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CAT 2023 Slot 1 — QA Question 11

Relative SpeedEasy

Arvind travels from town A to town B, and Surbhi from town B to town A, both starting at the same time along the same route. After meeting each other, Arvind takes 6 hours to reach town B while Surbhi takes 24 hours to reach town A. If Arvind travelled at a speed of 54 km/h, then the distance, in km, between town A and town B is

Answer & solution

Answer: 972

Solution

Easy

Classic "two travellers meet" result: if after meeting they take t1t_1 and t2t_2 hours to finish, the time to meet is t1t2\sqrt{t_1 t_2}. Get Arvind's total travel time, multiply by his speed to get the whole distance.

1

Time to meet using the geometric-mean property (after meeting they take 66 h and 2424 h):

tmeet=6×24=144=12 hourst_{\text{meet}}=\sqrt{6\times 24}=\sqrt{144}=12\ \text{hours}
2

Arvind's total time A → B = time until meeting + time after meeting:

12+6=18 hours12+6=18\ \text{hours}
3

Distance = Arvind's speed ×\times his total time:

d=54×18=972 kmd=54\times 18=972\ \text{km}
d=972 kmd=\mathbf{972}\ \text{km}

Where does t1t2\sqrt{t_1 t_2} come from? Speeds are inversely proportional to times for the same distance, so vAvS=tStA\dfrac{v_A}{v_S}=\dfrac{t_S}{t_A} both before and after meeting. Equating gives tmeet2=t1t2t_{\text{meet}}^2=t_1 t_2.

CAT 2023 Slot 1 QA Q11: Arvind travels from town A to town B, and Surbhi from town B to town A, both starting at the same time along t — Solution | TheCATExam