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Test statements separately, then together, using proof or two allowed cases to decide sufficiency.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
Test statements separately, then together, using proof or two allowed cases to decide sufficiency.
Read conditional statements precisely and keep ratios, percentages and units distinct. A fraction compares a part with a named whole; an implication states a relationship rather than its converse. Before combining quantities, check that they refer to the same population and time period. If an arithmetic operation is uncertain, use the linked mathematics lesson to repair it before adding a timer. The purpose of this lesson is to decide what follows from supplied information, not to test outside knowledge of the scenario.
| Term | What it means |
|---|---|
| Sufficient | Every allowed case gives the same requested answer. |
| Counterexample pair | Two allowed cases producing different answers to the target. |
| Intersection | Cases satisfying the stem and both statements together. |
A data sufficiency question asks whether the requested answer is determined. It does not necessarily ask you to calculate that answer. Separate the question stem, which applies throughout, from statements (1) and (2), which initially belong to different information sets. Evaluate the stem plus statement (1), then forget statement (1) while evaluating the stem plus statement (2). Combine them only after those separate judgments. A common error is to carry a value found from statement (1) into the test of statement (2); that changes the information being assessed.
There are two important targets. A value question is sufficient when every allowed case gives the same requested value. A yes/no question is sufficient when every allowed case gives the same yes/no answer. The underlying variables can vary while the requested answer remains fixed. If x is known to be negative, the answer to 'Is x greater than ten?' is no for every allowed x. You do not need its precise value. Conversely, knowing x squared is sixteen does not determine whether x is positive: both four and negative four satisfy that statement.
Treat a counterexample as a complete allowed case, not an isolated number. It must satisfy the stem and whichever statements are currently being tested. To prove insufficiency, exhibit two such cases that produce different requested answers. To establish sufficiency, show why variation cannot change the target. Testing two convenient examples that agree is not a proof that all allowed cases agree. The five fixed answer choices encode the outcomes of the separate and combined tests; their order is part of this response format.
Another way: steps
Test statements separately, then together, using proof or two allowed cases to decide sufficiency.
Draw three small spaces headed 1 alone, 2 alone, and together. Write the stem's restrictions above them so that they remain visible. For example, if x and y are positive integers with x+y=12, the integer and positivity conditions apply in every space. Under x>y, the pairs (7,5) and (8,4) both work, so x is not fixed. You cannot use a negative y as a counterexample because it violates the stem. You also cannot assume that two named variables are different unless that is stated. Record enough detail that another reader can verify each allowed case. This ledger prevents an answer obtained from both statements from being mistakenly attributed to one statement alone.
Suppose a batch contains only red and blue parts. Knowing that blue parts are twice as numerous as red parts fixes the red fraction at one third, even though the total number is unknown. A target involving a ratio can therefore be determined without determining either count. Likewise, if a+b=20, the sum is known even while a and b vary. Read the requested expression before trying to solve every unknown. Extra arithmetic can obscure a simple invariant. On the other hand, a median alone usually does not determine a mean: values far from the center can change the total without moving the middle observation.
Zero, negative numbers, fractions and equality cases often distinguish sufficiency from a plausible guess. 'Nonnegative' includes zero; 'positive' does not. An integer less than two can be zero or one if nonnegative, while a positive real less than two has infinitely many possibilities. An inequality with an inclusive endpoint can determine a yes/no answer exactly at a policy threshold. Squaring can remove a sign; division by an expression can exclude zero; a square root convention can restore a restriction. State each transformation with its domain. When using a verbal rule such as 'approved exactly when documented and insured,' translate both directions rather than treating a necessary condition as sufficient by itself.
If each statement independently fixes the target, the correct classification is each alone. If neither alone works but their intersection fixes the target, choose together only. If the intersection still permits different target values, choose neither. The paired statements in a well-formed question must describe compatible possibilities; a contradiction is not a shortcut to claiming sufficiency. You may calculate enough to establish uniqueness, but a full numerical solution is unnecessary when the structure already proves it. During review, identify whether the error arose from mixing statements, overlooking an allowed case, or answering a different target. Those errors call for different repairs.
Consider the stem 'x is a real number; is x greater than four?' Statement (1) says x is negative. This is sufficient because every negative real is at most zero and therefore below four. It would be a mistake to reject the statement because x might be negative one or negative twenty. Those are different values but they produce the same answer to the actual question. Now change the target to 'What is x?' The same statement becomes insufficient. This contrast isolates the role of the target: sufficiency is a relationship between information and a question, not an intrinsic property of a statement.
A second contrast involves a ratio. If revenue is twice cost and cost is positive, profit as a fraction of revenue is one half, regardless of the currency amounts. If cost could be zero, a ratio might be undefined, so the domain still matters. Write the target expression, substitute the relationship, and cancel only factors known to be nonzero. This gives a proof for all allowed values rather than a collection of examples. In a yes/no problem, two answers are possible; in a numerical problem, there may be many. In both cases, the same discipline applies: vary the free information and watch the requested target.
An analyst evaluating an incomplete report faces the same distinction between what is known and what can be inferred. Suppose sales revenue is 600 and the number of sales is a positive integer below twelve. Revenue alone leaves the average open; count alone leaves it open too. Together the average exceeds fifty. The exact count is unnecessary because every allowed denominator produces the same yes/no answer. If the question instead asks for the precise average, the information remains insufficient.
In the timed section, write the target and domain before testing statements. Mark a statement sufficient only with a proof or a genuinely fixed expression. Mark it insufficient with two verified cases. Select the fixed response label after this analysis, not before. For a missed item, retain your attempted cases and check which restriction each satisfies. Revisit a different target after a gap: a method that works only when you remember one numerical answer has not yet transferred.
Do not infer a result from an attractive label or the order of displayed alternatives. A complete answer must preserve every stated restriction and the scope of the requested conclusion. When a tempting response is wrong, identify the exact missing dependency or violated condition; merely saying that it feels too strong does not explain the error.
Read the target and its restrictions
What is x?
(1) x + 7 = 12. (2) x is positive.
Test statements separately, then together, using proof or two allowed cases to decide sufficiency.
Derive the deciding relationship
The first statement gives x=5. Positivity permits 1 or 5, so the second alone does not fix x.
The first statement gives x=5. Positivity permits 1 or 5, so the second alone does not fix x.
Compare competing responses
Statement (2) alone is sufficient; statement (1) alone is not. Together the statements are sufficient, but neither alone is sufficient. Each statement alone is sufficient. Even together the statements are not sufficient.
The first statement gives x=5. Positivity permits 1 or 5, so the second alone does not fix x.
Select and verify the complete response
Statement (1) alone is sufficient; statement (2) alone is not.
The first statement gives x=5. Positivity permits 1 or 5, so the second alone does not fix x.
Read the target and its restrictions
For real x, what is x?
(1) 2x=18. (2) x-4=5.
Test statements separately, then together, using proof or two allowed cases to decide sufficiency.
Derive the deciding relationship
Each equation independently gives x=9. Do not combine them before evaluating each alone.
Each equation independently gives x=9. Do not combine them before evaluating each alone.
Check the information boundary
Keep an information ledger
Draw three small spaces headed 1 alone, 2 alone, and together. Write the stem's restrictions above them so that they remain visible. For example, if x and y are positive integers with x+y=12, the integer and positivity conditions apply in every space. Under x>y, the pairs (7,5) and (8,4) both work, so x is not fixed. You cannot use a negative y as a counterexample because it violates the stem. You also cannot assume that two named variables are different unless that is stated. Record enough detail that another reader can verify each allowed case. This ledger prevents an answer obtained from both statements from being mistakenly attributed to one statement alone.
Compare competing responses
Statement (1) alone is sufficient; statement (2) alone is not. Statement (2) alone is sufficient; statement (1) alone is not. Together the statements are sufficient, but neither alone is sufficient. Even together the statements are not sufficient.
Each equation independently gives x=9. Do not combine them before evaluating each alone.
Select and verify the complete response
Each statement alone is sufficient.
Each equation independently gives x=9. Do not combine them before evaluating each alone.
Read the target and its restrictions
What is the mean of five numbers?
(1) Their median is 10. (2) Their sum is 60.
Test statements separately, then together, using proof or two allowed cases to decide sufficiency.
Derive the deciding relationship
A median does not fix a sum. The second gives mean 60/5=12.
A median does not fix a sum. The second gives mean 60/5=12.
Check the information boundary
Ask whether the target, rather than everything, is fixed
Suppose a batch contains only red and blue parts. Knowing that blue parts are twice as numerous as red parts fixes the red fraction at one third, even though the total number is unknown. A target involving a ratio can therefore be determined without determining either count. Likewise, if a+b=20, the sum is known even while a and b vary. Read the requested expression before trying to solve every unknown. Extra arithmetic can obscure a simple invariant. On the other hand, a median alone usually does not determine a mean: values far from the center can change the total without moving the middle observation.
Compare competing responses
Statement (1) alone is sufficient; statement (2) alone is not. Together the statements are sufficient, but neither alone is sufficient. Each statement alone is sufficient. Even together the statements are not sufficient.
A median does not fix a sum. The second gives mean 60/5=12.
Select and verify the complete response
Statement (2) alone is sufficient; statement (1) alone is not.
A median does not fix a sum. The second gives mean 60/5=12.
Verify the target under the original conditions
Statement (2) alone is sufficient; statement (1) alone is not.
Test statements separately, then together, using proof or two allowed cases to decide sufficiency. A median does not fix a sum. The second gives mean 60/5=12.
Read the target and its restrictions
What is the mean of five numbers?
(1) Their median is 10. (2) Their sum is 60.
Test statements separately, then together, using proof or two allowed cases to decide sufficiency.
Derive the deciding relationship
A median does not fix a sum. The second gives mean 60/5=12.
A median does not fix a sum. The second gives mean 60/5=12.
Check the information boundary
What is x? (1) x + 7 = 12. (2) x is positive.
For real x, is x positive? (1) x squared = 16. (2) x cubed = -64.
For real x, what is x? (1) 2x=18. (2) x-4=5.
For positive integers x and y, what is x? (1) x+y=12. (2) x>y.
For integer n, is n even? (1) n is divisible by 6. (2) n is divisible by 3.
For real x, what is x? (1) x squared=25. (2) x>0.
For real x, is x greater than 10? (1) x<0. (2) x<-4.
What is the mean of five numbers? (1) Their median is 10. (2) Their sum is 60.
A batch has red and blue parts only. What fraction are red? (1) There are 30 red parts. (2) There are twice as many blue parts as red parts.
A price rises and then falls. Is the final price below the original? (1) The rise is 20 percent. (2) The fall is 20 percent.
What is x+y? (1) x=8. (2) y=-3.
Answer three reserved questions without hints. These cases are separate from lesson practice.
Is a project's revenue greater than its cost? (1) Revenue is 120 units. (2) Cost is less than 120 units.
Is the integer n divisible by 12? (1) n is divisible by 3. (2) n is divisible by 4.
For nonnegative integer n, is n positive? (1) n is even. (2) n<2.
After a study gap, recover the method without notes, then solve a different case. Identify the exact condition that decides the answer before comparing feedback.
15. Complete the final evidence check, step 3
Ask whether the target, rather than everything, is fixed
Suppose a batch contains only red and blue parts. Knowing that blue parts are twice as numerous as red parts fixes the red fraction at one third, even though the total number is unknown. A target involving a ratio can therefore be determined without determining either count. Likewise, if a+b=20, the sum is known even while a and b vary. Read the requested expression before trying to solve every unknown. Extra arithmetic can obscure a simple invariant. On the other hand, a median alone usually does not determine a mean: values far from the center can change the total without moving the middle observation.