Question

Ann and Bob drive separately to a meeting. Ann's average driving speed is greater than Bob's average driving speed by one-third of Bob's average driving speed, and Ann drives twice as many miles as Bob. What is the ratio of the number of hours Ann spends driving to the meeting to the number of hours Bob spends driving to the meeting?

Short answer

The ratio is $\frac{3}{2}$.

Step-by-step solution

Define Variables

Let Bob's average speed be $v$ and the distance Bob drives be $d$.

Express Ann's Speed

Since Ann's speed is greater by one-third of Bob's speed, Ann's speed is $v+\frac{v}{3}=\frac{4v}{3}$.

Express Ann's Distance

Ann drives twice as many miles as Bob, so Ann's distance is $2d$.

Calculate Bob's Driving Time

Bob's driving time $t_b$ is given by $t_b=\frac{d}{v}$.

Calculate Ann's Driving Time

Ann's driving time $t_a$ is given by $t_a=\frac{2d}{\frac{4v}{3}}=\frac{3d}{2v}$.

Determine the Ratio

The ratio of Ann's driving time to Bob's driving time is $\frac{t_a}{t_b}=\frac{\frac{3d}{2v}}{\frac{d}{v}}=\frac{3}{2}$.

Key concepts

ratios in word problems

Understanding ratios is crucial for solving many mathematical word problems, especially those related to speed and distance. A ratio compares two quantities to show how much of one exists relative to the other.
In word problems, ratios help us understand relationships between different quantities. For example, in the exercise with Ann and Bob, we're asked to find the ratio of the time Ann spends driving to the time Bob spends driving.
To solve this, we need to find the amount of time each person spends driving and then compare these amounts. This involves creating expressions for their speeds and distances and then converting these into time.
Once we have these times, we form a ratio by dividing one by the other, simplifying where possible. This helps us see, in a straightforward way, how one quantity compares to another.

average speed calculations

Calculating average speeds is a common requirement in problems involving motion. The average speed is fundamentally the total distance traveled divided by the total time taken.
For example, if Bob's average speed is represented as $v$, the total distance he drives is $d$, and if Ann's average speed is calculated to be one-third faster than Bob's, the process involves determining her speed as $\frac{4v}{3}$.
When distances and speeds are given, calculating the time spent traveling is straightforward: divide the distance by the speed. For Bob, his driving time would be $\frac{d}{v}$. For Ann, who drives twice the distance at a different speed, her time would be given by $\frac{2d}{\frac{4v}{3}}=\frac{3d}{2v}$.
Understanding these average speed calculations enables students to solve for time, which is often a crucial step for comparing different quantities, like the driving times of Ann and Bob.

time, speed, and distance relationship

The relationship between time, speed, and distance is foundational in solving travel-related problems. The basic formula connecting these three variables is:

• Speed (v) = Distance (d) / Time (t)
• Distance (d) = Speed (v) * Time (t)
• Time (t) = Distance (d) / Speed (v)

In the given problem: Ann drives twice the distance of Bob and at a speed which is greater by one-third of Bob's speed. Using the relationships above, we can express these relationships in terms of variables and solve for the amount of time each person spends driving.
By understanding that increasing the distance while maintaining speed increases the driving time, or increasing the speed while maintaining distance decreases the driving time, students can better grasp how these variables interplay. Finally, determining the ratio of these resulting times gives us a deeper insight into their travel specifics and helps answer the problem effectively.