Practice Set — A Caselet and Data Sufficiency

Two set types that punish the same habit: answering before organising. The caselet is a three-language overlap written entirely as prose — nothing in it says "sets", and a candidate who does not draw the diagram will subtract the same students twice and produce a figure that cannot be checked. Draw three circles, fill the centre first, then the pairwise regions with the centre taken out, then the singles; the parts will add to 105 and the answer to every question in the set is then a reading rather than a calculation. The data-sufficiency questions are built so that the four verdicts are all different — one is answered by statement I alone, one by statement II alone, one needs both, and one cannot be answered even with both. That last one is the honest answer to a real question and not a trick, and knowing it exists is what stops a candidate forcing an answer out of insufficient data.

The Two Methods, in One Line Each

  1. Caselet: build the table or the diagram the prose describes, before answering anything. Then check that the parts add to the whole — if they do not, a figure has been read wrongly and every answer after it is wrong too.
  2. Data sufficiency: ask "could I answer this uniquely?" and stop there. Computing the value is work the format was designed to save you, and it is where the time goes.
🧠 Fill the centre first
In a three-set diagram, write the all-three figure in the middle, then subtract it from each pairwise figure to get the "exactly two" regions, then subtract everything from each single total. Working outward-in is what makes people count the middle three times, and the error is invisible until the parts fail to add up.

Key takeaways

  • A pairwise figure like "20 study French and German" includes the students who study all three. Subtract the centre before using it.
  • In a three-set union, pairwise overlaps subtract and the triple overlap adds back. Then check that all seven regions plus the outsiders equal the total.
  • Draw the diagram a caselet describes before answering. A caselet is a table written in sentences, and the sentences are the only reason it looks hard.
  • In data sufficiency, test each statement alone before testing them together — otherwise a statement that was sufficient by itself is credited to the pair.
  • Two surviving possibilities means insufficient. "Nearly determined" is not determined, and that is the whole judgement the format tests.
  • Sometimes both statements together are still not enough. That is a real answer, and refusing to accept it is how a candidate invents data.

Practice questions (8)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

  1. Set 1 — a caselet. In a class of 120 students, every student may study any number of foreign languages. 60 students study French, 50 study German and 40 study Spanish. 20 study both French and German, 15 study both German and Spanish, and 18 study both French and Spanish. 8 students study all three languages.

    ℹ️ Read the overlaps carefully
    Each pairwise figure — "20 study both French and German" — counts everyone in that pair, including the 8 who study all three. It is not the number who study exactly those two.

    How many students study none of the three languages?

    1. 15
    2. 23
    3. 8
    4. 0
    Show answer

    Answer: A — 15

    Union = 60 + 50 + 40 − 20 − 15 − 18 + 8 = 105. The pairwise figures subtract and the triple adds back, because the 8 all-three students were removed three times by those subtractions and one has to be returned. So 120 − 105 = 15 study none. Forgetting the final +8 gives 97 and therefore 23, which is option B — the standard error in three-set questions, and the reason the parts-add-to-the-whole check is worth thirty seconds.

  2. Set 1 — a caselet. In a class of 120 students, every student may study any number of foreign languages. 60 students study French, 50 study German and 40 study Spanish. 20 study both French and German, 15 study both German and Spanish, and 18 study both French and Spanish. 8 students study all three languages.

    ℹ️ Read the overlaps carefully
    Each pairwise figure — "20 study both French and German" — counts everyone in that pair, including the 8 who study all three. It is not the number who study exactly those two.

    How many students study exactly one of the three languages?

    1. 68
    2. 105
    3. 150
    4. 76
    Show answer

    Answer: A — 68

    French only = 60 − 20 − 18 + 8 = 30; German only = 50 − 20 − 15 + 8 = 23; Spanish only = 40 − 18 − 15 + 8 = 15. Total 68. Note the +8 in each line: subtracting both pairwise figures removes the all-three students twice, so one copy comes back. Check the whole diagram: 68 studying one, 29 studying exactly two, 8 studying three — that is 105, matching the union, and 15 outside makes 120 ✓

  3. Set 1 — a caselet. In a class of 120 students, every student may study any number of foreign languages. 60 students study French, 50 study German and 40 study Spanish. 20 study both French and German, 15 study both German and Spanish, and 18 study both French and Spanish. 8 students study all three languages.

    ℹ️ Read the overlaps carefully
    Each pairwise figure — "20 study both French and German" — counts everyone in that pair, including the 8 who study all three. It is not the number who study exactly those two.

    How many students study at least two of the three languages?

    1. 53
    2. 37
    3. 29
    4. 45
    Show answer

    Answer: B — 37

    "At least two" is (exactly two) + (all three). Exactly two = (20 − 8) + (15 − 8) + (18 − 8) = 12 + 7 + 10 = 29, and all three is 8, giving 37. Option A, 53, is 20 + 15 + 18 — adding the pairwise figures as printed, which counts the all-three students three times. Option C is "exactly two", which is the answer to a different question and is offered for exactly that reason: read "at least" and "exactly" as instructions.

  4. Set 1 — a caselet. In a class of 120 students, every student may study any number of foreign languages. 60 students study French, 50 study German and 40 study Spanish. 20 study both French and German, 15 study both German and Spanish, and 18 study both French and Spanish. 8 students study all three languages.

    ℹ️ Read the overlaps carefully
    Each pairwise figure — "20 study both French and German" — counts everyone in that pair, including the 8 who study all three. It is not the number who study exactly those two.

    How many students study French only?

    1. 22
    2. 30
    3. 38
    4. 60
    Show answer

    Answer: B — 30

    60 − 20 − 18 + 8 = 30. Option A, 22, is 60 − 20 − 18 without adding back the 8: subtracting both pairwise figures removed the all-three students twice, so one has to be returned. This single correction is the difference between a set that adds up and one that does not, and it appears in every region of the diagram.

  5. Set 2 — data sufficiency. Each question below gives a query and two statements. Decide whether the statements are sufficient to answer the query uniquely — do not compute the answer itself once you know that you could.

    • A statement that narrows the possibilities to two is not sufficient. The test is a unique answer.
    • Test each statement alone first, and only then both together. Reading them together from the start is how a statement that was sufficient on its own gets missed.

    What is the value of x? I. x is the smallest prime number greater than 20. II. x is odd.

    1. Statement I alone is sufficient
    2. Statement II alone is sufficient
    3. Both statements together are needed
    4. Neither, even together
    Show answer

    Answer: A — Statement I alone is sufficient

    Statement I names exactly one number: 21 is divisible by 3 and 22 by 2, so the smallest prime above 20 is 23. Unique, and therefore sufficient. Statement II allows every odd number, which is no help at all. Note that II is consistent with the answer — 23 is odd — and consistency is not sufficiency, which is the confusion this question is built on.

  6. Set 2 — data sufficiency. Each question below gives a query and two statements. Decide whether the statements are sufficient to answer the query uniquely — do not compute the answer itself once you know that you could.

    • A statement that narrows the possibilities to two is not sufficient. The test is a unique answer.
    • Test each statement alone first, and only then both together. Reading them together from the start is how a statement that was sufficient on its own gets missed.

    Is the triangle right-angled? I. Two of its sides are 3 cm and 4 cm. II. Its three sides are 3 cm, 4 cm and 5 cm.

    1. Statement I alone is sufficient
    2. Statement II alone is sufficient
    3. Both statements together are needed
    4. Neither, even together
    Show answer

    Answer: B — Statement II alone is sufficient

    Statement I leaves the third side open: 3, 4, 5 is right-angled (9 + 16 = 25) while 3, 4, 6 is not (9 + 16 ≠ 36), so the answer could be yes or no — insufficient. Statement II fixes all three sides and gives a definite yes, so it is sufficient by itself. Option C is the trap: statement I adds nothing that II does not already contain, and a statement that is a subset of the other is never "needed".

  7. Set 2 — data sufficiency. Each question below gives a query and two statements. Decide whether the statements are sufficient to answer the query uniquely — do not compute the answer itself once you know that you could.

    • A statement that narrows the possibilities to two is not sufficient. The test is a unique answer.
    • Test each statement alone first, and only then both together. Reading them together from the start is how a statement that was sufficient on its own gets missed.

    What is A's present age? I. A is twice as old as B. II. B is 12 years old.

    1. Statement I alone is sufficient
    2. Statement II alone is sufficient
    3. Both statements together are needed
    4. Neither, even together
    Show answer

    Answer: C — Both statements together are needed

    Statement I gives a relation with no anchor — A could be any even age. Statement II gives B's age but says nothing about A. Together, A = 2 × 12 = 24, uniquely. This is the textbook "both needed" shape: one statement supplies the relation, the other the value, and neither is any use without the other. Notice that you do not have to compute 24 to answer the question — knowing that you could is the whole task.

  8. Set 2 — data sufficiency. Each question below gives a query and two statements. Decide whether the statements are sufficient to answer the query uniquely — do not compute the answer itself once you know that you could.

    • A statement that narrows the possibilities to two is not sufficient. The test is a unique answer.
    • Test each statement alone first, and only then both together. Reading them together from the start is how a statement that was sufficient on its own gets missed.

    What is the two-digit number N? I. N is a multiple of 9. II. N is a perfect square.

    1. Statement I alone is sufficient
    2. Statement II alone is sufficient
    3. Both statements together are needed
    4. Neither, even together
    Show answer

    Answer: D — Neither, even together

    Statement I allows 18, 27, 36, 45, 54, 63, 72, 81 and 99 — insufficient. Statement II allows 16, 25, 36, 49, 64 and 81 — insufficient. Together, the two-digit numbers that are both multiples of 9 and perfect squares are 36 and 81 — two possibilities, so still insufficient. This is the verdict candidates avoid, because both statements clearly narrow things a great deal; but narrowing to two is not answering, and the only way to be sure is to list the survivors rather than to feel that the answer must now be pinned down.