CSAT Solved Papers/ 2024/Q45

2024 CSAT — Q45

Quant Number theory 2.5 marks Medium

What is the number of fives used in numbering a $260$ -page book?

A 55
B 56 Answer
C 57
D 60

Worked rationale

Count appearances of the digit $5$ across $1$ to $260$ , place by place.

Units place ( $\dots5$ ): $5, 15, 25, \dots, 255$ . That is $\frac{255 - 5}{10} + 1 = 26$ numbers.

Tens place ( $\dots5\_$ ): the blocks $50$ – $59$ , $150$ – $159$ , $250$ – $259$ each contribute $10$ fives: $10 + 10 + 10 = 30$ .

Hundreds place: the hundreds digit ranges over $0,1,2$ only (up to $260$ ), never $5$ : $0$ .

Total $= 26 + 30 + 0 = 56$ .

Answer: (b) 56.

Why the other options miss

A

the fence-and-posts miscount: drops one units-place five (e.g. counts the $5$ – $255$ run as $25$ by a fencepost slip).
C

off by one: over-counts the tens block (e.g. extends $250$ – $259$ to include a phantom $5$ in $260$ ) or double-counts $255$ .
D

missed a case: assumes a clean $6$ fives per hundred $\times$ … or rounds to ” $10$ per $50$ ,” ignoring that the third hundred stops at $260$ , not $299$ .

Specialist insight

Digit-frequency questions are won by separating the places and counting each independently — never try to count “numbers containing a $5$ ” (which double-counts $55$ , $155$ , $255$ ). Units fives recur every $10$ ; tens fives come in full blocks of $10$ per hundred only where the range reaches them. The cap at $260$ is the trap: the third hundred yields its tens block ( $250$ – $259$ ) but the units run still ends at $255$ , and the hundreds digit is never $5$ . Count $255$ exactly once in each place it legitimately appears.

The trap, in one line

Count fives place-by-place ( $26$ in units, $30$ in tens, $0$ in hundreds), respecting the $260$ cap — total $56$ , not " $10$ per fifty".

← All 2024 CSAT questions