Which of the following statements is/are correct? 1. The average of four numbers 10, 15, 20 and 25 is 17.5 2. If a, b and c are three different natural numbers such that a + b + c = abc, then the average of a, b and c is 3 Select the answer using the code given below:
- (a)1 only
- (b)2 only
- (c)Both 1 and 2
- (d)Neither 1 nor 2
Correct — A, 1 only. Statement 1 is a two-second check: 10 + 15 + 20 + 25 = 70, and 70 ÷ 4 = 17.5, so it is true. Statement 2 fails on the only case that satisfies its own condition. Three different natural numbers with a + b + c = abc must be 1, 2 and 3, since 1 + 2 + 3 = 6 = 1 × 2 × 3, and any other triple of distinct naturals has a product that races ahead of its sum. The average of 1, 2 and 3 is 2, not 3. The claim would hold only if the numbers were allowed to repeat, which the statement rules out.
- (b)2 only — This rejects a statement that is plainly true — 70 ÷ 4 is 17.5 — and accepts one that is false.
- (c)Both 1 and 2 — Statement 2 does not survive its own test case. With a = 1, b = 2, c = 3 the average is 6 ÷ 3 = 2.
- (d)Neither 1 nor 2 — Statement 1 is arithmetic that anyone can verify: the four numbers total 70 and there are four of them.
The identity a + b + c = abc has exactly one solution in distinct natural numbers, namely 1, 2 and 3. Once the numbers are pinned down, every derived quantity is fixed with them: the arithmetic mean is 2, the harmonic mean is 3 ÷ (1 + 1/2 + 1/3) = 18/11, and the geometric mean is the cube root of 6.
The item pairs a trivial statement with a subtle one, which is the usual way of separating a candidate who verifies from one who guesses. Statement 2 rewards the habit of hunting for a counterexample or, better, for the unique example. Small numbers are forced here: if the smallest of the three were 2 or more, the product would be at least 2 × 3 × 4 = 24 against a sum of 9, and the gap only widens. So one of them has to be 1, and the rest follows in a line.
- The average of 10, 15, 20 and 25 is 70 ÷ 4 = 17.5.
- 1, 2 and 3 are the only three distinct natural numbers whose sum equals their product.
- Their arithmetic mean is 2; their harmonic mean is 18/11, which lies between 1 and 2.
- For distinct naturals a < b < c with a ≥ 2, the product abc always exceeds a + b + c.
- Allowing repeats admits other cases in integers, which is why the word 'different' is doing real work in the statement.
One counterexample is enough, and here the counterexample is the only case the statement can be about.
- Reading 'different' as optional and admitting repeated values.
- Remembering that 1, 2, 3 is the special triple but reporting their sum, 6, or their product, 6, as the average.
- Spending time on statement 1 when it is a single division.
A paired-statement item where one half is arithmetic and the other is a number-theory fact that a well-prepared candidate has met before.
Suppose a, b and c are three distinct natural numbers such that a + b + c = abc. Consider the following statements: 1. The arithmetic mean of a, b and c is a natural number. 2. The harmonic mean of a, b and c lies between 1 and 2. Which of the statements given above is/are correct?
- (a) 1 only
- (b) 2 only
- (c) Both 1 and 2
- (d) Neither 1 nor 2
Answer(c) Both 1 and 2
The identical triple, one year earlier and pushed further. That paper asks whether the arithmetic mean is a natural number — it is, namely 2 — and this paper asks what the mean actually equals. Anyone who had worked the 2023 item knew the answer here was 2 and not 3.
- practice — not a real PYQ
Three distinct natural numbers satisfy a + b + c = abc. What is their harmonic mean?
- (a)18/11
- (b)11/18
- (c)2
- (d)3
Answer(a) 18/11 — the numbers must be 1, 2 and 3, and 3 ÷ (1 + 1/2 + 1/3) = 18/11.
- practice — not a real PYQ
The average of five consecutive even numbers is 20. What is the largest of them?
- (a)22
- (b)24
- (c)26
- (d)28
Answer(b) 24 — the numbers are 16, 18, 20, 22, 24, with the middle term equal to the mean.