SSC CGL 2025 Tier-I · 21 September 2025, Shift 3 · Quantitative Aptitude · Question 4 of 100

We transcribed this text from the pictures on a candidate's response sheet for this shift; the answer is the key SSC colour-marked on that sheet. Solution last reviewed 11 September 2026.

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SSC does not publish CGL question papers or a public answer key. From 2025 the candidate response sheet prints every question and option as a picture, so the text here is transcribed from those pictures for this shift and checked against them independently; where the layout carries meaning, the picture is shown as well. Each answer is the key SSC colour-marked on that sheet. SSC released a tentative key for challenge and later its final one, both only to candidates behind a login, and the sheet does not say which of the two it carries. The solutions are written by UnlockIAS.

Which of these is a number that is real but NOT rational?

(a)3⁄4
(b)5
(c)Square root of 7
(d)−2

Answer

Why

Correct — C.
Test: a number is rational if it can be written as p⁄q, with p and q integers and q ≠ 0.

3⁄4 is already p⁄q, so rational
5 = 5⁄1, so rational
−2 = −2⁄1, so rational

7 lies between the perfect squares 4 and 9, so √7 is not a whole number
The root of a positive integer that is not a perfect square is irrational
√7 ≈ 2.6458 is real, so Square root of 7 is real but NOT rational → option (c)

Why the others are wrong

(a)
3⁄4 — 3⁄4 is a ratio of two integers with a non-zero denominator, which is the definition of rational. Its decimal, 0.75, terminates.
(b)
5 — 5 = 5⁄1. Every integer n can be written as n⁄1, so every integer, 5 included, is rational as well as real.
(d)
−2 — −2 = −2⁄1, so it is rational. A minus sign changes the sign, not the type: negative integers are rational too.

Concept

The real numbers split into two groups that never overlap. Rational numbers are fractions p⁄q of integers with q ≠ 0; their decimals terminate or repeat. Irrational numbers cannot be written that way; their decimals never end and never repeat.

Integers, positive or negative, and ordinary fractions are all rational. The square root of a positive integer is either a whole number (√9 = 3) or irrational (√7).

Why √7 cannot be p⁄q: suppose √7 = p⁄q in lowest terms.
Square it: p² = 7q², so 7 divides p², and since 7 is prime, 7 divides p.
Write p = 7k: 49k² = 7q², so q² = 7k² and 7 divides q as well.
Both divisible by 7 contradicts lowest terms, so √7 is irrational.

Key facts

Rational: expressible as p⁄q with integers p, q and q ≠ 0
Every integer n is rational, since n = n⁄1
√n for a positive integer n is irrational unless n is a perfect square
A rational number's decimal terminates or repeats: 3⁄4 = 0.75

Study next

Whether products of irrationals can be rational, as in √2 × √2 = 2Converting a repeating decimal such as 0.333… into p⁄qNesting natural, whole, integer, rational and real numbers

Common traps

Treating a negative integer such as −2 as irrational because of its sign
Marking a fraction such as 3⁄4 as not rational because it is not a whole number

Here the four options are all real, and the root is spelt out in words as 'Square root of 7', so the test is purely whether each can be written as p⁄q.

Related PYQs

No directly related past PYQ was found.

Topic: Ratio & Proportion — all 64 SSC CGL questions on it, every solved shift.