Distance equals rate times time, written d = rt, and that one relationship carries almost every travel question on the ACT. Work problems run on a close cousin of it. Once you get used to labeling which quantity you have and which one you want, these stop being word problems and start being one-line algebra.
One relationship, three ways to use it
If you know any two of distance, rate, and time, you have the third. It is the same equation rearranged, so there is nothing extra to memorize, which matters because the ACT does not give you a formula sheet.
- Distance. A car drives 3 hours at 55 miles per hour, so d = 55 times 3 = 165 miles.
- Time. A trip covers 210 miles at 60 miles per hour, so t = 210 divided by 60 = 3.5 hours.
- Rate. A train covers 150 miles in 2.5 hours, so r = 150 divided by 2.5 = 60 miles per hour.
Rates and proportional relationships live in Integrating Essential Skills, which is 20 percent of the Math section, and they also feed the modeling questions that make up at least 20 percent. They are worth real practice time.
Two objects moving at once
When two things move toward each other, add the rates. Two trains start 300 miles apart and head toward one another at 50 and 70 miles per hour. The gap closes at 50 + 70 = 120 miles per hour, so they meet after 300 divided by 120 = 2.5 hours. Check it: 50 times 2.5 = 125 miles, 70 times 2.5 = 175 miles, and 125 + 175 = 300. When one object chases another, subtract the rates instead, since that difference is how fast the gap shrinks.
Average speed is not the average of the speeds
This is the single most reliable trap in the topic. Average speed is total distance divided by total time, full stop.
Suppose you drive 120 miles to a friend's house at 60 miles per hour, then return the same 120 miles at 40 miles per hour. The trip out takes 120 divided by 60 = 2 hours. The trip back takes 120 divided by 40 = 3 hours. Total distance is 240 miles and total time is 5 hours, so the average speed is 240 divided by 5 = 48 miles per hour.
It is not 50. Averaging 60 and 40 ignores the extra hour you spent at the slower speed, and when the distances are equal the true average always lands below that midpoint.
Make the units match first
Mismatched units create wrong answers that look perfectly reasonable. Convert before you compute.
- A printer runs at 25 pages per minute. In 4 hours it prints 4 times 60 = 240 minutes of work, so 240 times 25 = 6000 pages.
- To convert 30 miles per hour into feet per second, note that 30 miles is 30 times 5280 = 158,400 feet, and one hour is 3600 seconds. So the rate is 158,400 divided by 3600 = 44 feet per second.
Work problems
Work problems use the same logic with a job in place of a distance. If someone finishes a job in h hours, then in one hour that person completes 1/h of the job. Add the hourly fractions to get the combined hourly rate, then invert it to get the time.
Maya paints a room in 6 hours and Dev paints the same room in 3 hours. Per hour, Maya does 1/6 of the room and Dev does 1/3, and 1/6 + 1/3 = 1/6 + 2/6 = 3/6, which is 1/2 of the room per hour. At half a room per hour, the whole room takes 2 hours.
A second one: one pipe fills a tank in 4 hours and another fills it in 12 hours. Together they fill 1/4 + 1/12 = 3/12 + 1/12 = 4/12, which is 1/3 of the tank per hour, so the tank fills in 3 hours. The combined time is always shorter than the faster worker alone, which is a free sanity check.
Common traps
- Averaging speeds directly. Total distance over total time, every time.
- Adding the times in a work problem. You add the hourly fractions of the job, not the hours.
- Forgetting to invert.Getting 1/2 of a job per hour and answering "one half hour" instead of 2 hours.
- Leaving minutes as a decimal. A 90 minute trip is 1.5 hours, not 1.30 hours, and 45 minutes is 0.75 hours.
- Answering the wrong quantity. Some questions want the meeting time, others the distance one object covered by then.
A short drill
Three quick ones. A cyclist covers 45 miles in 3 hours, so the rate is 15 miles per hour, and 60 miles would take 4 hours. A driver goes 100 miles at 50 miles per hour and another 100 at 25 miles per hour, taking 2 hours and 4 hours, so the average speed is 200 divided by 6, or about 33.3 miles per hour. Two workers who finish a job alone in 10 and 15 hours together do 1/10 + 1/15 = 3/30 + 2/30 = 5/30, which is 1/6 of the job per hour, so the job takes 6 hours.
These set up as equations more often than people expect, so comfort with linear equations makes the algebra painless. Rates share their proportional backbone with percent problems, and you can see where both sit in the full list of ACT Math topics.
Start practicing
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