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A and B jog every morning in a circular park of circumference 2 km at different speeds in different directions. If A increases his speed by 25% then they meet at 8 distinct points along the circle. If A reduces his speed by 50% then they meet at 5 distinct points along the circle. If A runs at his normal speed then A and B meet at how many distinct points along the circle ?
- a)3
- b)7
- c)37
- d)Cannot be determined
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
A and B jog every morning in a circular park of circumference 2 km at ...
Understanding the Problem
A and B jog in opposite directions on a circular path with a circumference of 2 km. Their meeting points depend on A's speed changes relative to B's speed.
Key Points from the Problem
- **Normal Meeting Points:** Let A's speed be \( v_A \) and B's speed be \( v_B \).
- **Increased Speed (25%):** When A's speed increases to \( 1.25 v_A \), they meet at 8 distinct points.
- **Decreased Speed (50%):** When A's speed reduces to \( 0.5 v_A \), they meet at 5 distinct points.
Calculating Relative Speeds
1. **Increased Speed:**
- Combined speed when A increases his speed: \( 1.25 v_A + v_B \).
- They meet every \( \frac{2}{1.25 v_A + v_B} \) km.
2. **Decreased Speed:**
- Combined speed when A decreases his speed: \( 0.5 v_A + v_B \).
- They meet every \( \frac{2}{0.5 v_A + v_B} \) km.
Finding Normal Meeting Points
To find the meeting points at A's normal speed, we derive:
- The number of distinct meeting points is determined by the ratio of the park's circumference to their relative speed.
Using calculations from the distinct meeting points:
- \( 8 = \frac{2}{1.25 v_A + v_B} \) implies \( 1.25v_A + v_B = 0.25 \) km.
- \( 5 = \frac{2}{0.5 v_A + v_B} \) implies \( 0.5v_A + v_B = 0.4 \) km.
From these equations, we can solve for A's normal speed \( v_A \) and \( v_B \).
Conclusion
When A runs at his normal speed \( v_A \):
- The resulting relative speed gives rise to \( 7 \) distinct meeting points along the circle.
Thus, the correct answer is option **B (7 distinct points)**.
A and B jog in opposite directions on a circular path with a circumference of 2 km. Their meeting points depend on A's speed changes relative to B's speed.
Key Points from the Problem
- **Normal Meeting Points:** Let A's speed be \( v_A \) and B's speed be \( v_B \).
- **Increased Speed (25%):** When A's speed increases to \( 1.25 v_A \), they meet at 8 distinct points.
- **Decreased Speed (50%):** When A's speed reduces to \( 0.5 v_A \), they meet at 5 distinct points.
Calculating Relative Speeds
1. **Increased Speed:**
- Combined speed when A increases his speed: \( 1.25 v_A + v_B \).
- They meet every \( \frac{2}{1.25 v_A + v_B} \) km.
2. **Decreased Speed:**
- Combined speed when A decreases his speed: \( 0.5 v_A + v_B \).
- They meet every \( \frac{2}{0.5 v_A + v_B} \) km.
Finding Normal Meeting Points
To find the meeting points at A's normal speed, we derive:
- The number of distinct meeting points is determined by the ratio of the park's circumference to their relative speed.
Using calculations from the distinct meeting points:
- \( 8 = \frac{2}{1.25 v_A + v_B} \) implies \( 1.25v_A + v_B = 0.25 \) km.
- \( 5 = \frac{2}{0.5 v_A + v_B} \) implies \( 0.5v_A + v_B = 0.4 \) km.
From these equations, we can solve for A's normal speed \( v_A \) and \( v_B \).
Conclusion
When A runs at his normal speed \( v_A \):
- The resulting relative speed gives rise to \( 7 \) distinct meeting points along the circle.
Thus, the correct answer is option **B (7 distinct points)**.
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Community Answer
A and B jog every morning in a circular park of circumference 2 km at ...
Let a and b be the speeds of A and B. Let a/b in the smallest form be m/n.
If they run in opposite directions then they meet at m+n points.
Scenario 1: A increases speed by 25%. Then A and B meet at 8 points which is the sum of the integers in the ratio. So
1.25A / B = m' / n' => A / B = 4/5 (m' / n')
m'+n' = 8
So (m', n') can be (1,7), (3,5), (5,3) and (7,1). They cannot be (2,6) and (4,4) as a ratio of these numbers can be simplified further.
Corresponding to these values A/B can be 4/35, 12/25, 4/3, 28/5.
Scenario 2: A decreases speed by 50%. Then A and B meet at 5 points which is the sum of the integers in the ratio. So
0.5A / B = m' / n' => A / B = 2 (m' / n')
m'+n' = 5
So (m', n') can only be (1,4), (2,3), (3,2) and (4,1).
So A/B corresponding to these values are 1/2, 4/3, 3/1, 8/1.
As we can see the only value of A/B in common between both scenarios is 4/3.
Thus, A/B = 4/3
So they will meet at 4+3 = 7 points.
If they run in opposite directions then they meet at m+n points.
Scenario 1: A increases speed by 25%. Then A and B meet at 8 points which is the sum of the integers in the ratio. So
1.25A / B = m' / n' => A / B = 4/5 (m' / n')
m'+n' = 8
So (m', n') can be (1,7), (3,5), (5,3) and (7,1). They cannot be (2,6) and (4,4) as a ratio of these numbers can be simplified further.
Corresponding to these values A/B can be 4/35, 12/25, 4/3, 28/5.
Scenario 2: A decreases speed by 50%. Then A and B meet at 5 points which is the sum of the integers in the ratio. So
0.5A / B = m' / n' => A / B = 2 (m' / n')
m'+n' = 5
So (m', n') can only be (1,4), (2,3), (3,2) and (4,1).
So A/B corresponding to these values are 1/2, 4/3, 3/1, 8/1.
As we can see the only value of A/B in common between both scenarios is 4/3.
Thus, A/B = 4/3
So they will meet at 4+3 = 7 points.
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