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CAT 2023 Slot 1 — QA Question 4

Basics of TSD/ProportinalityEasy

Brishti went on an 8-hour trip in a car. Before the trip, the car had travelled a total of x km till then, where x is a whole number and is palindromic, i.e., x remains unchanged when its digits are reversed. At the end of the trip, the car had travelled a total of 26862 kms till then, this number again being palindromic. If Brishti never drove at more than 110 kmph, then the greatest possible average speed at which she drove during the rip, in kmph was?

Answer & solution

  • A

    110

  • B

    80

  • C

    90

  • 100

Solution

Easy

The odometer reads a palindrome xx before the trip and 2686226862 (also a palindrome) after 88 hours. To maximise the average speed we must maximise the trip distance 26862−x26862-x, i.e. make xx as small as possible — but the speed cap of 110110 kmph limits how far the trip could have been, which puts a floor on xx. Find the smallest valid palindrome above that floor.

1

Speed cap bounds the trip distance. At most 110110 kmph for 88 h:

trip distance≤110×8=880 km\text{trip distance}\le 110\times 8=880\ \text{km}
2

This forces a lower bound on xx:

x=26862−trip ≥ 26862−880=25982x=26862-\text{trip}\ \ge\ 26862-880=25982

So xx must be a palindrome with x≥25982x\ge 25982.

3

Find the smallest palindrome ≥25982\ge 25982. A 5-digit palindrome looks like abcba‾\overline{abcba}. Starting near 259 ⁣…259\!\ldots, the candidate 2595225952 is below 2598225982; the next palindrome is

26062(since 2 6 0 6 2 reads the same both ways)26062\quad(\text{since }2\,6\,0\,6\,2\text{ reads the same both ways})

and 26062≥2598226062\ge 25982, so this is the least admissible xx.

4

Maximum trip distance and speed:

trip=26862−26062=800 km ⇒ avg speed=8008=100 kmph\text{trip}=26862-26062=800\ \text{km}\ \Rightarrow\ \text{avg speed}=\frac{800}{8}=100\ \text{kmph}

(And 100≤110100\le 110, so the cap is respected.)

100 kmph(option d)\mathbf{100}\ \text{kmph}\quad\text{(option d)}
Why not just take xx = 25982?

2598225982 reversed is 28952≠2598228952\neq 25982, so it isn't a palindrome — xx is constrained to be one. The smallest palindrome at or above the floor is 2606226062, which is what caps the trip at 800800 km.

CAT 2023 Slot 1 QA Q4: Brishti went on an 8-hour trip in a car. Before the trip, the car had travelled a total of x km till then, whe — Solution | TheCATExam