Mathematics, three units

When do you use a distance-speed-time table versus the work rate method?

Quick answer

Use a distance-speed-time table whenever an object moves across a physical distance, applying the rule that distance equals speed times time. Use the work rate method when people or machines complete a job with no distance involved. In work problems, represent the rate as the fraction of the whole task completed in one hour, and add those hourly fractions together.

The topic explained

How to tell motion and work problems apart

Choosing between a distance-speed-time table versus the work rate method comes down to what is being completed. If a car, runner, or boat travels between locations, there is a measurable physical distance. You can track this clearly with a three-column table labeled distance, speed, and time.

A work problem involves completing an entire job, such as painting a wall, digging a trench, or filling a pool. Because there is no distance in kilometers or miles, a motion table fails. Instead, set the entire finished job equal to 1, and find what fraction of that job each worker completes in one hour.

Setting up the motion table

For motion problems, create rows for each traveler or each part of a journey. Use the core formula: distance equals speed times time. When you fill in two cells in any row, the third cell is automatic. For example, if a car covers 180 kilometers at an unknown speed x, its time cell is 180 divided by x.

Your equation comes from the sentence connecting the rows. If a faster trip took one hour less, write the slower time minus the faster time equals 1. Watch your units carefully: convert 45 minutes to 0.75 hours before entering it into the table, because dividing minutes by 60 is necessary whenever speed is in kilometers per hour.

Working with the part completed in one hour

In work problems, you cannot add the total hours together. If Worker A finishes a job in 6 hours and Worker B finishes in 3 hours, their combined time is not 9 hours. Working together must take less time, not more.

Instead, add their hourly production rates. Worker A does 1/6 of the job per hour, and Worker B does 1/3 of the job per hour. Together, they complete 1/6 plus 1/3, which equals 1/2 of the job in one hour. Since half the job takes one hour, the entire job takes the reciprocal: 2 hours.

Free printable worksheet

Motion Tables and Work Rates Practice

Decide whether each problem requires a distance-speed-time table or the part-done-in-one-hour method, set up the relationship, and solve.

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Personal AI lessons and practice, from preschool to university

Free printable worksheetlessoons.com

Motion Tables and Work Rates PracticeName:

Identify the method: A cyclist travels 60 km to a park and returns along the same route at a slower speed.
Identify the method: An inlet pipe fills a tank in 4 hours while a drain empties it in 6 hours.
Convert 40 minutes into an hourly value for a distance-speed-time equation.
A painter finishes a room in 4 hours. A second painter finishes the same room in 12 hours. What fraction of the room do they finish together in one hour?
Using the painter problem above, how many hours does it take both painters working together to finish the entire room?
A delivery van drives 120 km. At a speed of x km/h, write the expression for time. Then write the time expression if speed increases by 10 km/h.
A motorboat goes 30 km downstream in 2 hours and 30 km upstream in 3 hours. Find the boat speed in still water and the current speed.

Common questions

Why can you never add individual completion times in a work problem?

Adding completion times implies two people take longer than one person working alone. People working together combine their effort, which increases productivity and reduces total time. You must add rates (work per hour), not total hours.

How do you account for a scheduled rest stop in a motion problem?

A break has zero speed and zero travel distance. Create a separate row in your table for the rest stop, or subtract the break time from total journey time before calculating average travel speed. Otherwise, your calculated speed will be artificially low.

What should you do if an equation gives a positive and a negative solution?

Check the physical context of the problem. Speeds, travel times, and construction durations cannot be negative values. You discard the negative root and keep only the positive answer, explaining that physical rates must be greater than zero.

How do river currents and wind speeds affect travel speed in a table?

Moving with the current or a tailwind adds to the vehicle's still-water or still-air speed (v + c). Moving against the current or into a headwind subtracts from that speed (v - c).

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