📚 Year 9 Core Homework Answers Compressed | Year 9 核心作业答案精华速览
This compressed guide collects the most common Year 9 core homework questions and model answers into one place. It covers number, algebra, geometry, ratio, proportion, probability, statistics and sequences. Each section gives a typical question, a quick answer and a short method so you can check your work and revise efficiently.
这份精华速览把 Year 9 核心课程最常见的作业题目与标准答案集中在一起。内容涵盖数字、代数、几何、比与比例、概率、统计和数列。每个小节给出一道典型题、快速答案和简要方法，方便你核对作业并高效复习。
1. Number Operations and BIDMAS | 数字运算与运算顺序
)². The key is to follow BIDMAS: Brackets, Indices, Division and Multiplication, Addition and Subtraction. Work through each step in order and write down every line so mistakes are easy to find.
一道典型的 Year 9 核心作业题是计算 24 ÷ 6 + 2 × (5 − 3)²。关键是遵循 BIDMAS 运算顺序：括号、指数、除法和乘法、加法和减法。每一步都按顺序写出，这样容易发现错误。
Answer: 24 ÷ 6 = 4, 5 − 3 = 2, 2² = 4, 2 × 4 = 8, so 4 + 8 = 12. Final answer: 12.
答案：24 ÷ 6 = 4，5 − 3 = 2，2² = 4，2 × 4 = 8，因此 4 + 8 = 12。最终答案为 12。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
To convert a fraction to a percentage, find an equivalent fraction with denominator 100. To find a percentage of an amount, change the percentage to a decimal and multiply. A common homework question is: Convert 7/20 to a percentage and find 15% of 240.
要把分数转换成百分数，先找到一个分母为 100 的等价分数。要求一个数的百分之几，把百分数化成小数再相乘。常见的作业题是：把 7/20 转换成百分数，并求 240 的 15%。
Answer: 7/20 = 35% because 20 × 5 = 100 and 7 × 5 = 35. Also 15% of 240 = 0.15 × 240 = 36.
答案：7/20 = 35%，因为 20 × 5 = 100，7 × 5 = 35。另外 240 的 15% 等于 0.15 × 240 = 36。
3. Algebraic Simplification and Indices | 代数化简与指数
In algebra, you can only add or subtract like terms: terms that have exactly the same variable and the same power. For indices, when you multiply powers of the same base, keep the base and add the exponents.
在代数中，你只能合并同类项：变量和指数完全相同的项。对于指数，同底数幂相乘时，底数不变，指数相加。
ecause 3x − x = 2x and 5y + 2y = 7y. Also simplify x³ × x². Answer: x⁵, because 3 + 2 = 5.
化简 3x + 5y − x + 2y。答案：2x + 7y，因为 3x − x = 2x，5y + 2y = 7y。再化简 x³ × x²。答案：x⁵，因为 3 + 2 = 5。
4. Solving Linear Equations | 解一元一次方程
To solve a linear equation, move all variable terms to one side and all constant terms to the other side. Do the same operation to both sides so the equation stays balanced. Then isolate the variable by division.
解一元一次方程时，把所有含变量的项移到一边，把所有常数项移到另一边。等式两边同时进行相同运算，使等式保持平衡。然后通过除法求出变量的值。
ubtract 2x from both sides: 3x − 7 = 11. Add 7 to both sides: 3x = 18. Divide by 3: x = 6.
解方程 5x − 7 = 2x + 11。两边同时减去 2x：3x − 7 = 11。两边同时加 7：3x = 18。两边除以 3：x = 6。
5. Expanding Brackets and Factorising | 去括号与因式分解
Expanding brackets means multiplying each term inside the bracket by the term outside. Factorising is the reverse process: find the highest common factor and write it outside a bracket.
去括号就是用括号外面的项乘括号里的每一项。因式分解是相反的过程：找到最高公因式，把它写在括号外面。
Answer: 3x(2x + 3), since 3x is the highest common factor and 6x² ÷ 3x = 2x, 9x ÷ 3x = 3.
展开 4(2x − 3)。答案：8x − 12，因为 4 × 2x = 8x，4 × (−3) = −12。对 6x² + 9x 进行因式分解。答案：3x(2x + 3)，因为 3x 是最高公因式，且 6x² ÷ 3x = 2x，9x ÷ 3x = 3。
6. Ratio and Proportion | 比与比例
To divide an amount in a given ratio, add the parts to find the total number of parts, divide the amount by that total to find one part, then multiply by each ratio number. For proportion, use a scaling factor or find the unit rate.
按给定比例分配一个量时，先把比例的份数相加得到总份数，用总量除以总份数得到一份的量，再分别乘以比例的每个数。对于比例应用题，可以使用倍数或先求单位量。
Divide £56 in the ratio 3 : 5. Total parts = 3 + 5 = 8. One part = £56 ÷ 8 = £7. The amounts are 3 × £7 = £21 and 5 × £7 = £ s. For 20 cakes, the number of cakes is multiplied by 4, so flour needed = 2 × 4 = 8 cups.
把 56 英镑按 3 : 5 分配。总份数 = 3 + 5 = 8。一份 = 56 ÷ 8 = 7 英镑。两份分别是 3 × 7 = 21 英镑和 5 × 7 = 35 英镑。一个食谱做 5 个蛋糕需要 2 杯面粉。做 20 个蛋糕时，蛋糕数量乘以 4，所以需要面粉 2 × 4 = 8 杯。
7. Percentage Increase, Decrease and Reverse | 百分数增减与逆运算
For a percentage decrease, find the percentage of the original amount and subtract it from the original. For a reverse percentage, divide the final amount by the decimal multiplier that represents the increase or decrease.
计算百分数减少时，先求出原数的百分比，再从原数中减去。计算逆百分数时，用最终数量除以表示增加或减少的小数乘数。
A jacket costing £80 is reduced by 15%. Decrease = 0.15 × 80 = £12. Sale price = 80 − 12 = £68. After a 20% increase, a value is 120. The multiplier is 1.20, so original value = 120 ÷ 1.20 = 100.
一件夹克原价 80 英镑，降价 15%。降价额 = 0.15 × 80 = 12 英镑。售价 = 80 − 12 = 68 英镑。一个数增加 20% 后为 120。乘数是 1.20，所以原数 = 120 ÷ 1.20 = 100。
8. Area, Perimeter and Volume | 面积、周长与体积
The area of a triangle is half the base times the perpendicular height. The volume of a cuboid is length times width times height. Always include the correct unit: cm² for area and cm³ for volume.
三角形的面积等于底乘高再除以 2。长方体的体积等于长乘宽乘高。始终要写明正确单位：面积用 cm²，体积用 cm³。
d the volume of a cuboid with dimensions 4 cm, 3 cm and 2 cm. Volume = 4 × 3 × 2 = 24 cm³.
求底为 12 cm、高为 5 cm 的三角形面积。面积 = ½ × 12 × 5 = 30 cm²。求长、宽、高分别为 4 cm、3 cm、2 cm 的长方体体积。体积 = 4 × 3 × 2 = 24 cm³。
9. Pythagoras' Theorem | 勾股定理
Pythagoras' theorem only applies to right-angled triangles. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. The hypotenuse is always the longest side and is opposite the right angle.
勾股定理只适用于直角三角形。它指出斜边的平方等于另外两条边的平方和。斜边永远是最长的边，并且位于直角的对面。
des 6 cm and 8 cm. Find the hypotenuse. c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm.
一个直角三角形的两条较短边分别为 6 cm 和 8 cm。求斜边。c² = 6² + 8² = 36 + 64 = 100，所以 c = √100 = 10 cm。
10. Linear Graphs and Coordinates | 线性图像与坐标
here m is the gradient and c is the y-intercept. To plot the graph, substitute x-values into the equation to find matching y-values, then plot the coordinate pairs.
线性图像的一般方程是 y = mx + c，其中 m 是斜率，c 是 y 轴截距。画图时，把 x 值代入方程求出对应的 y 值，然后描出坐标点。
= 2 and y-intercept = 1, because the line is in the form y = mx + c with m = 2 and c = 1.
画出 y = 2x + 1 在 x = 0, 1, 2, 3 时的图像。坐标点为 (0, 1)、(1, 3)、(2, 5)、(3, 7)。斜率为 2，y 轴截距为 1，因为这条直线符合 y = mx + c 的形式，且 m = 2，c = 1。
11. Probability | 概率
Probability is a number between 0 and 1 that describes how likely an event is. To find the probability of an event, divide the number of favourable outcomes by the total number of possible outcomes. The probability of an event not happening is 1 minus the probability that it does happen.
概率是介于 0 到 1 之间的数，用来描述一个事件发生的可能性。求一个事件的概率，用有利结果的数量除以所有可能结果的总数。一个事件不发生的概率等于 1 减去它发生的概率。
otal = 3 + 2 + 5 = 10. P(red) = 3/10. P(not green) = 1 − P(green) = 1 − 5/10 = 5/10 = 1/2.
一个袋子里有 3 个红色、2 个蓝色和 5 个绿色筹码。总数 = 3 + 2 + 5 = 10。P(红) = 3/10。P(非绿) = 1 − P(绿) = 1 − 5/10 = 5/10 = 1/2。
12. Statistics: Mean, Median, Mode and Range | 统计：平均数、中位数、众数和极差
The mean is the sum of all values divided by the number of values. The median is the middle value when the data are ordered. The mode is the most frequent value. The range is the difference between the largest and smallest values.
平均数等于所有数值之和除以数值的个数。中位数是把数据排序后位于中间的数。众数是出现次数最多的数。极差是最大值与最小值之差。
. Ordered data: 2, 3, 4, 7, 7, 9. Median = (4 + 7) ÷ 2 = 5.5. Mode = 7. Range = 9 − 2 = 7.
对于数据集 4、7、2、9、7、3：平均数 = (4 + 7 + 2 + 9 + 7 + 3) ÷ 6 = 32 ÷ 6 = 5.33（保留两位小数）。排序后：2、3、4、7、7、9。中位数 = (4 + 7) ÷ 2 = 5.5。众数 = 7。极差 = 9 − 2 = 7。
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