So sánh định lượng và suy luận số học
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| quantitative comparison/ˈkwɒntɪteɪtɪv kəmˈpærɪsn/ | A Quant question type asking which of two quantities is greater, or whether it cannot be determined. |
| counterexample/ˈkaʊntəreɡzæmpl/ | A specific example that shows a general claim is false. |
Which is bigger — always?
- Quantity A: $x^2$. Quantity B: $x$. Is A greater? For $x = 2$, yes; for $x = \tfrac{1}{2}$, no; for $x = 1$, they are equal.
- The answer is "cannot be determined". Quantitative comparison 数量比较 asks about all allowed values, not one example.
The four answers
- (A) A is greater, (B) B is greater, (C) equal, (D) relationship cannot be determined from the information.
- A, B or C must hold for every allowed value; if two different cases give different results, choose D.
If two allowed values give different results, the answer is that the relationship cannot be ____.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Choice D.
Test special values
- Try numbers that behave differently: 0, 1, negatives, fractions between 0 and 1, and large numbers.
- Respect the conditions given: if $x > 1$, then $x^2 > x$ always → A.
$n$ is a positive integer. A: the remainder when $n^2$ is divided by 4. B: 2. Squares give remainder 0 or 1 when divided by 4, so A is always less → B.
Is x² bigger than x?
Test special values in a quantitative comparison.
Quantity A: $x^2$. Quantity B: $x$. (No other information.)
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
x = 2 gives A > B; x = 1/2 gives B > A.
$x > 1$. Quantity A: $x^2$. Quantity B: $x$.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
For x > 1, x² > x always.
$n$ is a positive integer. Quantity A: the remainder when $n^2$ is divided by 4. Quantity B: 2.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Squares leave remainder 0 or 1 when divided by 4.
Number reasoning tools
- Prime factorisation for divisibility; ratios and percentages as multipliers; powers of a fraction shrink: $(\tfrac{1}{2})^3 < \tfrac{1}{2}$.
- Simplify both quantities the same way (add or subtract the same number); do not multiply by a variable that might be negative.
Common traps
- Testing only positive whole numbers.
- Multiplying or dividing both quantities by an expression whose sign is unknown.
You should not divide both quantities by $x$ when $x$ could be negative.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. Dividing by a negative number reverses an inequality, so with unknown sign the comparison could flip.
Recap
- A, B or C must hold for every allowed value; one counterexample 反例 means D.
- Test 0, 1, negatives, fractions and large values within the stated conditions.
- quantitative comparison: A Quant question type asking which of two quantities is greater, or whether it cannot be determined.