Desk Postings and a Pension Return Rate: Two Claims to Test
At a sub-treasury, four clerks - P, Q, R and S - must be posted one each to four desks: Receipts, Payments, Pension and Stamp. The posting rules are: P will not sit at Pension; Q must sit at either Payments or Stamp; R will sit at Receipts only if S sits at Pension; S will not sit at Stamp. The supervisor has recorded an inference for the file: 'If R sits at Receipts, then Q sits at Stamp.'
Separately, the pension section received 1,250 vouchers during the month. Of these, 8% were returned for want of life certificates; one-fourth of the returned vouchers were re-submitted correctly within the week and the rest are still pending. The section head claims: 'If we halve the number of returns next month, our return rate will fall below 3%.'
1. List all postings that satisfy the four rules. On that basis, decide whether the supervisor's inference is necessarily true.
Read the worked answer
Work from Q, because Q has only two possible desks.
Case Q = Payments (Receipts, Pension, Stamp remain for P, R, S; P ≠ Pension; S ≠ Stamp):
| # | P | Q | R | S |
|---|---|---|---|---|
| 1 | Stamp | Payments | Receipts | Pension |
| 2 | Stamp | Payments | Pension | Receipts |
| 3 | Receipts | Payments | Stamp | Pension |
Case Q = Stamp (Receipts, Payments, Pension remain for P, R, S):
| # | P | Q | R | S |
|---|---|---|---|---|
| 4 | Payments | Stamp | Receipts | Pension |
| 5 | Receipts | Stamp | Pension | Payments |
| 6 | Payments | Stamp | Pension | Receipts |
| 7 | Receipts | Stamp | Payments | Pension |
Seven postings are possible. In each one where R sits at Receipts (#1 and #4), the rule 'R at Receipts only if S at Pension' is satisfied by posting S at Pension.
The inference is not necessarily true. Take posting #1: P = Stamp, Q = Payments, R = Receipts, S = Pension. All four rules hold, yet Q is at Payments, not Stamp. So the antecedent is true and the consequent false - a counterexample.
Note the trap: the inference happens to hold in #4, and a candidate who stops at the first valid arrangement will wrongly mark it true. Enumerating all arrangements is what settles the question.
2. How many pension vouchers are still pending, and what share of the month's receipts do they represent? Is the section head's claim about halving returns correct?
Read the worked answer
Returned vouchers: 8% of 1,250 = 100. Re-submitted correctly: one-fourth of 100 = 25. Pending: 100 − 25 = 75. Pending as a share of the month's receipts: 75 ÷ 1,250 = 0.06 = 6%.
So the actual outstanding defect rate is 6%, double the 3% threshold the section head is aiming at.
Testing the claim: halving the returns means 100 → 50. The rate then becomes 50 ÷ 1,250 = 4%, which is still above 3%. The claim is therefore false.
What would be needed: 3% of 1,250 = 37.5, so returns must fall to 37 or fewer, i.e. a cut of about 63 vouchers (roughly a two-thirds reduction), not a one-half reduction.
Watch the base: the denominator is the total vouchers received (1,250), not the 100 returned. A very common error is to halve 100 and compare 50 with 100, producing a meaningless 50%. Also note that the 25 re-submitted vouchers are still counted as defective for the month; re-submission corrects the voucher, not the initial return.
3. The supervisor wants the inference to become necessarily true. Does adding the rule 'S sits at Pension' achieve this? Give a rule that does, and justify it.
Read the worked answer
Adding 'S sits at Pension' does not work. With S fixed at Pension, look at posting #1: P = Stamp, Q = Payments, R = Receipts. This satisfies the new rule (S at Pension) and the original four rules, yet Q is at Payments. So the inference still fails. All the new rule does is guarantee that whenever R sits at Receipts, the 'only if' condition is automatically met - it says nothing about Q.
A rule that does work: 'P sits at Payments.' Then Payments is taken, and since Q must sit at Payments or Stamp, Q is forced to Stamp. In every remaining arrangement, therefore, R at Receipts implies Q at Stamp - the consequent no longer depends on the antecedent at all.
The principle being tested: an 'if ... then' statement is necessary only if it holds in every model of the rules. To make it necessary, add a condition that removes the counterexamples - in this case, a condition that removes Q from Payments. A condition that merely duplicates an existing requirement cannot remove a counterexample. Checking one or two arrangements and generalising is the reasoning error this question targets.