# CAT 2005 | Question: 07

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Answer the question based on the information given below.

Ran and Shyam run a race between points A and B, $5$ km apart. Ram starts at $9$ a.m. from A at the speed of  $5$ km per hour, reaches B and returns to A at the same speed. Shyam starts at $9:45$ a.m. from A at the speed of $10$ km per hour, reached B and come back to A at the same speed.

At what time did Shyam overtake Ram?

1. $10:20$ a.m.
2. $10:30$ a.m.
3. $10:40$ a.m.
4. $10:50$ a.m.

At the point of overtake, distance travelled by Ram and Shyam must be the same.

So, if this is assumed as $t$ hours after 9:00, we have

$5 \times t = 10 \times \left(t- \frac{45}{60} \right)$

$\implies 20 t = 40t - 30$

$\implies t = \frac{3}{2}$

So, after 1.5 hrs from 9:00 i.e., at 10:30 Shyam overtakes Ram.
Shyam starts after 45 min of Ram

60 min Ram covers 5 km

45  "       "       "    5/60 ⨉ 45 = 15/4 km

Shyam covers 5 km more than Ram in 1 hr

"            "      15/4 "  " 1/5⨉15/4 =3/4 hr

So Shyam overtakes Ram 9:45 +.45 =10:30 AM

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