Two people, A and B, are traveling towards each other from points P and Q, which are 396 km apart. They meet after 11 hours. If A's speed is 6 km/hr more than B's, what is the spee... Two people, A and B, are traveling towards each other from points P and Q, which are 396 km apart. They meet after 11 hours. If A's speed is 6 km/hr more than B's, what is the speed of B?

Understand the Problem

This is a word problem involving relative speeds. We are given the total distance between two points, the time it takes for two people traveling towards each other to meet, and the difference in their speeds. The goal is to find the speed of the slower person (B).

Answer

$v_B = 7$ km/hr
Answer for screen readers

The speed of the slower person (B) is 7 km/hr.

Steps to Solve

  1. Define variables

Let $v_A$ be the speed of person A and $v_B$ be the speed of person B. We are given that $v_A - v_B = 1$ km/hr. Let $d$ be the total distance between the two points, which is 21 km. Let $t$ be the time it takes for them to meet, which is 1 hour and 24 minutes, or $1 + \frac{24}{60} = 1 + \frac{2}{5} = \frac{7}{5}$ hours.

  1. Write equation for combined distance

Since they are traveling towards each other, their speeds add up. The total distance covered is the sum of the distances each person travels, which equals the total distance between the points. So, we have: $$v_A \cdot t + v_B \cdot t = d$$

  1. Substitute values into the equation

Substitute the given values into the equation: $$v_A \cdot \frac{7}{5} + v_B \cdot \frac{7}{5} = 21$$

  1. Simplify the equation

Multiply both sides of the equation by $\frac{5}{7}$: $$v_A + v_B = 21 \cdot \frac{5}{7} = 3 \cdot 5 = 15$$

  1. Express $v_A$ in terms of $v_B$

We know that $v_A - v_B = 1$, so $v_A = v_B + 1$.

  1. Substitute $v_A$ into the equation

Substitute $v_A = v_B + 1$ into the equation $v_A + v_B = 15$: $$(v_B + 1) + v_B = 15$$

  1. Solve for $v_B$

Combine like terms: $$2v_B + 1 = 15$$ Subtract 1 from both sides: $$2v_B = 14$$ Divide by 2: $$v_B = 7$$

The speed of the slower person (B) is 7 km/hr.

More Information

The problem involves understanding relative speeds and how they combine when two objects move towards each other. The key is setting up the correct equations based on the given information and solving for the unknown variable.

Tips

A common mistake is not converting the time to hours correctly (1 hour and 24 minutes). Another common mistake is not correctly setting up the equation representing the sum of the distances traveled by each person. Some may struggle with substitution and solving the equations.

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