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CAT 2024 Slot 1 — QA Question 9

Relative SpeedEasy

 Two places A and B are 45 kms apart and connected by a straight road. Anil goes from A to B while Sunil goes from B to A. Starting at the same time, they cross each other in exactly 1 hour 30 minutes. If Anil reaches B exactly 1 hour 15 minutes after Sunil reaches A, the speed of Anil, in km per hour, is

Answer & solution

  • A

    16

  • 12

  • C

    18

  • D

    14

Solution

Hard

Use the classic meeting result: after meeting, the time each takes to finish satisfies tAtB=(vBvA)2\dfrac{t_A}{t_B}=\left(\dfrac{v_B}{v_A}\right)^2. The 11h 1515m gap between their finishing times pins the speed ratio; the 1.51.5h-to-meet condition gives the actual speeds.

1

Post-meeting times. They meet after 1.51.5 h. After meeting, Anil covers the stretch Sunil had walked and vice-versa. With meeting point dividing distance in ratio vA:vBv_A:v_B,

tA=vB⋅1.5vA,tB=vA⋅1.5vB⇒ tA−tB=1.25 h(Anil finishes 1h15m later)\begin{aligned} &t_A=\dfrac{v_B\cdot1.5}{v_A},\qquad t_B=\dfrac{v_A\cdot1.5}{v_B}\\ &\Rightarrow\ t_A-t_B=1.25\ \text{h} \quad\text{(Anil finishes 1h15m later)} \end{aligned}
2

Let r=vAvBr=\dfrac{v_A}{v_B}. Substitute into step 1, with tA=1.5rt_A=\dfrac{1.5}{r} and tB=1.5 rt_B=1.5\,r.

1.5r−1.5 r=1.25⇒ 1.5−1.5r2=1.25 r(multiply by r)⇒ 6r2+5r−6=0(×4, rearrange)⇒ r=−5+25+14412=−5+1312=23\begin{aligned} &\dfrac{1.5}{r}-1.5\,r=1.25\\ &\Rightarrow\ 1.5-1.5r^2=1.25\,r \quad\text{(multiply by }r\text{)}\\ &\Rightarrow\ 6r^2+5r-6=0 \quad\text{(}\times4\text{, rearrange)}\\ &\Rightarrow\ r=\dfrac{-5+\sqrt{25+144}}{12}=\dfrac{-5+13}{12}=\dfrac23 \end{aligned}
3

Find actual speeds. Combined they cover 4545 km in 1.51.5 h, so vA+vB=30v_A+v_B=30. With vA=23vBv_A=\tfrac23 v_B from step 2,

23vB+vB=30⇒ 53vB=30(combine)⇒ vB=18 ⇒ vA=23×18=12\begin{aligned} &\tfrac23 v_B+v_B=30\\ &\Rightarrow\ \tfrac53 v_B=30 \quad\text{(combine)}\\ &\Rightarrow\ v_B=18\ \Rightarrow\ v_A=\tfrac23\times18=12 \end{aligned}
vAnil=12 km/hv_{\text{Anil}}=12\ \text{km/h}

Anil finishes later, so Anil is the slower one — confirms vA<vBv_A<v_B and the 23\tfrac23 ratio before you even solve.

CAT 2024 Slot 1 QA Q9: Two places A and B are 45 kms apart and connected by a straight road. Anil goes from A to B while Sunil goes f — Solution | TheCATExam