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.
Quantitative Reasoning
This chapter uses small-number models to practice numerical relationships. Its examples are shared foundations; they do not certify a grade-specific or standardized difficulty level.
Number Analogies
Two pairs are shown: the first is complete, and the second has a missing number. Find the rule that turns the first number into its partner, then apply the same rule to the first number of the second pair. Rules may be one step (multiply by 4) or two steps (add 2, then multiply by 3). Test your rule on the complete pair before using it.
Number Puzzles
An equation has a missing value, often shown inside a shape. Use inverse operations: if ? + 7 = 15, subtract 7. If 4 x ? = 24, divide by 4. Always substitute your answer back into the puzzle to check that both sides balance.
Number Series
Look at how the numbers change. Check differences first: 2, 5, 8, 11 adds 3 each time. If the differences are not constant, check whether the numbers double or halve, or whether two patterns alternate, such as add 4, subtract 1, add 4, subtract 1.
Worked example
Analogy: 3 -> 12, 5 -> 20, 7 -> ? 3 x 4 = 12 and 5 x 4 = 20, so the rule is "multiply by 4." Apply it: 7 x 4 = 28. Check: 28 makes the pair 7 -> 28 consistent with both earlier pairs. Choosing 11 would just add 4, which fits 3 -> 7 but not the completed pairs, so it fails.
Traps that cost marks
- Using addition when the completed pair shows multiplication.
- Assuming every series adds the same number.
- Missing a two-step rule and stopping after the first operation.
- Solving for the wrong side of the equation in a puzzle.
- Not checking the answer in the original equation.
Quantitative Reasoning
What the Quantitative battery measures
These exercises develop numerical reasoning through analogies, series and missing-value puzzles without claiming to reproduce a particular current form. The items are designed so that arithmetic alone is not enough: you must infer the rule that generates the numbers and apply it.
Core rules and concepts
- Number Analogies: Two complete pairs establish a rule; apply the same rule to the third pair. The rule can be one step (add 4, multiply by 3) or two steps (add 2, then double). Check the rule on both complete pairs before using it.
- Number Series: A sequence follows a consistent pattern. First look at differences; if they are not constant, look at ratios, alternating patterns (two interleaved sequences), square numbers, or differences that themselves change in a pattern.
- Number Puzzles: An equation has one missing element-a number or an operation sign. Use inverse operations or fact families, and check whether the blank is in the position of an operator.
Worked example
6 → 18, 4 → 12, 9 → ? Test the first pair: 6 × 3 = 18. Test the second: 4 × 3 = 12. The rule is “multiply by 3.” Apply it to 9: 9 × 3 = 27. Answer: 27. If the choices include 12 or 18, they are distractors built from the input numbers or a partial rule.
Traps that cost marks
- Assuming the first operation you notice is the whole rule; always verify it on the second complete pair.
- In Number Series, stopping after first differences when the sequence actually uses second-order differences, doubling differences, or two alternating rules.
- In Number Puzzles, solving for a missing number when the blank is an operation sign-check the form of the answer choices.
- Rushing because the numbers look simple; distractors often come from common calculation slips or from applying the rule only once.
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.