Calculating Speed, Distance, and Time

Welcome to our new lesson! In our previous session, we looked at how to divide a total amount according to a ratio. This was an example of proportional reasoning, where we compared different parts of a whole. Today, we'll explore another crucial real-world application of this kind of thinking: the relationship between speed, distance, and time.

This lesson will teach you how to calculate one of speed, distance, or time when the other two are known. This skill is fundamental not just in maths, but also in science (especially physics) and everyday life, whether you're planning a road trip, figuring out your running pace, or understanding how fast things move in the world around you.

1. What is Speed?

At its heart, speed is a measure of how quickly an object covers a certain distance. It's a type of rate that connects two different measurements: distance and time. The basic formula is:

Common units for speed in New Zealand are kilometres per hour (km/h) or metres per second (m/s). The units you use are important, and they must be consistent.

The following short video provides an excellent introduction to this concept, using the exciting examples of sprinter Usain Bolt and a Formula 1 car to show the formula in action and highlight why consistent units are so important.

2. The Speed-Distance-Time Triangle

Remembering how to calculate speed is straightforward, but what about when you need to find distance or time? You could rearrange the formula algebraically, but there's a much easier way: the formula triangle. This is a simple visual tool that helps you find the correct formula every time.

The video below gives a very clear explanation of how to draw and use the formula triangle. It's a trick that will make these problems much simpler to solve.

As you just saw, the triangle gives us three formulas:

• Speed = Distance ÷ Time
• Distance = Speed × Time
• Time = Distance ÷ Speed

3. Solving Problems: A Step-by-Step Guide

Now that you have the formula triangle, you're ready to solve problems. A successful calculation always follows a clear process. The most important step is ensuring your units are compatible before you calculate. For example, if your speed is in km/h, your time must be in hours and your distance in kilometres.

The following resource provides a fantastic guide. It not only gives a 5-step checklist for solving these problems but also includes several worked examples.

4. Practice Time

Let's put what you've learned into practice. Use the formula triangle and the 5-step method to solve these problems. Remember to check your units!

Questions:

1. A cyclist travels 90 km in 4 hours. What is her average speed in km/h?
2. The Tūī bird can fly at an average speed of 60 km/h. If it flies continuously for 20 minutes, how far will it travel? (Hint: Your speed is in hours, but your time is in minutes!)
3. A ferry travels between Wellington and Picton, a distance of about 92 km. If the ferry's average speed is 50 km/h, how long will the journey take? Give your answer in hours and minutes.

Solutions:

1. Calculate Speed:

   • Knowns: Distance = 90 km, Time = 4 hours.
   • Formula: Speed = Distance ÷ Time.
   • Calculation: Speed = 90 km ÷ 4 hours = 22.5 km/h.
   • Answer: The cyclist's average speed is 22.5 km/h.

2. Calculate Distance:

   • Knowns: Speed = 60 km/h, Time = 20 minutes.
   • Unit Conversion: The units are not compatible. We need to convert the time to hours.
     Time in hours = 20 minutes ÷ 60 minutes/hour = 1/3 hours (or approx. 0.333 hours).
   • Formula: Distance = Speed × Time.
   • Calculation: Distance = 60 km/h × (1/3) h = 20 km.
   • Answer: The Tūī will travel 20 km.

3. Calculate Time:

   • Knowns: Distance = 92 km, Speed = 50 km/h.
   • Formula: Time = Distance ÷ Speed.
   • Calculation: Time = 92 km ÷ 50 km/h = 1.84 hours.
   • Unit Conversion: The question asks for hours and minutes.
     • We have 1 full hour.
     • We need to convert the 0.84 hours into minutes: 0.84 × 60 = 50.4 minutes. Let's round that to 50 minutes.
   • Answer: The journey will take approximately 1 hour and 50 minutes.

Key Takeaways

Excellent work! You can now confidently solve for speed, distance, or time. These calculations are a great example of how mathematical formulas describe relationships in the physical world.

• The relationship between speed, distance, and time can be remembered using the formula triangle.
• The three formulas are:
  • Speed = Distance ÷ Time
  • Distance = Speed × Time
  • Time = Distance ÷ Speed
• Always check that your units are compatible before calculating. Pay special attention when converting time.

In our next and final lesson for this module, we will return to financial mathematics and learn how to calculate GST, discounts, and simple interest in New Zealand dollar contexts.