) 展开，用 5 乘以括号内每一项，得到 10x − 15。
5(2x − 3) = 10x − 15
equation 3x + 7 = 22. First subtract 7 from both sides: 3x = 15. Then divide by 3: x = 5.
解方程 3x + 7 = 22。先从两边减去 7：3x = 15。然后除以 3：x = 5。
3x + 7 = 22 → x = 5
t. For 2(x + 3) = 14, expand to 2x + 6 = 14, then subtract 6 and divide by 2 to get x = 4.
解带括号的方程时，先展开。对于 2(x + 3) = 14，展开为 2x + 6 = 14，然后减去 6 再除以 2，得到 x = 4。
5. Linear Graphs and Coordinates | 线性图像与坐标
and the y-intercept is 1. The gradient means y increases by 2 for every increase of 1 in x.
对于直线 y = 2x + 1，斜率是 2，y 截距是 1。斜率表示 x 每增加 1，y 增加 2。
the value of y when x = 3, substitute x = 3 into y = 2x + 1. This gives y = 2 × 3 + 1 = 7.
要求当 x = 3 时 y 的值，将 x = 3 代入 y = 2x + 1。得到 y = 2 × 3 + 1 = 7。
y = 2(3) + 1 = 7
To plot a linear graph, choose three x-values, calculate the matching y-values, and plot the points. The points should lie on a straight line; if not, check your arithmetic.
绘制线性图像时，选择三个 x 值，计算对应的 y 值，然后描点。这些点应在一条直线上；如果不是，检查计算。
6. Angles and Polygons | 角与多边形
ith n sides is (n − 2) × 180°. For a pentagon, n = 5, so the sum is (5 − 2) × 180° = 540°.
n 边形内角和为 (n − 2) × 180°。对于五边形，n = 5，所以内角和为 (5 − 2) × 180° = 540°。
Sum of interior angles = (n − 2) × 180°
xagon, each exterior angle is 360° ÷ 6 = 60°, so each interior angle is 180° − 60° = 120°.
任意多边形的外角和总是 360°。对于正六边形，每个外角为 360° ÷ 6 = 60°，因此每个内角为 180° − 60° = 120°。
Angles on a straight line add up to 180°, and angles around a point add up to 360°. These facts are often used in multi-step geometry questions.
直线上的角相加为 180°，围绕一点的角相加为 360°。这些事实常用于多步几何题中。
7. Perimeter, Area and Volume | 周长、面积与体积
cle is A = πr². If the radius is 3 cm, then A = π × 3² = 9π ≈ 28.3 cm² to 1 decimal place.
圆的面积为 A = πr²。如果半径为 3 cm，则 A = π × 3² = 9π ≈ 28.3 cm²（精确到 1 位小数）。
es and h is the height. For a = 5 cm, b = 9 cm and h = 4 cm, A = ½ × (5 + 9) × 4 = 28 cm².
梯形面积为 A = ½(a + b)h，其中 a 和 b 是平行边，h 是高。当 a = 5 cm、b = 9 cm、h = 4 cm 时，A = ½ × (5 + 9) × 4 = 28 cm²。
A cylinder with radius 2 cm and height 10 cm has volume V = π × 2² × 10 = 40π ≈ 125.7 cm³.
棱柱体积 = 横截面积 × 长度。半径为 2 cm、高为 10 cm 的圆柱体体积为 V = π × 2² × 10 = 40π ≈ 125.7 cm³。
8. Pythagoras' Theorem | 勾股定理
Pythagoras' ngled triangle with legs 6 cm and 8 cm, c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm.
勾股定理为 a² + b² = c²，其中 c 是斜边。对于直角边分别为 6 cm 和 8 cm 的直角三角形，c² = 6² + 8² = 36 + 64 = 100，所以 c = √100 = 10 cm。
a² + b² = c²
To find a shorter side, rearrange the formula. If the hypotenuse is 13 cm and one leg is 5 cm, then the other leg is √13² − 5² = √169 − 25 = √144 = 12 cm.
求直角边时，重新整理公式。如果斜边为 13 cm，一条直角边为 5 cm，则另一条直角边为 √13² − 5² = √169 − 25 = √144 = 12 cm。
Always identify the hypotenuse first: it is the side opposite the right angle and is always the longest side.
始终先找出斜边：它是直角对面的边，并且总是最长的边。
9. Probability and Statistics | 概率与统计
The probability of rolling a 5 on a fair six-sided die is 1/6, because there is one favourable outcome out of six possible outcomes.
在一枚均匀的六面骰子上掷出 5 的概率是 1/6，因为六个可能结果中只有一个有利结果。
P(5) = 1/6
number of values. For 3, 7, 8, 10, 12, the mean is (3 + 7 + 8 + 10 + 12) ÷ 5 = 40 ÷ 5 = 8.
平均数通过将所有数值相加再除以数值个数求得。对于 3、7、8、10、12，平均数为 (3 + 7 + 8 + 10 + 12) ÷ 5 = 40 ÷ 5 = 8。
ata, the range is 12 − 3 = 9. The median is the middle value, 8, when the data is ordered.
极差是最大值与最小值之差。对于同一组数据，极差为 12 − 3 = 9。中位数是排序后位于中间的值，即 8。
10. Transformations and Congruence | 变换与全等
A translation moves a shape without turning or flipping it. A shape translated by the vector (3, −2) moves 3 units right and 2 units down.
平移是移动图形而不旋转或翻转它。图形按向量 (3, −2) 平移，即向右移动 3 个单位、向下移动 2 个单位。
A rotation of 90° clockwise around the origin turns a shape one quarter turn. A reflection in the x-axis flips the shape over the x-axis, changing the sign of each y-coordinate.
绕原点顺时针旋转 90° 将图形转动四分之一圈。关于 x 轴反射将图形翻转到 x 轴另一侧，改变每个 y 坐标的符号。
An enlargement with scale factor 2 doubles every side length. The shape remains similar, meaning angles stay the same and side lengths are multiplied by 2.
比例因子为 2 的放大将每条边长加倍。图形保持相似，即角度不变，边长乘以 2。
11. Indices and Standard Form | 指数与标准形式
wers with the same base, add the indices: aᵐ × aⁿ = aᵐ⁺ⁿ. For example, 2³ × 2⁴ = 2⁷ = 128.
同底数幂相乘时，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。例如，2³ × 2⁴ = 2⁷ = 128。
2³ × 2⁴ = 2⁷
the same base, subtract the indices: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. For example, 10⁵ ÷ 10² = 10³ = 1000.
同底数幂相除时，指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ。例如，10⁵ ÷ 10² = 10³ = 1000。
where 1 ≤ a < 10. The number 4500 in standard form is 4.5 × 10³, and 0.0062 is 6.2 × 10⁻³.
标准形式将一个数写成 a × 10ⁿ，其中 1 ≤ a < 10。数字 4500 的标准形式为 4.5 × 10³，0.0062 为 6.2 × 10⁻³。
12. Answer Checking and Common Errors | 答案检查与常见错误
Always check units. If a length is in cm and a volume is in cm³, avoid mixing cm and m without converting first.
始终检查单位。如果长度用 cm，体积用 cm³，在没有先换算的情况下不要混用 cm 和 m。
Check negative signs carefully. In a calculation like −5 + 3, the result is −2, not −8. Writing each step clearly reduces sign errors.
仔细检查负号。在 −5 + 3 这样的计算中，结果是 −2，而不是 −8。清晰写出每一步可以减少符号错误。
te 30 × 3 = 90, then compare with the calculator answer to spot accidental button presses.
使用计算器时，先估算。对于 29.8 × 3.05，估算为 30 × 3 = 90，然后与计算器答案比较，以发现误按按钮。
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