Quantitative comparison and number reasoning · 定量比较与数字推理
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| constraint/kənˈstreɪnt/ | 约束条件 | yuē shù tiáo jiàn |
| indeterminate/ˌɪndɪˈtɜːmɪnət/ | 不能确定的 | bù néng què dìng de |
A decision before an answer
- When x²=9, comparing x with zero has two possible answers. One convenient substitution can hide the other.
- Your goal: Compare quantities across all allowed values.
答案前的判断
- 当 x²=9 时，将 x 与零比较会有两种可能的结果。一种简便的代入法可能会掩盖另一种情况。
- 你的目标：对所有允许范围内的数值进行数量比较。
Read the relationship
- Quantitative Comparison has four fixed relationships: A greater, B greater, equal, or cannot determine.
- Use arithmetic, ratios, divisibility and powers.
阅读关系
- 数量比较题有四种固定的关系：A 较大、B 较大、相等或无法确定。
- 运用算术、比率、整除性和幂运算。
x²=4; compare x and 1.
x=2 makes A greater; x=−2 makes B greater.
Use the defining rule
- Check all constraints. A variable can be negative, zero or fractional unless the stem restricts it.
- Recognise when the relationship cannot be determined.
运用定义规则
- 检查所有约束条件。除非题干加以限制，否则变量可以是负数、零或分数。
- 识别何时无法确定两者关系。
x>0; compare x² and 0.
The square of a positive nonzero number is positive.
Check the conditions
- Use strategic substitutions to disprove a fixed relation. They can show indeterminacy but a few examples do not prove universality.
- Recognise when the relationship cannot be determined.
Given x²=9, x can be 3 or −3. Quantity A=x; B=0. A is greater for 3 and smaller for −3, so the relationship cannot be determined. If x is positive, only 3 is allowed and A is greater.
检查条件
- 运用策略性代入来反驳固定关系。这可以证明不确定性，但仅凭几个例子并不能证明普遍性。
- 识别何时无法确定两者关系。
已知 x²=9，x 可以是 3 或 −3。量 A=x；B=0。对于 3，A 较大；对于 −3，A 较小，因此关系无法确定。如果 x 为正数，则只允许取 3，此时 A 较大。
The greatest common divisor of 18 and 24 is ____.
Both divide by 6; no greater integer divides both.
Apply the task format
- Simplify both quantities without silently dividing by a value that could be zero or negative.
- Recognise when the relationship cannot be determined.
Figures are not necessarily drawn to scale. Given labels and constraints carry the evidence.
应用题目格式
- 简化两边的数量，但不要无声地除以可能为零或负数的值。
- 识别何时无法确定两者关系。
图形未必按比例绘制。标签和约束条件才是提供证据的依据。
Which answer fits this case? · 哪个答案符合此案例？
Compare quantities across all allowed values · 在所有允许取值范围内比较数量
A real variable is always a positive integer unless otherwise stated.
Real values can be negative, zero or fractional.
Keep the distinctions
- constraint 约束条件 — A condition limiting allowed values.
- indeterminate 不能确定的 — Not fixed by the given information.
- Compare quantities across all allowed values.
- Use arithmetic, ratios, divisibility and powers.
- Recognise when the relationship cannot be determined.
保持区别
- constraint 约束条件 — 限制允许取值的条件。
- indeterminate 不能确定的 — 未由给定信息确定。
- 在所有允许取值范围内比较数量。
- 运用算术、比率、整除性和幂运算。
- 识别何时无法确定两者关系。
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.