CAT 2024 Slot 1QAArithmetic • Time, Speed & DistanceHard
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
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Official Correct Answer: B. First, the problem says that Anil goes from A to B, and Sunil goes from B to A. They start at the same time and cross each other after exactly 1 hour and 30 minutes. That's 1.5 hours. The distance between A and B is 45 km. So, when they meet, they've covered the entire 45 km together.
Let Anil's speed as 'a' km/h and Sunil's speed as 's' km/h. Since they start at the same time and meet after 1.5 hours, the distance Anil covers before meeting is a * 1.5, and the distance Sunil covers is s * 1.5. Together, these distances add up to 45 km. So, We write:
a * 1.5 + s * 1.5 = 45
Hmm, maybe Factoring out the 1.5:
1.5(a + s) = 45
To find (a + s), I'll divide both sides by 1.5:
a + s = 45 / 1.5
Calculating that, 45 divided by 1.5 is the same as 45 multiplied by 2/3, which is 30. So,
a + s = 30
Okay, that's one equation. Now, the second part of the problem says that Anil reaches B exactly 1 hour 15 minutes after Sunil reaches A. So, Sunil starts from B and goes to A. The time he takes is the distance divided by his speed, which is 45 / s hours. Similarly, Anil starts from A and goes to B, taking 45 / a hours.
But Anil takes 1 hour 15 minutes longer than Sunil. 1 hour 15 minutes is 1.25 hours. So, the time Anil takes is equal to the time Sunil takes plus 1.25 hours. So, We write:
45 / a = (45 / s) + 1.25
Hmm, that's another equation. Now I have two equations:
1. a + s = 30
2. 45/a = 45/s + 1.25
I need to solve these two equations to find 'a', Anil's speed.
. From the first equation, I can express 's' in terms of 'a':
s = 30 - a
Now, substitute this into the second equation:
45/a = 45/(30 - a) + 1.25
This looks a bit complicated, but I can solve it step by step. Let me write it down:
45/a - 45/(30 - a) = 1.25
To combine the fractions, I'll find a common denominator, which is a(30 - a). So,
[45(30 - a) - 45a] / [a(30 - a)] = 1.25
Expanding the numerator:
45*30 - 45a - 45a = 1350 - 90a
So, the equation becomes:
(1350 - 90a) / [a(30 - a)] = 1.25
Multiply both sides by [a(30 - a)] to eliminate the denominator:
1350 - 90a = 1.25 * a * (30 - a)
Compute the right side:
1.25 * a * (30 - a) = 1.25*(30a - a²) = 37.5a - 1.25a²
So, the equation is:
1350 - 90a = 37.5a - 1.25a²
Let me bring all terms to one side to form a quadratic equation:
1.25a² - 90a - 37.5a + 1350 = 0
Combine like terms:
1.25a² - 127.5a + 1350 = 0
Hmm, dealing with decimals can be messy. Maybe I can multiply the entire equation by 4 to eliminate the decimal:
4*(1.25a²) = 5a²
4*(-127.5a) = -510a
4*1350 = 5400
So, the equation becomes:
5a² - 510a + 5400 = 0
Simplifying by dividing all terms by 5:
a² - 102a + 1080 = 0
Now, I have a quadratic equation: a² - 102a + 1080 = 0
Factoring this. I need two numbers that multiply to 1080 and add up to 102.
Hmm, 1080 is a big number. Factors of 1080: 10 and 108, 12 and 90, 15 and 72, 18 and 60, 20 and 54, 24 and 45, 27 and 40, 30 and 36.
Looking for a pair that adds up to 102. 30 and 36: 30 + 36 = 66, too low.
24 and 45: 24 + 45 = 69, still low.
18 and 60: 18 + 60 = 78.
15 and 72: 15 + 72 = 87.
12 and 90: 12 + 90 = 102. So, the equation factors as:
(a - 12)(a - 90) = 0
So, the solutions are a = 12 or a = 90.
But wait, Anil is going from A to B, which is 45 km. If his speed were 90 km/h, he would take 45/90 = 0.5 hours, which is 30 minutes. But according to the problem, Anil takes longer than Sunil. Let's check the times.
If a = 12, then s = 30