If the difference of two numbers is greater than the sum of the numbers, then

Correct — C, (c) At least one of the numbers is negative.

THE STEP THAT DECIDES THIS QUESTION is to turn the condition into an inequality and cancel. Nothing about "difference" and "sum" needs to stay in words.

Take the two numbers as x and y and read the difference in the ordinary ordered sense, x − y. The stem says

x − y > x + y

Subtract x from both sides, and the larger term disappears at once:

−y > y 0 > 2y y < 0

So the condition "the difference exceeds the sum" is EXACTLY the condition that the number being subtracted is negative. Nothing at all is required of the other number, which may be positive, negative or zero. At least one of the two is therefore negative, and that is option (c).

READ THE DIFFERENCE AS A MAGNITUDE INSTEAD and the result is even cleaner. Let M be the larger of the two numbers and m the smaller, so that the difference is M − m and the sum is M + m. The stem says

M − m > M + m −m > m m < 0

that is, the SMALLER of the two numbers is negative. And "the smaller of the two is negative" says precisely the same thing as "at least one of the two is negative". Under this reading option (c) is not merely implied by the condition; it is equivalent to it. Whichever way a candidate reads the word "difference", the answer is the same, which is a mark of a well-set item.

Notice what the strict inequality rules out. If the smaller number were zero, the difference and the sum would be equal, not greater — so zero is excluded and the number really must be below it.

NOW TEST THE OTHER CLAIMS, because on a question of this shape the work is done by counterexamples. The stem asks what MUST follow, so a statement is defeated by a single pair of numbers that satisfies the condition and contradicts the statement.

Take 5 and −1. The difference is 5 − (−1) = 6 and the sum is 4, so the condition holds. Here just one number is negative, which kills the claim that BOTH must be. Take −3 and −5. The difference is −3 − (−5) = 2 and the sum is −8, so the condition holds again. Here both numbers are negative, which kills the claim that EXACTLY ONE must be.

Both of the stronger claims fall to one example each, while the weaker claim survives every example because it is a theorem. That is the general shape of these items: among several statements about the same thing, the one that asserts least is the hardest to falsify, and here it happens to be provable.

A WORD ABOUT OPTION (d). This is the only escape option printed anywhere in this booklet, and an escape option is not to be dismissed on sight — it is genuinely the answer whenever all three substantive options fail. It fails here, and it fails for a definite reason: option (c) has been shown to hold for every pair of numbers satisfying the stem's condition, so there is something on the list that is true and "None of the above" cannot be. The way to handle such an option is always the same — settle the other three on their own merits first, and let the escape option be decided by what is left.

Why the others are wrong

• (a)Both the numbers are negative — This asserts too much. The condition in the stem constrains only ONE of the two numbers — the one being subtracted, or equivalently the smaller of the pair — and says nothing whatever about the other. A single pair of numbers disposes of the claim: take 5 and −1, where the difference is 5 − (−1) = 6 and the sum is 5 + (−1) = 4, so the difference does exceed the sum while one of the numbers is comfortably positive. Take 100 and −1 and the point is starker still. The option is attractive because a candidate who tries one example, and happens to try a pair of negatives such as −3 and −5, will find that it works and may generalise from it. Working means the pair satisfies the condition, not that the condition forces the pair; testing a claim of this kind requires looking for a case that BREAKS it, not one that fits it.
• (b)Exactly one of the numbers is negative — This also asserts too much, and it fails in the opposite direction from the previous option. "Exactly one" excludes the case where both numbers are negative, and that case is perfectly consistent with the stem. Take −3 and −5: the difference is −3 − (−5) = 2, the sum is −8, and 2 is greater than −8 by a wide margin. Both numbers are negative and the condition holds, so "exactly one" is false. Notice how easy this case is to overlook, because a reader's mental picture of "difference bigger than sum" tends to involve one large positive number and one small negative one. In fact the condition is easiest of all to satisfy when both numbers are negative, since the sum then plunges while the difference stays modest. The printed word here is set as "Exactly one of the numbers is negative", and it is the exactness that destroys it.
• (d)None of the above — This is the only escape option in the entire booklet, and it deserves a careful answer rather than a reflex one. An option of this kind is right precisely when every other option fails, and there is nothing improper about it being right; it is a legitimate answer that a well-prepared candidate should be willing to choose. It is wrong here for a specific and demonstrable reason: the third option is not merely plausible but provable. Rearranging the stem's inequality gives y < 0 under the ordered reading of "difference", and m < 0 under the magnitude reading, and each of those says that at least one of the two numbers lies below zero. Since a true statement is present on the list, "None of the above" cannot stand. The working rule for any escape option is to leave it until last: establish or refute the substantive options first, and let this one be decided by the result rather than used as a way of avoiding the work.

THE IDEA BEING TESTED IS THE DIFFERENCE BETWEEN WHAT IS POSSIBLE AND WHAT IS NECESSARY, dressed up as a piece of school algebra. The stem gives a condition and asks what follows FROM it — not what is consistent with it, and not what usually accompanies it. That distinction governs how the options must be handled.

A statement that MUST follow is a universally quantified claim: for every pair of numbers satisfying the condition, the statement holds. Such a claim is established by proof and destroyed by a single counterexample. A statement that MAY follow is an existential claim and is established by a single example. Candidates lose these questions by testing an option with one convenient pair, finding that it fits, and treating the fit as confirmation. It is not; it is only the absence of a refutation from one trial.

THE ALGEBRA ITSELF is a small lesson in cancellation. The expressions x − y and x + y share the term x, so comparing them compares −y with +y and nothing else. That is why the condition constrains only one of the two numbers. Many inequality questions collapse in the same way once both sides are written out: whatever is common to the two sides is irrelevant to which is larger, and removing it usually leaves a statement about a single quantity.

THE LOGICAL LADDER of the three substantive options is worth seeing as a ladder. "Both are negative" is the strongest claim. "Exactly one is negative" is a different strong claim, incompatible with the first. "At least one is negative" is weaker than both and is implied by each of them. When several options make claims of differing strength about the same thing, the weaker ones are harder to falsify, and a question asking what must follow is often testing whether the candidate will settle for the weakest claim that the evidence actually supports rather than the most specific-sounding one.

FINALLY, THE VOCABULARY. "At least one" means one or more, and is consistent with both. "Exactly one" means one and not the other. "Both" means two. "At most one" means one or none. These four phrases are the standard furniture of logical-reasoning items, and misreading one of them for another accounts for a large share of the errors on this type of question.

Key facts

• Writing the condition as x − y > x + y and cancelling the common x leaves −y > y, hence y < 0: the number being subtracted must be negative.
• Reading the difference as a magnitude gives M − m > M + m for the larger M and smaller m, hence m < 0 — the smaller of the two numbers is negative.
• Under the magnitude reading the condition is exactly equivalent to "at least one of the numbers is negative", so the two statements are interchangeable.
• The strict inequality excludes zero: if the smaller number were zero the difference and the sum would be equal rather than one being greater.
• The pair 5 and −1 satisfies the condition with only one negative number, which refutes any claim that both must be negative.
• The pair −3 and −5 satisfies the condition with both numbers negative, which refutes any claim that exactly one must be negative.
• A statement that MUST follow is destroyed by a single counterexample, so these options are tested by hunting for cases that break them, not cases that fit them.
• Option (d) is the only "None of the above" printed anywhere in this booklet; it fails here only because a provable statement is present on the list.

Common traps

• Testing an option with one pair of numbers that fits it and treating the fit as proof. A claim about all cases is refuted by cases, not confirmed by them.
• Overlooking the both-negative case, where the sum plunges while the difference stays small and the condition is easiest of all to satisfy.
• Reading "at least one" as "exactly one". The first is consistent with both being negative; the second is not.
• Dismissing "None of the above" on principle. It is a legitimate answer whenever the other three fail, and it fails here only because one of them is provable.
• Forgetting that the strict inequality excludes zero, so the smaller number must be below zero and not merely not above it.

• practice — not a real PYQ

  If the difference of two numbers is exactly equal to their sum, then which one of the following statements must be true of those two numbers ?

  • (a)Both the numbers are zero
  • (b)One of the numbers is zero
  • (c)The two numbers are equal
  • (d)None of the above

  Answer(b) One of the numbers is zero — writing x − y = x + y and cancelling x gives −y = y, so y = 0. The other number is unconstrained, which is why neither "both are zero" nor "the two are equal" need hold; 5 and 0 satisfies the condition and contradicts both.

• practice — not a real PYQ

  If the sum of two numbers is negative while their product is positive, then which one of the following statements must be true of those two numbers ?

  • (a)Both the numbers are negative
  • (b)Exactly one of the numbers is negative
  • (c)At least one of the numbers is zero
  • (d)None of the above

  Answer(a) Both the numbers are negative — a positive product forces the two to share a sign and rules out zero, and a negative sum then rules out their both being positive, so both must lie below zero. Here the strongest claim is the one that is entailed.