Objects R and S each travel at a constant speed. The speed of object R is half the speed of...

1. TRANSLATE the problem information

• Given information:
  • Object R travels \(\mathrm{4x}\) inches in \(\mathrm{y}\) seconds
  • Speed of R is half the speed of S
  • Need time for S to travel \(\mathrm{24x}\) inches
• What this tells us: We have R's distance and time, so we can find R's speed. The speed relationship will help us find S's speed.

2. INFER the solution approach

• Key insight: We need S's speed to find S's time
• Strategy: Find R's speed first → use relationship to get S's speed → calculate S's time

3. Calculate object R's speed

Using Speed = Distance/Time:

Speed of R = \(\mathrm{\frac{4x\ inches}{y\ seconds} = \frac{4x}{y}}\) inches per second

4. INFER object S's speed from the relationship

Since "speed of R is half the speed of S":

\(\mathrm{Speed\ of\ R = \frac{1}{2} \times Speed\ of\ S}\)

\(\mathrm{\frac{4x}{y} = \frac{1}{2} \times Speed\ of\ S}\)

\(\mathrm{Speed\ of\ S = 2 \times \frac{4x}{y} = \frac{8x}{y}}\) inches per second

5. SIMPLIFY to find the time for object S

Using Time = Distance/Speed:

\(\mathrm{Time\ for\ S = 24x\ inches \div \frac{8x}{y}\ inches\ per\ second}\)

\(\mathrm{Time\ for\ S = 24x \times \frac{y}{8x}}\)

\(\mathrm{= \frac{24xy}{8x}}\)

\(\mathrm{= 3y}\) seconds

Answer: B. \(\mathrm{3y}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "speed of R is half the speed of S" as "speed of S is half the speed of R," leading them to calculate S's speed as \(\mathrm{\frac{1}{2} \times \frac{4x}{y} = \frac{2x}{y}}\) instead of \(\mathrm{\frac{8x}{y}}\).

With this incorrect speed, they calculate:

\(\mathrm{Time = 24x \div \frac{2x}{y}}\)

\(\mathrm{= 24x \times \frac{y}{2x}}\)

\(\mathrm{= 12y}\)

This may lead them to select Choice A (\(\mathrm{12y}\)).

Second Most Common Error:

Poor INFER reasoning: Students correctly find both speeds but then apply S's speed to find the time for R to travel \(\mathrm{24x}\) inches instead of the time for S to travel \(\mathrm{24x}\) inches.

Using R's speed:

\(\mathrm{Time = 24x \div \frac{4x}{y}}\)

\(\mathrm{= 24x \times \frac{y}{4x}}\)

\(\mathrm{= 6y}\)

This may lead them to select Choice D (\(\mathrm{6y}\)).

The Bottom Line:

This problem requires careful attention to which object does what, and precise interpretation of comparative language about speeds. Students must systematically work through finding speeds before calculating times.