Question

Youssef lives \(x\) blocks from his office. It takes him 1 minute per block to walk to work and 20 seconds per block to ride his bicycle to work. If it takes him exactly 10 minutes more to walk to work than to ride his bicycle, then \(x\) equals A. 4 B. 7 C. 10 D. 15 E. 20

Short answer

The value of \( x \) is 15.

Step-by-step solution

Define the Variables

Let the number of blocks Youssef lives from his office be represented by the variable \( x \).

Express Walking Time in Terms of x

Since it takes Youssef 1 minute to walk each block, the total walking time in minutes is \( x \) minutes.

Express Bicycling Time in Terms of x

It takes Youssef 20 seconds per block to ride his bicycle. Convert 20 seconds to minutes: \( 20 \text{ seconds} = \frac{20}{60} = \frac{1}{3} \) minutes. Therefore, the total bicycling time in minutes is \( \frac{x}{3} \).

Set Up the Equation

According to the problem, it takes Youssef exactly 10 minutes more to walk than to ride his bicycle. Establish the equation based on this information: \( x = \frac{x}{3} + 10 \).

Solve the Equation

To solve for \( x \), first eliminate the fraction by multiplying the entire equation by 3: \[ 3x = x + 30 \] Next, subtract \( x \) from both sides to get all \( x \)-terms on one side: \[ 2x = 30 \] Finally, divide both sides by 2: \[ x = 15 \]

Verify the Solution

To ensure the solution is correct, check if the values satisfy the original condition. Walking time: \( 15 \) minutes. Bicycling time: \( \frac{15}{3} = 5 \) minutes. Difference: \( 15 - 5 = 10 \), which matches the problem statement.

Key concepts

Distance and Time Calculations

To solve this problem, we first need to understand the relationship between distance and time. Distance can often be expressed in terms of how long it takes to travel, which in turn depends on the mode of transportation. In the given problem, Youssef either walks or rides his bicycle to work. The walking time per block is 1 minute, so if he travels x blocks, the total walking time is simply x minutes. On the other hand, the bicycling time per block is 20 seconds. Since there are 60 seconds in a minute, 20 seconds is equivalent to 1/3 of a minute. Hence, if he rides his bicycle for x blocks, the total biking time is x/3 minutes.

Linear Equations

Linear equations are crucial in finding the values of unknown variables. In our problem, we are given that it takes Youssef exactly 10 minutes more to walk to work than to ride his bicycle. We set up the equation to express this situation: \ \( x = \frac{x}{3} + 10 \). This equation states that Youssef's walking time (x minutes) is equal to his bicycling time (x/3 minutes) plus an additional 10 minutes. Solving this equation helps us find the value of x.

Speed and Rate Problems

Understanding speed and rate is essential in problems involving distance and time. Speed is defined as distance divided by time. In our problem, Youssef's walking speed is 1 block per minute and his bicycling speed is 3 blocks per minute (since \( \frac{1}{\frac{1}{3}} = 3 \)). Using these rates, we calculate the time it takes him to travel x blocks using different forms of transportation. By comparing these times, we figure out the time difference stated in the problem.

Algebraic Expressions

Algebraic expressions allow us to represent relationships between different quantities. Here, they help express the total walking and bicycling times in terms of x. The equation \( x = \frac{x}{3} + 10 \) represents the given relationship. Solving this equation involves algebraic manipulations such as eliminating fractions by multiplying both sides by 3, resulting in \( 3x = x + 30 \). Subtracting x from both sides gives \( 2x = 30 \), and dividing by 2, we find \( x = 15 \). This result is then verified by plugging it back into the original expressions for walking and bicycling times to ensure they meet the given condition.