Understanding the Speed Ratio in a Chase Problem

This problem involves calculating the ratio of speeds of two individuals, A and B, based on their jumping rates and the distance covered per jump. The core concept is that speed is directly proportional to the distance covered in a given time interval. Since A and B are observed over the same time interval, the ratio of their speeds will be equal to the ratio of the total distances they cover in that interval.

Relating Distance per Jump

We are given that the distance covered by A in 7 jumps is equal to the distance covered by B in 5 jumps. Let's denote the distance covered by A in one jump as \(d_A\) and the distance covered by B in one jump as \(d_B\).

According to the problem statement:

\(7 \times d_A = 5 \times d_B\)

From this equation, we can find the ratio of the distance covered in a single jump for A compared to B:

\(\frac{d_A}{d_B} = \frac{5}{7}\)

This means that for every 5 units of distance A covers in a jump, B covers 7 units in a jump.

Calculating Total Distance Covered

In the given interval of time, A jumps 8 times and B jumps 6 times. Let the total distance covered by A in this interval be \(D_A\) and the total distance covered by B be \(D_B\).

The total distance is the number of jumps multiplied by the distance per jump:

• Total distance covered by A, \(D_A = (\text{Number of jumps by A}) \times d_A = 8 \times d_A\)
• Total distance covered by B, \(D_B = (\text{Number of jumps by B}) \times d_B = 6 \times d_B\)

Determining the Ratio of Speeds

The speed of an object is the distance covered per unit time. Since the time interval for A and B is the same, the ratio of their speeds (\(S_A : S_B\)) is equal to the ratio of the total distances they cover in that time (\(D_A : D_B\)).

\(\frac{S_A}{S_B} = \frac{D_A}{D_B}\)

Substitute the expressions for \(D_A\) and \(D_B\):

\(\frac{S_A}{S_B} = \frac{8 \times d_A}{6 \times d_B}\)

We know from the previous step that \(\frac{d_A}{d_B} = \frac{5}{7}\). Substitute this ratio into the speed ratio equation:

\(\frac{S_A}{S_B} = \frac{8}{6} \times \frac{d_A}{d_B} = \frac{8}{6} \times \frac{5}{7}\)

Simplify the expression:

\(\frac{S_A}{S_B} = \frac{4}{3} \times \frac{5}{7} = \frac{4 \times 5}{3 \times 7} = \frac{20}{21}\)

Thus, the ratio between the speeds of A and B is 20 : 21.

The final ratio of speeds is 20 : 21.

Revision Table: Key Concepts in Speed Ratio Problems

Additional Information: Understanding Relative Speed and Chase Problems

Chase problems often involve the concept of relative speed. However, in this specific question, we are asked for the ratio of individual speeds, not their relative speed (which would be relevant if we wanted to find out when A catches B). Understanding the fundamental definition of speed (distance/time) and how to calculate total distance based on repetitive actions (like jumping) is crucial.

In problems where time is constant, the ratio of speeds simplifies directly to the ratio of distances covered. If distance were constant, the ratio of speeds would be the inverse ratio of the times taken. If speed were constant, the ratio of times would be the ratio of distances covered.

Always break down such problems into:

• What is given? (Jumps per interval, distance relation per jump)
• What needs to be found? (Ratio of speeds)
• How can speed be related to the given information? (Speed proportional to total distance if time is constant)
• Calculate the necessary intermediate values or ratios (like the ratio of distance per jump).
• Combine everything to find the final ratio.