Finding Distance

A car travels at 80 km/h for 3 hours. How far does it go?

Identify the known values: Rate $r = 80$ km/h, Time $t = 3$ hours = We need to find distance $d$

Write the formula: $d = r \times t$ = Distance equals rate times time

Substitute the values: $d = 80 \times 3$ = Plug in the numbers

Calculate: $d = 240$ km = The car travels 240 km

Answer: The car travels **240 kilometers**.

Finding Rate (Speed)

A cyclist covers 45 km in 3 hours. What is the cyclist's average speed?

Identify the known values: Distance $d = 45$ km, Time $t = 3$ hours = We need to find rate $r$

Rearrange the formula: $r = \frac{d}{t}$ = Rate equals distance divided by time

Substitute the values: $r = \frac{45}{3}$ = Plug in the numbers

Calculate: $r = 15$ km/h = The cyclist's speed is 15 km/h

Answer: The cyclist's average speed is **15 km/h**.

Finding Time

A train travels at 120 km/h. How long will it take to travel 360 km?

Identify the known values: Rate $r = 120$ km/h, Distance $d = 360$ km = We need to find time $t$

Rearrange the formula: $t = \frac{d}{r}$ = Time equals distance divided by rate

Substitute the values: $t = \frac{360}{120}$ = Plug in the numbers

Calculate: $t = 3$ hours = The journey takes 3 hours

Answer: The train will take **3 hours** to complete the journey.

Unit Conversion Required

A plane flies at 600 km/h. How far does it travel in 90 minutes?

Identify the known values: Rate $r = 600$ km/h, Time $t = 90$ minutes = Units don't match!

Convert time to hours: $t = 90 \div 60 = 1.5$ hours = 90 minutes = 1.5 hours

Apply the formula: $d = r \times t = 600 \times 1.5$ = Now units are consistent

Calculate: $d = 900$ km = The plane travels 900 km

Answer: The plane travels **900 kilometers** in 90 minutes.

Two-Part Journey

Sarah drives 100 km at 50 km/h, then another 80 km at 40 km/h. What is her total travel time?

Find time for first part: $t_1 = \frac{100}{50} = 2$ hours = First part: 2 hours

Find time for second part: $t_2 = \frac{80}{40} = 2$ hours = Second part: 2 hours

Add the times: $t_{total} = t_1 + t_2 = 2 + 2$ = Total time = 4 hours

State the answer: Total travel time is 4 hours = Complete!

Answer: Sarah's total travel time is **4 hours**.

Mistake: Forgetting to convert units

Why: If rate is in km/h but time is in minutes, the formula won't work correctly.

Correct: Always convert so units match: km/h with hours, m/s with seconds, etc.

Mistake: Using the wrong formula rearrangement

Why: Students sometimes multiply when they should divide, or vice versa.

Correct: Remember the triangle: cover what you need, and the remaining operation is what you do. Or: $d = rt$, $r = d/t$, $t = d/r$.

Mistake: Adding speeds instead of finding total time

Why: For a two-part trip, you can't just average the speeds.

Correct: Calculate time for each part separately, then add the times together.

Mistake: Confusing rate with distance

Why: 60 km/h means traveling 60 km every hour, not that the total distance is 60 km.

Correct: Rate describes how fast (per unit time), distance describes how far (total).

Road Trip Planning

Drivers use distance-rate-time calculations to plan journeys and estimate arrival times.

A 450 km trip at an average speed of 90 km/h takes $\frac{450}{90} = 5$ hours.

Sports and Fitness

Athletes use these calculations to track performance and set training goals.

A runner who completes 10 km in 50 minutes has an average pace of $\frac{10}{50/60} = 12$ km/h.

Aviation and Flight Planning

Pilots calculate flight times and fuel requirements using distance, speed, and time.

A flight covering 2400 km at 800 km/h takes $\frac{2400}{800} = 3$ hours.

Q: What's the difference between speed and rate?

A: In most math problems, speed and rate mean the same thing - how fast something is moving. Rate is the more general term (it could apply to anything that happens over time), while speed specifically refers to motion.

Q: How do I know which formula to use?

A: Identify what you're solving for: If you need distance, use $d = rt$. If you need rate/speed, use $r = d/t$. If you need time, use $t = d/r$. A helpful trick: draw a triangle with $d$ on top and $r$ and $t$ on the bottom - cover what you need!

Q: What if the units don't match?

A: Convert them so they're consistent. If speed is in km/h, convert minutes to hours (divide by 60) or seconds to hours (divide by 3600). The time unit should match the 'per' part of the rate.