Numerical analogies, series, puzzles and justified comparisons
Quantitative Reasoning: Analogies, Series, and Puzzles
Independent quantitative-reasoning practice uses numerical analogies, series, comparisons and balanced puzzles. Infer a simple consistent intended rule from all supplied examples, state it clearly, and test the answer. No official score distribution, age norm, form-equivalence or timing rule is asserted here.
The three item types
- Number analogy: two pairs governed by the same rule. Find the rule on the complete pair, then apply it to the incomplete one. Rules may be one step (×3) or two (×2 then +1).
- Number series: a list following a rule; supply the next or missing term.
- Number puzzle: a short statement or equation with a missing value; translate it into an equation, or back-solve from the answer choices.
Rules that decide the answer
- Test the simplest rule first on all terms; if it breaks, escalate to a two-step rule.
- Write the differences between consecutive terms. If the differences change, look at how they change.
- When terms alternate high/low, check for two interleaved sequences.
- Back-solving from choices is often faster and safer than algebra under time pressure.
Worked example
3, 6, 11, 18, 27, ? Differences: 3, 5, 7, 9 - increasing by 2 each time, so the next difference is 11 and the term is 38. Confirm with a second rule: each term equals n² + 2 for n = 1, 2, 3, 4, 5, giving 3, 6, 11, 18, 27; the sixth term is 36 + 2 = 38. These are consistent descriptions of the intended simple pattern, not proof that a finite list admits no other mathematical continuation.
Traps
- Committing to a constant difference after seeing only one pair of terms.
- Off-by-one counting: the number of integers from 12 to 30 inclusive is 30 − 12 + 1 = 19, not 18.
- Reversing a comparison: "how many more than" is not "how many fewer than".
- Dropping a remainder in division items or ignoring units.
- Order of operations in puzzles: 4 + 3 × 5 = 19, not 35.
Proportions, percentages and units
Compute 25 percent of 240 as 0.25 times 240, giving 60. For a price of 80 raised by 15 percent, calculate 80 times 1.15=92. The reverse problem uses the original base: if a 20 percent discount leaves 64, solve 0.80 x=64 and obtain 80. Adding 20 percent to 64 would use the wrong base.
For constant-rate work, retain the scenario assumptions. If five identical workers complete a task in six days and output scales with worker-days, the task requires 30 worker-days. Three such workers then need 10 days. This model assumes equal productivity and no coordination loss; it is not a prediction about every real workplace.
Write a unit beside every result. A distance of 600 metres at a 1:1000 scale becomes 0.6 metres on the drawing, or 60 centimetres. A percentage and a percentage-point change answer different questions, just as total cost and cost per item do. Check whether the requested answer is a number, rate, duration or share before selecting an option.
Evidence, sufficiency and quantitative comparison
Information is sufficient when it determines the answer to the question being asked, within the stated domain. A statement need not identify an exact value to answer a yes-or-no question consistently. Conversely, finding one example that gives a yes does not establish that every permitted example does. First rewrite the target question precisely; then test each supplied statement independently before combining them.
For example, if x is a real number and x²=49, its value could be 7 or-7. This is insufficient to determine x uniquely, but it is sufficient to answer whether x is nonzero. Adding x>0 determines x=7. The question matters as much as the numerical equation. Do not import a second statement when checking the first alone, and do not assume positivity, integrality or distinctness unless it is actually given.
Tables and charts introduce source discipline. Read the label, unit, date or period, denominator and any note before comparing values. A total in thousands is not directly comparable to a single-unit count without conversion. A percentage based on all responses can differ from the percentage based on respondents to one item. A share of a total, a percent increase from a baseline and a percentage-point difference answer different questions. State which one the problem requests.
In a multi-source model, assign each fact to its source and time. An original estimate and a revised estimate are not necessarily contradictory: they may answer different versions of the same planning question. If three people cost 120 per hour and two cost 90 per hour, the combined hourly rate is 540. At 200 hours the modeled labor cost is 108000; at 250 hours it is 135000. A budget note cannot silently replace the requested original duration. These are fictional calculations, not labor-market facts.
Finally, distinguish consistency from proof. A rule may fit every supplied term of a finite sequence without being the only mathematical continuation possible. Such exercises ordinarily ask for the intended simple pattern represented by the available choices. Explain that rule and test it on all supplied pairs. In an argument question, similarly identify the premise, proposed conclusion and any missing assumption. A plausible conclusion is not automatically entailed. A counterexample can disprove a universal claim, while several supporting examples alone do not establish one.