📚 Year 9 SQA Maths: Unit Test Mock Paper Analysis | Year 9 SQA 数学：单元测试模拟卷解析
This mock paper is designed to help Year 9 students in Scotland prepare for their SQA maths unit tests. Each question targets a key skill from the Curriculum for Excellence, and the detailed solutions will strengthen your understanding and boost your confidence. Work through each section carefully — you’ve got this!
这份模拟卷旨在帮助苏格兰 Year 9 学生备战 SQA 数学单元测试。每道题都针对卓越课程中的一项关键技能，详细的解析将加深你的理解并提升信心。请认真完成每一部分——你一定可以的！
1. Simplifying Algebraic Expressions | 代数表达式化简
Simplify the expression: 4a + 3b − 2a + 5b − a.
化简表达式：4a + 3b − 2a + 5b − a。
a. For b we have 3b + 5b = 8b. The simplified expression is a + 8b. Always write 1a as a.
我们将同类项分组：含 a 的项有 4a − 2a − a = (4 − 2 − 1)a = 1a，即 a。含 b 的项有 3b + 5b = 8b。化简后为 a + 8b。记得将 1a 写作 a。
2. Solving Linear Equations | 解一元一次方程
Solve for x: 5x − 7 = 3x + 9.
解方程求 x：5x − 7 = 3x + 9。
e both sides by 2: x = 8. Always check: 5(8) − 7 = 40 − 7 = 33 and 3(8) + 9 = 24 + 9 = 33.
首先，两边同时减去 3x，将含 x 的项移到左边：5x − 3x − 7 = 9，得 2x − 7 = 9。然后，两边同时加 7：2x = 16。最后两边除以 2：x = 8。检验：5(8) − 7 = 40 − 7 = 33，3(8) + 9 = 24 + 9 = 33。
3. Fractions and Decimals | 分数与小数
Calculate ¾ + ½ and express the answer as both an improper fraction and a decimal.
计算 ¾ + ½，并将答案表示为假分数和小数。
o get 1.25. The improper fraction is ⁵⁄₄ (or 1¼ as a mixed number, but the question asks for improper fraction).
找到公分母：4 和 2 的最小公分母是 4。½ 等于 ²⁄₄。因此 ¾ + ²⁄₄ = ⁵⁄₄。转换为小数，5 除以 4 得 1.25。假分数形式为 ⁵⁄₄（带分数形式为 1¼，但题目要求假分数）。
4. Percentages and Percentage Change | 百分数及其变化
A jacket originally costs £80. In a sale, its price is reduced by 15%. What is the sale price? Then, after the sale, the price is increased by 20% of the sale price. Find the final price.
一件夹克原价 £80。促销时降价 15%。促销价是多少？之后，在此基础上又涨价 20%。求最终价格。
15% of £80 = 0.15 × 80 = £12. Sale price = 80 − 12 = £68. Next, 20% of £68 = 0.20 × 68 = £13.60. Final price = 68 + 13.60 = £81.60. Notice the final price is not simply the original plus 5% — percentage increases and decreases are not symmetric.
£80 的 15% = 0.15 × 80 = £12。促销价 = 80 − 12 = £68。接着，£68 的 20% = 0.20 × 68 = £13.60。最终价格 = 68 + 13.60 = £81.60。注意最终价格并不是原价加 5%——百分比的增减并不对称。
5. Area and Perimeter of Composite Shapes | 组合图形的面积与周长
ttached to one of the shorter sides. Find the total area and the perimeter. (Use π ≈ 3.14)
一个组合图形由一个长 10 米、宽 4 米的矩形，以及一个直径为 4 米的半圆组成（半圆连接在矩形的一条短边上）。求总面积和周长。（π 取 3.14）
er, add three sides of the rectangle (10 + 4 + 10 = 24 m) and the semicircular arc length = ½ × 2πr = π × 2 = 6.28 m. Total perimeter = 24 + 6.28 = 30.28 m. Do not include the side where the semicircle is attached.
矩形面积 = 10 × 4 = 40 m²。半圆半径 = 2 m。半圆面积 = ½ × π × 2² = ½ × 3.14 × 4 = 6.28 m²。总面积 = 40 + 6.28 = 46.28 m²。周长方面，加上矩形的三条边（10 + 4 + 10 = 24 m），以及半圆弧长 = ½ × 2πr = π × 2 = 6.28 m。总周长 = 24 + 6.28 = 30.28 m。不要计入半圆与矩形相连的那条边。
6. Interpreting Bar Charts and Pie Charts | 柱状图与饼图的解读
A bar chart shows the number of books read by four friends: Ali 12, Ben 8, Chloe 15, Dana 10. If a pie chart were drawn to represent this data, what angle would represent Chloe’s books?
一个柱状图显示四位朋友阅读的书籍数量：Ali 12，Ben 8，Chloe 15，Dana 10。如果要绘制一个饼图来表示这些数据，表示 Chloe 所读书籍的扇区角度是多少？
whole. So Chloe’s angle = ⅓ × 360° = 120°. Always reduce the fraction before multiplying.
总书籍数 = 12 + 8 + 15 + 10 = 45。Chloe 所占比例 = 15/45 = ⅓。在饼图中，360° 代表整体。因此 Chloe 的扇区角度 = ⅓ × 360° = 120°。计算时先约分会更简便。
7. Ratio and Direct Proportion | 比与正比例
A recipe uses flour, sugar and butter in the ratio 5:2:3. If 400 g of flour is used, how much sugar and butter are needed?
一份食谱中面粉、糖和黄油的比例为 5:2:3。如果使用了 400 克面粉，需要多少糖和黄油？
then 1 part = 400 ÷ 5 = 80 g. Sugar = 2 parts × 80 g = 160 g; Butter = 3 parts × 80 g = 240 g. Check total: 400 + 160 + 240 = 800 g.
比例 5:2:3 意味着每 5 份面粉对应 2 份糖和 3 份黄油。面粉为 5 份，总份数 = 5 + 2 + 3 = 10。如果 5 份为 400 克，则 1 份 = 400 ÷ 5 = 80 克。糖 = 2 份 × 80 克 = 160 克；黄油 = 3 份 × 80 克 = 240 克。检验总重：400 + 160 + 240 = 800 克。
8. Pythagoras’ Theorem | 勾股定理
A right-angled triangle has legs of length 6 cm and 8 cm. Find the length of the hypotenuse, and then calculate the triangle’s perimeter.
一个直角三角形的两条直角边分别为 6 cm 和 8 cm。求斜边长度，并计算三角形的周长。
0 cm. Perimeter = 6 + 8 + 10 = 24 cm. This is a classic 3-4-5 triangle scaled by factor 2.
根据勾股定理：c² = a² + b²，因此 c² = 6² + 8² = 36 + 64 = 100。开方得 c = √100 = 10 cm。周长 = 6 + 8 + 10 = 24 cm。这是一个经典的 3-4-5 直角三角形按比例 2 放大后的结果。
9. Coordinates and Midpoints | 坐标与中点
Points A(2, 5) and B(8, 1) are plotted. Find the coordinates of the midpoint of AB, and then the distance between A and B. (Leave distance in surd form.)
点 A(2, 5) 和 B(8, 1) 已标出。求线段 AB 中点的坐标，并计算 A 与 B 之间的距离。（距离结果保留根式形式。）
ance formula: d = √[(x₂−x₁)² + (y₂−y₁)²] = √[(8−2)² + (1−5)²] = √[6² + (−4)²] = √[36+16] = √52 = 2√13. Always simplify surds.
中点公式：((x₁+x₂)/2, (y₁+y₂)/2)。所以 M = ((2+8)/2, (5+1)/2) = (5, 3)。距离公式：d = √[(x₂−x₁)² + (y₂−y₁)²] = √[(8−2)² + (1−5)²] = √[6² + (−4)²] = √[36+16] = √52 = 2√13。记得化简根式。
10. Word Problems and Mathematical Reasoning | 应用题与数学推理
Liam is thinking of a number. He multiplies it by 3, adds 10, and then divides by 2. The result is 20. Write an equation and find the number.
Liam 想了一个数。他先将它乘以 3，再加 10，然后除以 2，结果是 20。请列出方程并求出这个数。
×10+10)/2 = (30+10)/2 = 40/2 = 20, correct. Forming the correct equation is the key skill.
设这个数为 n。根据操作可得：(3n + 10) ÷ 2 = 20。两边乘以 2：3n + 10 = 40。减去 10：3n = 30。除以 3：n = 10。检验：(3×10+10)/2 = (30+10)/2 = 40/2 = 20，正确。列出正确的方程是关键技能。
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