📚 Grade 8 Math Test Answers | 八年级数学测试答案解析
This article provides a complete answer key and step-by-step explanations for a Grade 8 mathematics practice test. It covers algebraic expressions, linear equations, slope, the Pythagorean theorem, scientific notation, ratios, inequalities, volume, statistics, and probability. Use the explanations to see where marks are earned and where common errors usually happen.
本文提供一套八年级数学模拟测试的完整答案与逐步解析，涵盖代数式、一元一次方程、斜率、勾股定理、科学计数法、比例、不等式、体积、统计和概率等核心考点。通过解析可以明确得分点，并发现常见错误通常出现在哪里。
1. How to Use This Answer Key | 如何使用本答案与解析
Use this key as a model answer, not just as a final result. In Grade 8 math tests, method marks are often awarded for correct steps, even when the final number is wrong.
请把这份答案当作示范答案，而不只是最终结果。在八年级数学测试中，只要步骤正确，即使最后数字算错，通常也能拿到方法分。
Each section below shows the question, the answer, the working, and an exam tip. The working shows the order of operations, the formula used, and the unit required.
下面每一节都包含题目、答案、解题过程与考试提示。解题过程会展示运算顺序、所用公式以及必须写出的单位。
When you review, cover the answer first and try the question again. Then compare your working line by line with the model solution.
复习时，先遮住答案，自己重新做一遍。然后再把自己的步骤与示范解答逐行对照。
2. Question 1 – Simplifying Algebraic Expressions | 第1题 – 化简代数式
Question: Simplify 3x + 5x − 2x + 7.
题目：化简 3x + 5x − 2x + 7。
Answer: 6x + 7.
答案：6x + 7。
Explanation: 3 + 5 − 2 = 6. The x-term becomes 6x. The constant 7 has no like term, so it stays as +7.
解析：合并含 x 的同类项。将系数相加减：3 + 5 − 2 = 6，所以含 x 的项变为 6x。常数 7 没有同类项，因此保持不变。
3x + 5x − 2x + 7 = 6x + 7
Exam tip: Only terms with exactly the same variable part can be combined. Do not add x-terms to constant terms.
考试提示：只有变量部分完全相同的项才能合并。不要把含 x 的项与常数项相加。
Common error: A frequent mistake is writing 13x or 6x + 7x. The constant 7 is not a like term with x, so it cannot be combined with 6x.
常见错误：常见错误是写成 13x 或 6x + 7x。常数 7 与 x 不是同类项，不能和 6x 合并。
3. Question 2 – Solving a Linear Equation | 第2题 – 解一元一次方程
Question: Solve 4x − 7 = 13.
题目：解方程 4x − 7 = 13。
Answer: x = 5.
答案：x = 5。
Explanation: = 20. Then divide both sides by 4: x = 5. Check by substitution: 4 × 5 − 7 = 20 − 7 = 13.
解析：两边同时加 7，抵消减 7：4x = 20。然后两边同时除以 4：x = 5。代入检验：4 × 5 − 7 = 20 − 7 = 13。
4x − 7 = 13 → 4x = 20 → x = 5
Exam tip: Keep the equation balanced. Whatever operation you do to one side, do the same to the other side.
考试提示：保持方程两边平衡。对一边做什么运算，就要对另一边做同样的运算。
Common error: Some students subtract 7 instead of adding 7, or divide only one term by 4. Both mistakes break the balance of the equation.
常见错误：有些同学会减 7 而不是加 7，或者只把其中一项除以 4。这两种错误都会破坏方程平衡。
4. Question 3 – Slope of a Line | 第3题 – 直线的斜率
Question: A line passes through the points (2, 5) and (6, 13). Find the slope of the line.
题目：一条直线经过点 (2, 5) 和 (6, 13)。求这条直线的斜率。
Answer: m = 2.
答案：m = 2。
Explanation: ) and (x₂, y₂) = (6, 13). Then m = (y₂ − y₁) / (x₂ − x₁) = (13 − 5) / (6 − 2) = 8 / 4 = 2.
解析：使用斜率公式。设 (x₁, y₁) = (2, 5)，(x₂, y₂) = (6, 13)。则 m = (y₂ − y₁) / (x₂ − x₁) = (13 − 5) / (6 − 2) = 8 / 4 = 2。
m = (y₂ − y₁) / (x₂ − x₁) = 8 / 4 = 2
Interpretation: The slope is positive, so the line rises from left to right. For every 1 unit moved to the right, the line goes up 2 units.
解释：斜率为正，因此直线从左向右上升。每向右移动 1 个单位，直线向上移动 2 个单位。
Exam tip: Subtract the coordinates in the same order. If you use (6, 13) first, then use (2, 5) first as well.
考试提示：相减的顺序必须一致。如果你先用 (6, 13)，那么后面也要先用 (2, 5)。
5. Question 4 – Pythagorean Theorem | 第4题 – 勾股定理
Question: A right triangle has legs of length 9 cm and 12 cm. Find the length of the hypotenuse.
题目：一个直角三角形的两条直角边分别为 9 cm 和 12 cm。求斜边长度。
Answer: 15 cm.
答案：15 cm。
Explanation: ypotenuse. Substitute the values: 9² + 12² = c², so 81 + 144 = 225. Then c = √225 = 15 cm.
解析：使用勾股定理 a² + b² = c²，其中 c 为斜边。代入数值：9² + 12² = c²，所以 81 + 144 = 225。因此 c = √225 = 15 cm。
a² + b² = c² → 9² + 12² = 15²
Exam tip: The hypotenuse is always the longest side and is opposite the right angle. Write the unit, cm, in your final answer.
考试提示：斜边永远是最长边，并且与直角相对。最终答案要写单位 cm。
Common error: A common mistake is adding 9 and 12 first and then squaring. The square must be applied before adding.
常见错误：常见错误是先把 9 和 12 相加再平方。必须先平方，再相加。
6. Question 5 – Scientific Notation | 第5题 – 科学计数法
Question: Write 0.000047 in scientific notation.
题目：将 0.000047 写成科学计数法。
Answer: 4.7 × 10⁻⁵.
答案：4.7 × 10⁻⁵。
Explanation: Move the decimal point to the right until the number is between 1 and 10. Moving from 0.000047 to 4.7 requires 5 places. Because the original number is less than 1, the exponent is negative.
解析：将小数点向右移动，直到数字在 1 和 10 之间。从 0.000047 移到 4.7 需要移动 5 位。因为原数小于 1，所以指数为负。
0.000047 = 4.7 × 10⁻⁵
Extra example: 3.2 × 10⁴ = 32000. Also, (2 × 10³) × (3 × 10⁻²) = 6 × 10¹ = 60.
补充示例：3.2 × 10⁴ = 32000。另外，(2 × 10³) × (3 × 10⁻²) = 6 × 10¹ = 60。
Exam tip: ast 1 and less than 10. So 47 × 10⁻⁶ is not correct, even though it equals the same value.
考试提示：系数必须大于或等于 1，且小于 10。所以 47 × 10⁻⁶ 虽然数值相同，但不是标准科学计数法。
7. Question 6 – Ratios and Proportions | 第6题 – 比例与比例式
Question: A recipe uses flour and sugar in the ratio 5 : 2. If a baker uses 15 cups of flour, how many cups of sugar are needed?
题目：一份食谱中面粉与糖的比例为 5 : 2。如果烘焙师使用 15 杯面粉，需要多少杯糖？
Answer: 6 cups of sugar.
答案：6 杯糖。
Explanation: up the proportion 5 / 2 = 15 / x. Cross multiply: 5x = 30. Divide both sides by 5: x = 6.
解析：列出比例式 5 / 2 = 15 / x。交叉相乘：5x = 30。两边同时除以 5：x = 6。
5 / 2 = 15 / x → 5x = 30 → x = 6
Exam tip: Keep the same units in the same position. Flour is on top in both fractions, and sugar is on the bottom in both fractions.
考试提示：相同单位要放在相同位置。两个分数中，面粉都在分子，糖都在分母。
Common error: Do not confuse part-to-part ratios with part-to-whole ratios. Here 5 : 2 compares flour to sugar, not flour to the total mixture.
常见错误：不要把部分与部分之比和部分与整体之比混淆。这里的 5 : 2 是面粉与糖之比，不是面粉与总量之比。
8. Question 7 – Solving Inequalities | 第7题 – 解不等式
Question: Solve −3x + 4 > 19.
题目：解不等式 −3x + 4 > 19。
Answer: x < −5.
答案：x < −5。
Explanation: Subtract 4 from both sides: −3x > 15. Then divide both sides by −3. Because you are dividing by a negative number, reverse the inequality sign: x < −5.
解析：两边同时减 4：−3x > 15。然后两边同时除以 −3。因为除以负数，所以要改变不等号方向：x < −5。
−3x + 4 > 19 → −3x > 15 → x < −5
Check: Test x = −6: −3 × (−6) + 4 = 18 + 4 = 22, and 22 > 19 is true.
检验：取 x = −6：−3 × (−6) + 4 = 18 + 4 = 22，而 22 > 19 成立。
Exam tip: The inequality sign flips only when you multiply or divide by a negative number. Adding or subtracting a number does not flip it.
考试提示：只有当乘以或除以负数时，不等号方向才改变。加或减一个数不会改变方向。
9. Question 8 – Volume of a Cylinder | 第8题 – 圆柱体积
Question: A cylinder has radius 4 cm and height 10 cm. Find its volume in terms of π.
题目：一个圆柱的半径为 4 cm，高为 10 cm。用 π 表示它的体积。
Answer: 160π cm³.
答案：160π cm³。
Explanation: e formula V = πr²h. Substitute r = 4 and h = 10: V = π × 4² × 10 = π × 16 × 10 = 160π cm³.
解析：使用公式 V = πr²h。代入 r = 4，h = 10：V = π × 4² × 10 = π × 16 × 10 = 160π cm³。
V = πr²h = π × 4² × 10 = 160π cm³
