📚 Essential Maths Book 7 Answers: Year 7 Key Topics Worked Solutions | Year 7 数学 Essential Maths Book 7 答案解析与复习指南
This guide gives worked solutions for the most common Year 7 topics in Essential Maths Book 7. It does not copy the full answer key, but it shows the methods you need to check your own work and build confidence.
本指南提供 Essential Maths Book 7 中 Year 7 最常见主题的解题示例。它不复制完整答案，但展示你检查作业和建立信心所需的方法。
1. Number and Place Value | 数与位值
In Year 7, you must read, write and order whole numbers up to at least 10 000 000. Place value tells you the value of each digit, so in 4 305 721 the digit 4 has value 4 000 000.
在 Year 7，你需要会读、写和比较至少到 10 000 000 的整数。位值表示每个数字的值，因此在 4 305 721 中，数字 4 的值是 4 000 000。
Worked example: Write 7 056 309 in words and find the value of the digit 6.
例题：用文字写出 7 056 309 并找出数字 6 的值。
56 309 = seven million fifty-six thousand three hundred and nine; the value of 6 is 6 000.
The answer is easy if you group the digits into millions, thousands and units.
如果把数字按百万、千和个位分组，答案就很容易得出。
2. Factors, Multiples and Primes | 因数、倍数与质数
A factor divides exactly into a number. A multiple is the result of multiplying by an integer. A prime number has exactly two factors: 1 and itself.
因数能整除一个数。倍数是整数相乘的结果。质数只有两个因数：1 和它本身。
Worked example: Find all the factors of 36 and identify which are prime.
例题：找出 36 的所有因数，并指出哪些是质数。
Factors of 36 = {1, 2, 3, 4, 6, 9, 12, 18, 36}; prime factors: 2 and 3.
Remember that 1 is not prime because it has only one factor.
请记住 1 不是质数，因为它只有一个因数。
3. Fractions: Simplifying and Equivalents | 分数：约分与等值分数
To simplify a fraction, divide the numerator and the denominator by their highest common factor. Equivalent fractions show the same value but have different numerators and denominators.
要化简分数，用分子和分母的最大公因数同时除以它们。等值分数表示相同的值，但分子和分母不同。
Worked example: Simplify 18/24 fully.
例题：将 18/24 化为最简分数。
18/24 = (18 ÷ 6)/(24 ÷ 6) = 3/4
The highest common factor of 18 and 24 is 6, so dividing both terms by 6 gives 3/4.
18 和 24 的最大公因数是 6，因此分子分母同时除以 6 得到 3/4。
4. Decimals and Rounding | 小数与四舍五入
When ordering decimals, compare place values from left to right. To round to a given number of decimal places, look at the next digit: if it is 5 or more, round up.
比较小数时，从左到右比较位值。将小数四舍五入到指定小数位时，看下一位数字：如果是 5 或更大，则进位。
Worked example: Round 4.276 to 2 decimal places.
例题：将 4.276 四舍五入到两位小数。
4.276 ≈ 4.28 (to 2 d.p.)
The third decimal place is 6, which is 5 or more, so the second decimal place increases from 7 to 8.
第三位小数是 6，大于或等于 5，因此第二位小数从 7 增加到 8。
5. Percentages and Conversions | 百分数与转换
Percent means out of 100. To convert a fraction to a percentage, first change it to a decimal, then multiply by 100.
百分数表示每一百份中的多少。要将分数转换为百分数，先把它化为小数，再乘以 100。
Worked example: Write 3/5 as a percentage.
例题：将 3/5 写成百分数。
3/5 = 0.6 = 60%
5 × 20 = 100, you can also multiply the numerator by 20: 3 × 20 = 60, so 3/5 = 60/100 = 60%.
因为 5 × 20 = 100，也可以将分子乘以 20：3 × 20 = 60，所以 3/5 = 60/100 = 60%。
6. Algebraic Expressions | 代数表达式
Algebra uses letters to stand for unknown numbers. Like terms have the same letter part, so they can be collected together by adding or subtracting the coefficients.
代数用字母表示未知数。同类项具有相同的字母部分，因此可以通过加减系数来合并。
Worked example: Simplify 3a + 5b + 2a - b.
例题：化简 3a + 5b + 2a - b。
3a + 5b + 2a - b = (3a + 2a) + (5b - b) = 5a + 4b
Group the a terms together and the b terms together, then add or subtract their coefficients.
将含 a 的项和含 b 的项分别分组，然后加减它们的系数。
7. Solving Simple Equations | 解一元一次方程
To solve an equation, find the value of the unknown that makes the statement true. Use inverse operations to undo the steps around the unknown.
要解方程，就要找到使等式成立的未知数的值。使用逆运算来逐步解除未知数周围的运算。
Worked example: Solve x + 5 = 12.
例题：解方程 x + 5 = 12。
x + 5 = 12; x = 12 - 5 = 7
Subtracting 5 from both sides isolates x, so the solution is x = 7.
两边同时减去 5 可以将 x 单独分离出来，因此解为 x = 7。
8. Perimeter, Area and Volume | 周长、面积与体积
ea is the space inside a 2D shape. For a rectangle, perimeter = 2(l + w) and area = l × w.
周长是图形外缘的长度。面积是二维图形内部的空间。对于长方形，周长 = 2(长 + 宽)，面积 = 长 × 宽。
Worked example: A rectangle has length 8 cm and width 5 cm.
例题：一个长方形长 8 厘米，宽 5 厘米。
Perimeter = 2(8 + 5) = 26 cm; Area = 8 × 5 = 40 cm²
Always include the correct units: cm for perimeter, cm² for area, and cm³ for volume.
务必写对单位：周长用 cm，面积用 cm²，体积用 cm³。
9. Angles and Shapes | 角与图形
Angles on a straight line add up to 180°. Angles around a point add up to 360°. Vertically opposite angles are equal.
直线上的角之和为 180°。围绕一点的角之和为 360°。对顶角相等。
Worked example: One angle on a straight line is 65°. Find the missing angle.
例题：直线上一个角为 65°，求另一个角。
Missing angle = 180° - 65° = 115°
Since the two angles lie on a straight line, they must add up to 180°.
因为两个角在同一直线上，它们的和必定为 180°。
10. Coordinates and Graphs | 坐标与图像
Coordinates are written as (x, y), where x is the horizontal position and y is the vertical position. The midpoint of two points is found by averaging the x-values and the y-values.
坐标写为 (x, y)，其中 x 表示水平位置，y 表示垂直位置。两点中点的求法是分别对 x 值和 y 值取平均。
Worked example: Find the midpoint of (2, 5) and (8, 11).
例题：求 (2, 5) 和 (8, 11) 的中点。
Midpoint = ((2 + 8)/2, (5 + 11)/2) = (5, 8)
Add the two x-coordinates and divide by 2, then do the same for the y-coordinates.
将两个 x 坐标相加后除以 2，再对 y 坐标进行同样操作。
11. Averages and Range | 平均数与极差
The mean is the sum of values divided by how many there are. The median is the middle value when data are ordered. The mode is the most common value. The range is the difference between the largest and smallest values.
平均数是一组数值的总和除以数值的个数。中位数是将数据排序后位于中间的值。众数是出现次数最多的值。极差是最大值与最小值之差。
Worked example: Find the mean, median, mode and range of 4, 7, 7, 9, 11.
例题：求 4、7、7、9、11 的平均数、中位数、众数和极差。
Mean = (4 + 7 + 7 + 9 + 11) ÷ 5 = 7.6; Median = 7; Mode = 7; Range = 11 - 4 = 7
Order the data first: 4, 7, 7, 9, 11. The middle number is 7, and 7 appears twice.
先将数据排序：4、7、7、9、11。中间的数是 7，且 7 出现了两次。
12. Checking Answers and Avoiding Errors | 检查答案与避免常见错误
Always check your work by using the inverse operation. For addition, subtract. For subtraction, add. For multiplication, divide.
始终使用逆运算检查作业。加法用减法检查，减法用加法检查，乘法用除法检查。
Worked example: Check whether 42 - 17 = 25 is correct.
例题：检查 42 - 17 = 25 是否正确。
17 + 25 = 42, therefore 42 - 17 = 25 is correct.
Common errors include forgetting to carry in column addition or misreading decimal place value, so write each step clearly.
常见错误包括列竖式加法时忘记进位或看错小数位值，因此每一步都要写清楚。
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