Question

sarah, tom, and alex, are competing in a \\(12\text{ km}\\) walking race. their speeds are recorded in the table.

\

$$\begin{array}{|c|c|} \\hline \\textbf{racer} & \\textbf{speed} \\\\ \\hline \\text{sarah} & 4\\frac{1}{4}\\text{ km/h} \\\\ \\hline \\text{tom} & 5\\frac{1}{4}\\text{ km/h} \\\\ \\hline \\text{alex} & 6\\frac{1}{3}\\text{ km/h} \\\\ \\hline \\end{array}$$

select the two correct statements about the race.

alex will finish the race in exactly 2 hours.
sarah will finish the race in less than 3 hours.
alex will finish the race in exactly 2 hours.
the difference between sarah and alexs finish times is less than 30 minutes.
tom will finish the race faster than sarah but slower than alex.

Explanation:

Calculate individual finish times

Using the Rate Word Problems and Mixed Numbers knowledge points:

• Distance \(d = 12\text{ km}\)
• Sarah's speed \(v_S = 4\frac{1}{4} = \frac{17}{4}\text{ km/h}\)
• Tom's speed \(v_T = 5\frac{1}{4} = \frac{21}{4}\text{ km/h}\)
• Alex's speed \(v_A = 6\frac{1}{3} = \frac{19}{3}\text{ km/h}\)

Calculate times using \(t = \frac{d}{v}\):

• Sarah's time: \(t_S = \frac{12}{17/4} = \frac{48}{17} \approx 2.82\text{ hours}\)
• Tom's time: \(t_T = \frac{12}{21/4} = \frac{48}{21} = \frac{16}{7} \approx 2.29\text{ hours}\)
• Alex's time: \(t_A = \frac{12}{19/3} = \frac{36}{19} \approx 1.89\text{ hours}\)

Evaluate each statement

• Statement 1 & 3 (Alex finishes in exactly 2 hours):

\(t_A \approx 1.89\text{ hours}
eq 2\text{ hours}\). (Incorrect)

• Statement 2 (Sarah finishes in less than 3 hours):

\(t_S \approx 2.82\text{ hours} < 3\text{ hours}\). (Correct)

• Statement 4 (Difference between Sarah and Alex's times is less than 30 minutes):

\(t_S - t_A = \frac{48}{17} - \frac{36}{19} = \frac{912 - 612}{323} = \frac{300}{323} \approx 0.929\text{ hours} \approx 55.7\text{ minutes}\). This is not less than 30 minutes. (Incorrect)

• Statement 5 (Tom finishes faster than Sarah but slower than Alex):

Since \(t_A < t_T < t_S\) (\(1.89 < 2.29 < 2.82\)), Tom is faster than Sarah and slower than Alex. (Correct)

Answer:

• Alex will finish the race in exactly 2 hours.
• Sarah will finish the race in less than 3 hours. (Correct answer)
• Alex will finish the race in exactly 2 hours.
• The difference between Sarah and Alex's finish times is less than 30 minutes.
• Tom will finish the race faster than Sarah but slower than Alex. (Correct answer)