All PSAT Math Resources
Example Question #1 : How To Find Rate
Bob can build a house in 3 days. Gary can build a house in 5 days. How long does it take them to build a house together?
DO NOT pick 4 days, which would be the middle number between Bob and Gary's rates of 3 and 5 days respectively. The middle rate is the answer that students always want to pick, so the SAT will provide it as an answer to trick you!
Let's think about this intuitively before we actually solve it, so hopefully you won't be tempted to pick a trick answer ever again! Bob can build the house in 3 days if he works by himself, so with someone else helping him, it has to take less than 3 days to build the house! This will always be true. Never pick the middle rate on a combined rates problem like this!
Now let's look at the problem computationally. Bob can build a house in 3 days, so he builds 1/3 of a house in 1 day. Similarly, Gary can build a house in 5 days, so he builds 1/5 of a house in 1 day. Then together they build 1/3 + 1/5 = 5/15 + 3/15 = 8/15 of the house in 1 day.
Now, just as we did to see how much house Gary and Bob can build separately in one day, we can take the reciprocal of 8/15 to see how many days it takes them to build a house together. (When we took the reciprocal for Bob, 3 days/1 house = 1/3 house per day.) The reciprocal of 8/15 is 15/8, so they took 15/8 days to build the house together. 15/8 days is almost 2 days, which seems like a reasonable answer. Make sure your answer choices make sense when you are solving word problems!
Example Question #2 : How To Find Rate
A family is on a road trip from Cleveland to Virginia Beach, totaling 600 miles. If the first half of the trip is completed in 6.5 hours and the second half of the trip is completed in 5.5 hours, what is the average speed in miles per hour of the whole trip?
Take the total distance travelled (600 miles) and divide it by the total time travelled (6.5 hrs + 5.5 hrs = 12 hours) = 50 miles/hour
Example Question #3 : How To Find Rate
Two electric cars begin moving on circular tracks at exactly 1:00pm. If the first car takes 30 minutes to complete a loop and the second car takes 40 minutes, what is the next time they will both be at the starting point?
Call the cars “Car A” and “Car B”.
The least common multiple for the travel time of Car A and Car B is 120. We get the LCM by factoring. Car A’s travel time gives us 3 * 2 * 5; Car B’s time gives us 2 * 2 * 2 * 5. The smallest number that accommodates all factors of both travel times is 2 * 2 * 2 * 3 * 5, or 120. There are 60 minutes in an hour, so 120 minutes equals two hours. Two hours after 1:00pm is 3:00pm.
Example Question #23 : Proportion / Ratio / Rate
If Jon is driving his car at ten feet per second, how many feet does he travel in 30 minutes?
If Jon is driving at 10 feet per second he covers 10 * 60 feet in one minute (600 ft/min). In order to determine how far he travels in thirty minutes we must multiply 10 * 60 * 30 feet in 30 minutes.
Example Question #24 : Proportion / Ratio / Rate
An arrow is launched at 10 meters per second. If the arrow flies at a constant velocity for an hour, how far has the arrow gone?
There are 60 seconds in a minute and 60 minutes in an hour, therefore 3600 seconds in an hour. The arrow will travel 3600x10= 36,000 meters in an hour.
Example Question #1 : How To Find Rate
If Jack ran at an average rate of 7 miles per hour for a 21 mile course, and Sam ran half as fast for the same distance, how much longer did it take for Sam to run the course than Jack?
Using the rate formula: Distance = Rate x Time,
Since Jack’s speed was 7 mph, Jack completed the course in 3 hours
21 = 7 x t
t = 3
Sam’s speed was half of Jack’s speed: 7/2 = 3.5
21 = 3.5 x t
t = 6
Therefore it took Sam 3 hours longer to run the course.
Example Question #4 : How To Find Rate
If a pail collects x ounces of dripping water every 15 minutes, how many ounces will it collect in h hours?
Algebraic solution: First, convert minutes to hours.
60/15 = 4, so there are 4 15-minute increments in each hour. Therefore, 4x ounces of water are collected each hour. Multiply by h to get 4xh as the solution
Plug-in method: Just choose numbers.
x = 2
h = 3
If 2 ounces drip in 15 minutes, how many ounces will drip in one hour?
2/15 = x/60
15x = 120
x = 8
If 8 ounces drip in one hour, how many ounces will drip in 3 hours? (remember we chose that h = 3)
3 x 8 = 24
This is the answer we are looking for.
Plug x = 2, and h = 3 into each answer choice, to determine which will work. Remember you must plug into every answer choice in case more than one works. In that case, choose different values for x and h, and plug into only the choices that worked the first time.
Example Question #5 : How To Find Rate
Mary can make 20 snowballs in an hour. Mark can make 15 snowballs in 30 minutes. If they work together, how long will it take them to make 150 snowballs?
If Mark makes 15 snowballs in 30 minutes, he can make 30 snowballs in an hour. Working together they can 50 snowballs in one hour. 150 snowballs divided by the amount they can make in one hour (50) will give us the total time it will take them to make 150 snowballs. In this case, 3 hours.
Example Question #32 : Proportion / Ratio / Rate
Car X used 4 gallons of gas in one week, and gets 10 miles to the gallon. If car Y went the same number of miles but only gets 8 miles to the gallon, how much gas did car Y use?
We first use the data for car X to conclude that car X went 40 miles (4gallons*10mi/gallon). We then use 40 miles for car Y, and divide 40 by 8, to give us 5 gallons of gas.
Example Question #33 : Proportion / Ratio / Rate
Bob and Sally are doing chores. It takes them 10 hours to do one of their chores. Assuming everyone works at the same rate, how many of their friends would they need to get to help them to do their chores in 2 hours?
None of the above
Since the kids are trying to do their chores in one fifth of the time, they need five times as many people. Since they have two, five times as many would be ten. We subtract the two of them and that would mean they need 8 more people, giving us answer 8.