ISEE Upper Level

Distance, Speed & Time

Q: Why do we use fractions for time and rate?

Because division is the same thing as a fraction! Writing `\frac{10}{2}` is just a cleaner way for mathematicians to write '10 divided by 2'. It makes simplifying the math much easier.

Solving problems using d = r × t, average speed, and multi-leg journey calculations

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Have you ever wondered how long it would take a cheetah riding a skateboard to travel to your favorite pizza place? 🐆🍕 That's exactly what distance, speed, and time problems are all about! On the ISEE, you will act like a math detective figuring out how far someone traveled, how fast they were going, or how much time their epic journey took.

To solve these mysteries, we use a super famous math rule: Distance equals Rate times Time. Think of "Rate" as just a fancy test-word for "Speed." If you ride your bike at a speed of 10 miles per hour for 2 hours, you just multiply them together to find out you traveled 20 miles! Easy peasy, right? 🚲💨

The ISEE Quantitative Reasoning and Mathematics Achievement sections love to test this. Sometimes they will give you the distance and the time, and ask you to find the speed. Other times, they might ask about a tricky multi-leg journey, like walking to the park and then running back home. Just remember to keep your units (like miles and hours) matching, and you'll be zooming through these questions in no time! Plus, there is no penalty for guessing on the ISEE, so if you ever get stuck on a speed bump, just pick your favorite letter and keep on rolling!

Key Formula

The ultimate magic spell for these problems is the D-R-T formula:

d = r \cdot t

(Distance = Rate × Time). If you need to find Rate or Time instead, just use fractions! Rate is

r = \frac{d}{t}

and Time is

t = \frac{d}{r}

Practice Questions

3 practice questions for ISEE Upper Level

Q1 Hard

A freight train leaves a station traveling at a constant speed of

40

miles per hour. Two hours later, an express train leaves the same station traveling in the same direction on a parallel track at a constant speed of

60

miles per hour. How many hours will the express train need to travel to catch up to the freight train?

2

3

4

6

Show Solution

First, determine how far the freight train traveled before the express train started. The freight train traveled for $2$ hours at $40$ miles per hour, giving it a head start of $40 \times 2 = 80$ miles. The express train travels at $60$ miles per hour, which is $20$ miles per hour faster than the freight train ( $60 - 40 = 20$ ). This means the express train closes the $80$ -mile gap at a rate of $20$ miles per hour. To find the time it takes to catch up, divide the distance of the head start by the difference in speeds: $\frac{80}{20} = 4$ hours.

Answer: C

Q2 Hard

Marcus drives from his home to a mountain cabin at a constant speed of

30

miles per hour. He drives back home along the exact same route at a constant speed of

60

miles per hour. What is his average speed, in miles per hour, for the entire round trip?

40

45

48

50

Show Solution

Average speed is defined as the total distance divided by the total time. It is a common trap to simply average the two speeds ( $\frac{30 + 60}{2} = 45$ ), but Marcus spends more time driving at the slower speed, so the true average will be lower. To solve, pick a convenient distance for the one-way trip, such as a multiple of both $30$ and $60$ . Let the distance to the cabin be $60$ miles. The trip there takes $\frac{60}{30} = 2$ hours. The return trip takes $\frac{60}{60} = 1$ hour. The total distance for the round trip is $60 + 60 = 120$ miles, and the total time is $2 + 1 = 3$ hours. The average speed is $\frac{120}{3} = 40$ miles per hour.

Answer: A

Q3 Hard

Two drivers start at the same time from opposite ends of a straight

300

-mile highway and drive toward each other at constant speeds. Which of the following pieces of information is sufficient to determine the exact location on the highway where the two cars will meet?

A the average of the two cars' speeds

B the ratio of the two cars' speeds

C the sum of the two cars' speeds

D the difference in the two cars' speeds

Show Solution

To find the exact meeting location, we need to know what fraction of the $300$ miles each car travels. Since both cars start at the same time and meet at the same time, their travel times are equal. The distance formula is $d = r t$ . If time ( $t$ ) is constant, the distance traveled by each car is directly proportional to its speed. Therefore, the ratio of the distances they travel is equal to the ratio of their speeds ( $\frac{d _{1}}{d _{2}} = \frac{r _{1}}{r _{2}}$ ). If we know the ratio of their speeds, we can divide the total $300$ miles into proportional parts to find exactly how far each car traveled. Knowing the sum, difference, or average of their speeds does not give us the individual proportions needed to find the specific meeting point.

Answer: B

Tips & Strategies

Draw a 'DIRT' triangle to remember the formulas! Write 'D' at the top of a triangle, and 'R' and 'T' at the bottom. To find one, cover it up! Cover D, and you see R next to T ( $r \cdot t$ ). Cover T, and you see D over R ( $\frac{d}{r}$ ).
Watch your units! If your speed is in miles per hour but your time is in minutes, you must convert the minutes into a fraction of an hour (like 30 minutes = $\frac{1}{2}$ hour) before doing any math.
For Quantitative Comparison questions, check if you even need to do the math. If Column A has a longer distance but a much faster speed than Column B, you might be able to estimate the answer without doing long division.

Common Mistakes

Watch out for 'Average Speed' tricks! You CANNOT just add two speeds together and divide by 2. To find average speed for a whole trip, you must use $\frac{Total Distance}{Total Time}$ .
Don't forget to check that your units match. Multiplying miles per hour by minutes without converting will give you a wacky, wrong answer that the test makers might put as a trap answer choice!

Frequently Asked Questions

What does the word 'Rate' mean on the ISEE?

Rate is just a fancy test-word for 'speed'. If a question asks for the rate of a car, it just wants to know how fast the car is going in miles per hour or feet per second!

What if a trip has two different parts?

Break it into chunks! Create a mini D-R-T chart for Part 1 of the trip, and another chart for Part 2. Solve them separately before adding the distances or times together.

Is there a penalty if I guess on a really hard speed question?

Nope! The ISEE does not have a guessing penalty. If a multi-leg journey question is taking up too much time, take your best guess, bubble it in, and move on to an easier question.

Why do we use fractions for time and rate?

Because division is the same thing as a fraction! Writing $\frac{10}{2}$ is just a cleaner way for mathematicians to write '10 divided by 2'. It makes simplifying the math much easier.

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