X, Y and Z jump forward 4', 6' and 5', respectively. At 8 AM, they all land on mark 199'. How many times will they all land on the same mark (need not be at the same moment) between mark 195' and 1000', if all of them cross mark 1000' by 9 AM?

EXPLANATION

The correct option is D - 14.

[as per provisional answerkey]

Solution

We are given the jump lengths of three individuals: X = 4', Y = 6', and Z = 5'. We need to find how many marks between 195' and 1000' (inclusive) are reachable by all three individuals.

1. Identify the condition for a common mark: For a mark to be stepped on by X, Y, and Z, the mark must be a multiple of their respective jump lengths. However, the question states they all land on mark 199' at 8 AM. This implies that 199' is a common landing point for their current sequences of jumps.
2. Determine the starting point: If they all land on 199', the next common mark they will all land on depends on the Least Common Multiple (LCM) of their jump lengths.
$L C M (4, 6, 5)$ :
$4 = 2^{2}$
$6 = 2 \times 3$
5 = 5
$L C M = 2^{2} \times 3 \times 5 = 60$ .
3. Find the general formula for common marks: Since 199' is a common mark, all other common marks will be of the form: $199 + 60 n$ (where n is an integer).
4. Calculate marks between 195' and 1000':
For n = 0: $199 + 60 (0) = 199$ ' (Valid, as 195 < 199 < 1000)
For n = -1: $199 - 60 = 139$ ' (Invalid, below 195)
To find the maximum n: $199 + 60 n <= 1000$
$60 n <= 1000 - 199$
$60 n <= 801$
$n <= 801 / 60 = 13.35$
So, n can range from 0 to 13.
5. Count the values: The values of n are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}.
Total count = 14.

Why the other options are incorrect

Option (a) - 11: This result might be reached if one incorrectly calculates the LCM or starts the sequence from a higher number, missing the initial common marks near 199.
Option (b) - 12: This is a common error if the student calculates the number of jumps after 199 but forgets to include the mark 199 itself, or makes a minor division error ( $801 / 60$ ).
Option (c) - 13: This occurs if the student correctly identifies the range but fails to account for the "zero" term (n=0) in the sequence $199 + 60 n$ , effectively missing one occurrence.

Key Concept

The common points in multiple periodic sequences are determined by the Least Common Multiple (LCM) of the periods, starting from a known common offset point.