Inequality Questions and Answers

Answer: B

Correct answer: Only conclusion b follows

What survives of the statements reads \( M < S \;?\; A > J > R \), \( A < B \;?\; Z \) and \( Z > F \), where each ? marks a relation lost from the record.

Conclusion (a) claims \( Z < S \). Z is reached only through B, and S only through the first statement, and the link between them is one of the lost ones - so nothing in the surviving text orders them, and the key is right that (a) does not follow.

Conclusion (b) claims \( C > R \), and C appears nowhere in the surviving statements at all. For the key to hold, the lost text must have placed C above one of A, J or R - \( C > A \) would do it, since \( A > J > R \) then carries down to R.

A note on this item: its statements have been damaged in this record. A raw < in the stored HTML swallows everything up to the next >, and in a question written entirely out of < and > signs that removes whole relations. The keyed answer is retained because it is the item’s own, but the derivation above is as far as the surviving text allows. Restoring the original statements would be needed to work it in full.

Answer: D

Correct answer: Neither conclusion a nor b follows

What survives reads \( R > A > H < D \;?\; C < E \;?\; H \) and \( H < J \;?\; K < L \;?\; \), with several relations lost.

Conclusion (b) can still be tested, and it fails. It claims \( R = E \). R stands above H by way of A, and E stands above C, which is itself above H - so both are above H on separate branches. Two elements above a common third are never ordered by that fact, still less forced into equality. Does not follow.

Conclusion (a) claims \( F > J \), and F appears nowhere in the statements. A conclusion naming an element the statements never mention cannot follow, which is consistent with the key.

A note on this item: its statements have been damaged in this record. A raw < in the stored HTML swallows everything up to the next >, and in a question written entirely out of < and > signs that removes whole relations. The keyed answer stands, and conclusion (b) is genuinely demonstrable from what remains.

Answer: D

Correct answer: Neither conclusion a nor b follows

What survives reads \( F > G = H < I \;?\; J \), \( K < L \;?\; M > N > H \) and \( H < B \;?\; \).

Conclusion (a) can be tested in full, and it fails. It claims \( F > M \). From the first statement \( F > G = H \), so F stands above H; from the second \( M > N > H \), so M stands above H as well. They are on separate branches rising from H, and nothing orders them: taking \( H = 1 \), both \( F = 2, M = 9 \) and \( F = 9, M = 2 \) satisfy every surviving statement. Does not follow.

Conclusion (b) claims \( C > J \), and C appears nowhere in the statements, so it cannot follow from them.

A note on this item: its statements have been damaged in this record. A raw < in the stored HTML swallows everything up to the next >, and in a question written entirely out of < and > signs that removes whole relations. The keyed answer stands, and conclusion (a) is demonstrable exactly as shown.

Answer: A

Correct answer: Only conclusion a follows

What survives of the statements reads \( F < D \;?\; \ldots \;?\; O > P = E \), with two relations lost from the middle of the chain.

Conclusion (b) claims \( J > O \), and J appears nowhere in the surviving statements. Nothing in what remains connects it to O, which is consistent with the key treating (b) as not following.

Conclusion (a) claims \( L > F \), and L is likewise absent from the surviving text - so the lost relations must have carried it. The only way (a) can follow is if the missing text placed L at or above D, since \( F < D \) would then give \( L > F \) by the usual rule that a chain with one strict link is strict throughout.

A note on this item: its statements have been damaged in this record. A raw < in the stored HTML swallows everything up to the next >, and in a question written entirely out of < and > signs that removes whole relations. That is why the earlier version of this explanation was wrong to say neither conclusion follows: it was written from the damaged text alone, and contradicted the key. The item needs its original statements restored before it can be worked properly.

Answer: D

Correct answer: Neither conclusion a nor b follows

What survives reads \( M < V \;?\; E > O < A \;?\; U > Z = Y < J \;?\; \).

Conclusion (a) can be tested, and it fails. It claims \( M = Z \). M is known only to lie below V; Z is known to lie below U and to equal Y. The two sit in different parts of the arrangement with no chain between them, and even if one existed, an equality conclusion needs the two values pinned together, which nothing here does. Does not follow.

Conclusion (b) claims \( B = J \), and B appears nowhere in the statements, so it cannot follow.

A note on this item: its statements have been damaged in this record. A raw < in the stored HTML swallows everything up to the next >, and in a question written entirely out of < and > signs that removes whole relations. The keyed answer stands, and conclusion (a) is demonstrable from what remains.

Answer: B

Correct answer: Conclusion (c) Only

This set’s code reads \(\&\) as “not greater than”, so \(\le\); \(*\) as “not smaller than”, so \(\ge\); \(\%\) as “neither greater than nor equal”, so \(<\); \(\$\) as “neither smaller than nor equal”, so \(>\); and \(@\) as \(=\). The statements become \( I \le J \), \( V \ge W \), \( K < V \), \( J = K \), \( L \le M \) and \( K < L \).

(c) \( M > J \): J equals K, and \( K < L \le M \), so \( M > K = J \). True.

(d) \( I \ge V \): the chain gives \( I \le J = K < V \), so \( I < V \). Definitely false - the comparison can be made and comes out the other way.

(a) \( I < W \): I sits below K, and W below V, with \( K < V \) joining the two groups from below. V is a high point, so I and W are not ordered. Not determined.

(b) \( L = W \): L stands above K and W below V, and nothing pins them together. Not determined.

Only (c) follows.

Answer: E

Correct answer: None of these

This set’s code reads \(\&\) as “not greater than”, so \(\le\); \(*\) as “not smaller than”, so \(\ge\); \(\%\) as “neither greater than nor equal”, so \(<\); \(\$\) as “neither smaller than nor equal”, so \(>\); and \(@\) as \(=\). The statements become \( A > R \), \( R < S \), \( T \le U \), \( V > J \), \( U = V \), \( U > K \) and \( R \le U \).

Look at the shape before testing anything. R is a low point - A stands above it, S stands above it, U stands above it - while U is a high point, with T, K and R all at or below it. An arrangement in which almost everything hangs below a common element settles almost nothing.

(a) \( A \ge U \): A and U both stand above R. Not determined.

(b) \( S > J \): S stands above R; J stands below V, which equals U. The two are joined only through R and U, and the direction reverses. Not determined.

(c) \( T \le K \): T and K both sit below U. Not determined.

(d) \( A \ge K \): A is above R, K is below U, and R is below U - again the wrong way round to carry anything. Not determined.

None of the four is definite, so the answer is none of these.

Answer: E

Correct answer: None of these

This set’s code reads \(\&\) as “not greater than”, so \(\le\); \(*\) as “not smaller than”, so \(\ge\); \(\%\) as “neither greater than nor equal”, so \(<\); \(\$\) as “neither smaller than nor equal”, so \(>\); and \(@\) as \(=\). The statements become \( U < B \), \( B < T \), \( U < W \), \( H \le C \), \( K > U \) and \( C = K \).

Every relation here has U at the bottom: B, W and K all stand above it, and T stands above B while H sits below C, which equals K. So the diagram is a fan of branches rising out of U, and elements on different branches of such a fan are never ordered.

(a) \( H \le T \): H hangs below the K branch, T stands at the top of the B branch. Not determined.

(b) \( K = T \): different branches again, and nothing forces equality. Not determined.

(c) \( C \ge B \): C equals K, which is on one branch, and B is on another. Not determined.

(d) \( W < K \): W and K are both above U on separate branches. With \( U = 1 \), \( W = 2, K = 9 \) and \( W = 9, K = 2 \) both satisfy every statement. Not determined.

None of the four follows.

Answer: A

Correct answer: Conclusion (d) Only

This set’s code reads \(\&\) as “not greater than”, so \(\le\); \(*\) as “not smaller than”, so \(\ge\); \(\%\) as “neither greater than nor equal”, so \(<\); \(\$\) as “neither smaller than nor equal”, so \(>\); and \(@\) as \(=\). The statements become \( P < R \), \( S > T \), \( R \le S \), \( K \le S \), \( G \ge K \) and \( S > M \).

(d) \( S \ge K \): this is the fourth statement read the other way round - \( K \le S \) is exactly \( S \ge K \). True.

The other three all founder on the same feature: S is a high point, with R, K, T and M all at or below it, so any two of them are on separate branches beneath it.

(a) \( P \le K \): P sits below R, K sits below S. Not determined.

(b) \( R \ge M \): both are below S. Not determined.

(c) \( S = G \): G stands above K, which sits below S, so G could be anywhere from K upwards - below S, equal to it, or above. Not determined.

Only (d) follows, and it follows because it merely restates a premise. Scanning the conclusions for a restated statement before doing any chaining is usually the fastest first move on these.

Answer: C

Correct answer: None of these

This set’s code reads \(\&\) as “not greater than”, so \(\le\); \(*\) as “not smaller than”, so \(\ge\); \(\%\) as “neither greater than nor equal”, so \(<\); \(\$\) as “neither smaller than nor equal”, so \(>\); and \(@\) as \(=\). The statements become \( V > R \), \( W < K \), \( R = A \), \( A \le W \), \( C \le I \), \( D > C \) and \( A \ge D \). Six of these assemble into one long chain: \[ C < D \le A = R \le W < K, \qquad V > R. \]

(a) \( C < V \): from \( C < D \le A = R \) and \( V > R \), so \( V > R \ge D > C \). True.

(c) \( K > R \): from \( R \le W < K \). True.

(d) \( K \ge C \): run the whole chain - \( C < D \le A = R \le W < K \) gives \( C < K \), which certainly satisfies \( K \ge C \). Also true.

(b) \( D \ge R \): the chain gives \( D \le A = R \), so D is at or below R; the reverse relation holds only in the equality case and is not forced. Not determined.

So three conclusions follow - (a), (c) and (d) - and no option offers that combination, which makes the answer “none of these”. This record was keyed to “(a) and (c) only”, which overlooks that the chain runs all the way from C to K; the key has been corrected.