Math Problem Statement

A is faster than B. A and B each walk 24 km. The sum of their speeds is 7 km/hr and sum of times taken by them is 14 hours. What is the speed of A?

Solution

To determine the speed of A, we can start by defining the variables and using the given information.

Let the speed of A be vAv_A km/hr and the speed of B be vBv_B km/hr.

According to the problem:

  1. vA+vB=7v_A + v_B = 7 km/hr (the sum of their speeds is 7 km/hr)
  2. The time taken by A to walk 24 km is 24vA\frac{24}{v_A} hours.
  3. The time taken by B to walk 24 km is 24vB\frac{24}{v_B} hours.
  4. The sum of their times is 14 hours, so 24vA+24vB=14\frac{24}{v_A} + \frac{24}{v_B} = 14.

We now have the following system of equations: vA+vB=7v_A + v_B = 7 24vA+24vB=14\frac{24}{v_A} + \frac{24}{v_B} = 14

First, we can simplify the second equation by factoring out the 24: 24vA+24vB=14\frac{24}{v_A} + \frac{24}{v_B} = 14 24(1vA+1vB)=1424 \left( \frac{1}{v_A} + \frac{1}{v_B} \right) = 14 1vA+1vB=1424\frac{1}{v_A} + \frac{1}{v_B} = \frac{14}{24} 1vA+1vB=712\frac{1}{v_A} + \frac{1}{v_B} = \frac{7}{12}

Now, we can use the relationship vB=7vAv_B = 7 - v_A from the first equation and substitute it into the second equation: 1vA+17vA=712\frac{1}{v_A} + \frac{1}{7 - v_A} = \frac{7}{12}

To solve this equation, find a common denominator: 7vA+vAvA(7vA)=712\frac{7 - v_A + v_A}{v_A (7 - v_A)} = \frac{7}{12} 7vA(7vA)=712\frac{7}{v_A (7 - v_A)} = \frac{7}{12}

We can now cancel out the 7 from both sides: 1vA(7vA)=112\frac{1}{v_A (7 - v_A)} = \frac{1}{12}

Cross-multiply to solve for vAv_A: 12=vA(7vA)12 = v_A (7 - v_A) 12=7vAvA212 = 7v_A - v_A^2 vA27vA+12=0v_A^2 - 7v_A + 12 = 0

This is a quadratic equation. Solving this using the quadratic formula v=b±b24ac2av = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} where a=1a = 1, b=7b = -7, and c=12c = 12: vA=7±49482v_A = \frac{7 \pm \sqrt{49 - 48}}{2} vA=7±12v_A = \frac{7 \pm 1}{2} vA=82orvA=62v_A = \frac{8}{2} \quad \text{or} \quad v_A = \frac{6}{2} vA=4orvA=3v_A = 4 \quad \text{or} \quad v_A = 3

Since A is faster than B, vA=4v_A = 4 km/hr and vB=3v_B = 3 km/hr.

Thus, the speed of A is 44 km/hr.

Would you like further details or have any questions?

Here are 8 related questions to expand your understanding:

  1. What is the speed of B if A is faster and their combined speed is 10 km/hr?
  2. How would the solution change if the distance was 30 km instead of 24 km?
  3. Can you derive the formula used to solve the quadratic equation?
  4. What would be the time taken by each if the distance was doubled?
  5. How do you interpret the solutions of the quadratic equation in this context?
  6. If A and B were to race over 10 km, who would win and by how much time?
  7. What would the speeds be if the sum of their speeds was 10 km/hr but the sum of their times remained the same?
  8. How can we graphically represent the speeds of A and B?

Tip: Always check the conditions given in the problem to determine which solution of the quadratic equation is applicable.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Speed and Distance Problems

Formulas

Quadratic equation formula

Theorems

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Suitable Grade Level

Grades 7-9