QuanticaPrealgebraOpen in the courseA rate compares two quantities in different units, like dollars per pound or miles per hour. The word per signals a rate. A unit rate is the per-one value, how much of the first thing goes with exactly one unit of the second.
Problem
Dried cherries cost 20 dollars for 4 pounds. What is the price per pound, in dollars?
Show a hint
- The sign packs four pounds together into one price. To find what just one pound costs, share the \(20\) dollars equally among the \(4\) pounds.
- Divide the dollars by the pounds, \(\tfrac{20 \text{ dollars}}{4 \text{ pounds}}\). That single quotient is the price for one pound.
Show the full solution
Divide the dollars by the pounds. \(\tfrac{20 \text{ dollars}}{4 \text{ pounds}} = 5\) dollars per pound, so one pound costs \(\boxed{5}\) dollars. A unit rate is always the first quantity divided by the second, which here splits the total price evenly across the pounds.
Problem
A reader finishes a 426-page novel in 6 days at a steady pace. How many pages per day did she read?
Show a hint
- A unit rate is the amount that goes with one of the second quantity. Here the second quantity is days, so you want the pages that go with one day. That means sharing all the pages equally among the 6 days.
- Divide the total pages by the number of days. Compute \(426 \div 6\), and the result is the pages she reads each day.
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Divide the pages by the days. $$\frac{426 \text{ pages}}{6 \text{ days}} = 71 \text{ pages per day}$$ So she read \(\boxed{71}\) pages per day. Going backward checks it, since \(71 \times 6 = 426\) pages, the whole novel.
Problem
Two olive-oil bottles: 15 oz for \$6.00, or 24 oz for \$8.40. Find the price per ounce of each and identify the better buy. What is the price per ounce of the better buy, in dollars?
Show a hint
- A unit rate is a cost for one of something. For each bottle, divide the total dollars by the number of ounces to get dollars for one ounce. Do this twice, once per bottle.
- First bottle: \(\tfrac{\$6.00}{15 \text{ oz}} = \$0.40\) per ounce. Second bottle: \(\tfrac{\$8.40}{24 \text{ oz}} = \$0.35\) per ounce. The better buy is the smaller of these two numbers.
Show the full solution
Divide dollars by ounces for each bottle. First bottle, \(\tfrac{\$6.00}{15 \text{ oz}} = \$0.40\) per ounce. Second bottle, \(\tfrac{\$8.40}{24 \text{ oz}} = \$0.35\) per ounce. The second bottle is cheaper per ounce, so it is the better buy at \(\boxed{0.35}\) dollars per ounce. The larger total price does not settle a best buy, only the price for one unit does.
Problem
Printer A outputs 1,800 pages in 40 minutes. Printer B outputs 1,450 pages in 25 minutes. Find each printer's pages-per-minute rate. What is the rate of the faster printer?
Show a hint
- The page totals are over different times, so you cannot compare them head to head yet. Knock each printer down to one minute first. For each printer divide its pages by its minutes.
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Divide pages by minutes for each printer. Printer A, \(\tfrac{1{,}800 \text{ pages}}{40 \text{ minutes}} = 45\) pages per minute. Printer B, \(\tfrac{1{,}450 \text{ pages}}{25 \text{ minutes}} = 58\) pages per minute. Printer B is faster, at \(\boxed{58}\) pages per minute. The raw page totals cannot be compared until both machines are measured over the same single minute.
Problem
A drive covers 195 miles in 3 hours at a steady pace. What is the speed in miles per hour?
Show a hint
- The unit miles per hour is a recipe. It says take the miles and divide by the hours, which lands you on the miles covered in one hour.
- Compute \(195 \div 3\). That tells you how many miles fit into each single hour of driving.
Show the full solution
Divide the miles by the hours. $$\frac{195 \text{ miles}}{3 \text{ hours}} = 65 \text{ miles per hour}$$ The speed is \(\boxed{65}\) miles per hour. The unit names the operation, since miles per hour means miles divided by hours.
Problem
A van cruises at 48 miles per hour for 7 hours straight. How many miles did it travel?
Show a hint
- Speed tells you the miles gained in one hour. To get the miles for all the hours, you do not divide here, you build up, so distance is speed times time.
- Multiply the \(48\) miles in each hour by the \(7\) hours the van kept driving.
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Distance is speed times time, so \(48 \tfrac{\text{miles}}{\text{hour}} \times 7 \text{ hours} = 336\) miles. The van covered \(\boxed{336}\) miles. The hours in the rate cancel the hours of driving and leave plain miles, the unit a distance should have.
Problem
An e-bike can hold 56 mph on a 392-mile route. How many hours does the ride take?
Show a hint
- Speed connects distance, time, and miles per hour. You have the distance and the speed, so flip the relationship around. Time is distance divided by speed.
- Each hour covers 56 miles, so count how many 56 mile chunks fit in 392 miles. Compute \(392 \div 56\).
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Time is distance divided by speed, so \(\tfrac{392 \text{ miles}}{56 \text{ miles per hour}} = 7\) hours. The ride takes \(\boxed{7}\) hours. Running it forward checks the answer, since \(56 \times 7 = 392\) miles.
Problem
A scooter covers 21 miles in 45 minutes. Convert 45 minutes to hours first, then find the speed in miles per hour.
Show a hint
- The unit you want is miles per hour, so the time underneath the miles has to be in hours, not minutes. Turn 45 minutes into a fraction of an hour first by writing it over the 60 minutes in one hour.
- You have \(\tfrac{45}{60} = \tfrac{3}{4}\) of an hour, which is \(0.75\) hours. Now divide the distance by that time, \(21 \div 0.75\), and the result comes out in miles per hour.
Show the full solution
Put the time in hours first. \(45\) minutes is \(\tfrac{45}{60} = \tfrac{3}{4}\) of an hour, which is \(0.75\) hours. Now divide. $$\frac{21 \text{ miles}}{0.75 \text{ hours}} = 28 \text{ miles per hour}$$ The speed is \(\boxed{28}\) miles per hour. Miles per hour needs hours underneath the miles, so leaving the time in minutes gives the wrong number.
Problem
A commuter train travels at 64 mph for 105 minutes. Convert 105 minutes to hours, then find the distance in miles.
Show a hint
- The speed is per hour, so the time must be in hours too. Turn \(105\) minutes into hours by dividing by \(60\), since one hour is \(60\) minutes. You should get \(\tfrac{105}{60} = 1.75\) hours.
- Now that the units match, use distance \(=\) speed \(\times\) time. Multiply \(64\) miles per hour by \(1.75\) hours and watch the "hours" cancel, leaving miles.
Show the full solution
The speed is per hour, so convert the time. $$105 \text{ minutes} = \tfrac{105}{60} \text{ hours} = 1.75 \text{ hours}$$ Then use distance \(=\) speed \(\times\) time. $$64 \tfrac{\text{miles}}{\text{hour}} \times 1.75 \text{ hours} = 112 \text{ miles}$$ The train traveled \(\boxed{112}\) miles. The hours cancel and leave miles, which is the check that the units were set up right.
Problem
A road trip has two legs: 84 miles in 2 hours, then 161 miles in 3 hours. What is the average speed for the whole trip in miles per hour?
Show a hint
- Average speed is not the average of the two separate speeds. It is the total distance you traveled divided by the total time you spent. First add up all the miles, then add up all the hours.
- The miles add to \(84 + 161 = 245\) miles and the hours add to \(2 + 3 = 5\) hours. Now divide the total distance by the total time, \(245 \div 5\).
Show the full solution
Add the miles, add the hours, then divide. The trip covers \(84 + 161 = 245\) miles in \(2 + 3 = 5\) hours, so $$\frac{245 \text{ miles}}{5 \text{ hours}} = 49 \text{ miles per hour}.$$ The average speed is \(\boxed{49}\) miles per hour. Averaging the leg speeds of \(42\) and about \(53.7\) would give something else, because the two legs took different amounts of time.
Problem
A courier rides 210 miles out at 30 mph (headwind) and 210 miles back at 70 mph (tailwind). The simple midpoint of 50 mph is wrong. What is the average speed for the whole round trip?
Show a hint
- Average speed is never the average of the two speeds. It is total distance divided by total time, so your first job is to find each leg's time. For the trip out, time is distance divided by speed, which is \(\tfrac{210 \text{ miles}}{30 \text{ miles per hour}}\). Do the same for the trip back at \(70\) miles per hour.
- The trip out takes \(\tfrac{210}{30} = 7\) hours and the trip back takes \(\tfrac{210}{70} = 3\) hours, so the total time is \(10\) hours. The total distance is the two equal legs added together, \(210 + 210 = 420\) miles. Now divide \(420\) miles by \(10\) hours.
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Find each leg's time with time equals distance divided by speed. The leg out takes \(\tfrac{210}{30} = 7\) hours and the leg back takes \(\tfrac{210}{70} = 3\) hours, so the round trip is \(420\) miles in \(10\) hours. $$\frac{420 \text{ miles}}{10 \text{ hours}} = 42 \text{ miles per hour}$$ The average speed is \(\boxed{42}\) miles per hour. It sits below \(50\) because \(7\) of the \(10\) hours were spent on the slow leg.
Problem
Two trucks 300 miles apart drive toward each other at 45 mph and 55 mph. How many hours until they meet?
Show a hint
- You do not have to track each truck on its own. Think about the gap between them. Every hour it gets smaller by both speeds added together, so the gap closes at \(45 + 55\) miles per hour.
- Once you know how fast the gap closes, treat it like any time question. Time equals the distance to close divided by the closing speed, so divide the \(300\) mile gap by that combined speed.
Show the full solution
The gap closes at the combined speed, \(45 + 55 = 100\) miles per hour. The gap starts at \(300\) miles, so they meet after \(\tfrac{300 \text{ miles}}{100 \text{ miles per hour}} = \boxed{3}\) hours. Watching the gap is easier than tracking each truck, and the check works, since \(45 \times 3 = 135\) and \(55 \times 3 = 165\) add to \(300\) miles.
Practice these ideas
Practice
A fruit cart sells 7 pounds of apples for 28 dollars. What is the price per pound, in dollars?
Show the solution
Divide the dollars by the pounds. \(\tfrac{28 \text{ dollars}}{7 \text{ pounds}} = 4\), so one pound costs \(\boxed{4}\) dollars. Every pound costs the same, so splitting the total evenly across the pounds gives the price of one.
Practice
A resting heart beats 312 times in 4 minutes. What is the heart rate in beats per minute?
Show the solution
Divide the beats by the minutes. $$\frac{312 \text{ beats}}{4 \text{ minutes}} = 78 \text{ beats per minute}$$ The heart rate is \(\boxed{78}\) beats per minute.
Practice
Two bags of brown rice: 12 oz for \$3.60, or 20 oz for \$5.00. Find the price per ounce of each and identify the better buy. What is that price per ounce, in dollars?
Show the solution
Turn each bag into a price for one ounce by dividing the cost by the number of ounces. The smaller bag gives \(\tfrac{\$3.60}{12 \text{ ounces}} = \$0.30\) per ounce. The larger bag gives \(\tfrac{\$5.00}{20 \text{ ounces}} = \$0.25\) per ounce. Now compare. Since \(0.25\) is less than \(0.30\), the larger bag is the better buy, and its price per ounce is \(\boxed{0.25}\) dollars.
Practice
Machine A makes 240 parts in 8 hours. Machine B makes 350 parts in 10 hours. What is the rate of the faster machine, in parts per hour?
Show the solution
Find the unit rate for each machine by dividing its parts by its hours. For machine A, \(\tfrac{240 \text{ parts}}{8 \text{ hours}} = 30\) parts per hour. For machine B, \(\tfrac{350 \text{ parts}}{10 \text{ hours}} = 35\) parts per hour. Now compare the two per-hour numbers. Machine B makes 35 parts every hour and machine A makes only 30, so machine B is the faster one. Its rate is \(\boxed{35}\) parts per hour.
Practice
A commuter train covers 280 miles in 5 hours at a steady pace. What is the train's speed in miles per hour?
Show the solution
Divide the distance by the time. $$\frac{280 \text{ miles}}{5 \text{ hours}} = 56 \text{ miles per hour}$$ The train's speed is \(\boxed{56}\) miles per hour. A quick check, \(56 \times 5 = 280\) miles, matches the distance given.
Practice
A maglev shuttle holds a steady 62 miles per hour for 4 hours. How many miles does it travel?
Show the solution
Distance is speed times time. $$62 \text{ miles per hour} \times 4 \text{ hours} = 248 \text{ miles}$$ The shuttle covers \(\boxed{248}\) miles. The hours cancel against the hours in the rate, leaving miles.
Practice
A tour bus has 414 miles ahead and cruises at 46 miles per hour. How many hours does the trip take?
Show the solution
Time is distance divided by speed. $$\text{time} = \frac{414 \text{ miles}}{46 \text{ miles per hour}} = 9 \text{ hours}$$ The trip takes \(\boxed{9}\) hours. The miles cancel against the miles in "miles per hour" and leave hours, exactly the unit we want.
Practice
A cyclist covers 35 miles in 50 minutes. Convert 50 minutes to hours (\(\tfrac{5}{6}\) hr), then find the speed in miles per hour.
Show the solution
First put the time into hours. Fifty minutes is \(\tfrac{50}{60} = \tfrac{5}{6}\) of an hour. Speed is distance divided by time, so $$\frac{35 \text{ miles}}{\tfrac{5}{6} \text{ hour}} = 35 \times \frac{6}{5} = \frac{210}{5} = 42 \text{ miles per hour}.$$ The speed is \(\boxed{42}\).
Practice
A survey drone flies at 72 mph for 25 minutes. Convert 25 minutes to hours first, then find the distance covered in miles.
Show the solution
The speed is given per hour, so first change \(25\) minutes into hours. One hour is \(60\) minutes, so \(25\) minutes is \(\tfrac{25}{60}=\tfrac{5}{12}\) of an hour. Distance is speed times time, so \(72 \text{ miles per hour} \times \tfrac{5}{12} \text{ hour} = \tfrac{72 \times 5}{12} = \tfrac{360}{12} = 30\) miles. The drone covers \(\boxed{30}\) miles.
Practice
A scooter rides at 36 mph to a drop-off 9 miles away. Find the travel time in minutes.
Show the solution
Start with time equals distance divided by speed. The distance is \(9\) miles and the speed is \(36\) miles per hour, so the time in hours is $$\frac{9 \text{ miles}}{36 \text{ miles per hour}} = \tfrac{1}{4} \text{ hour}.$$ Now turn that quarter of an hour into minutes. One hour is \(60\) minutes, so multiply. $$\tfrac{1}{4} \text{ hour} \times 60 \text{ minutes per hour} = 15 \text{ minutes}.$$ The trip takes \(\boxed{15}\) minutes.
Practice
A van drives two legs: 90 miles in 2 hours through hills, then 140 miles in 3 hours on the highway. What is the average speed for the whole route in miles per hour?
Show the solution
Add the miles, add the hours, then divide. The route is \(90 + 140 = 230\) miles in \(2 + 3 = 5\) hours, so $$\frac{230 \text{ miles}}{5 \text{ hours}} = 46 \text{ miles per hour}.$$ The average speed is \(\boxed{46}\) miles per hour. The mean of the leg speeds would give \(45\tfrac{5}{6}\) instead, since the van spent different amounts of time on each leg.
Practice
A kayaker paddles 60 miles upstream at 20 mph, then 60 miles downstream at 30 mph. The simple midpoint of 25 mph is wrong. What is the average speed for the whole round trip?
Show the solution
Find each leg's time with time equals distance divided by speed. Upstream takes \(\tfrac{60 \text{ miles}}{20 \text{ miles per hour}} = 3\) hours and downstream takes \(\tfrac{60 \text{ miles}}{30 \text{ miles per hour}} = 2\) hours, so the round trip is \(120\) miles in \(5\) hours. That gives \(\tfrac{120 \text{ miles}}{5 \text{ hours}} = \boxed{24}\) miles per hour. It lands below the midpoint of \(25\) because the slow upstream leg took more of the clock.
Practice
Two drones start from opposite ends of a 264-mile tunnel and fly toward each other at 38 mph and 50 mph. How many hours until they meet?
Show the solution
The gap closes at the combined speed, \(38 + 50 = 88\) miles per hour. The tunnel is \(264\) miles long, so they meet after \(\tfrac{264 \text{ miles}}{88 \text{ miles per hour}} = \boxed{3}\) hours. Checking, the drones cover \(38 \times 3 = 114\) miles and \(50 \times 3 = 150\) miles, which add to the full \(264\).
Practice
Two cyclists start 350 miles apart and pedal toward each other at 30 mph and 40 mph. Find when they meet, then calculate how far the faster cyclist has traveled.
Show the solution
The gap closes at the combined speed, \(30 + 40 = 70\) miles per hour, so the cyclists meet after \(\tfrac{350 \text{ miles}}{70 \text{ miles per hour}} = 5\) hours. The faster cyclist rides \(40 \times 5 = \boxed{200}\) miles. As a check, the slower one covers \(30 \times 5 = 150\) miles, and \(200 + 150 = 350\) miles, the whole starting gap.