Rate Problems: – Calculating simple speed, distance, and time relationships.

RATE PROBLEMS Calculating simple speed, distance, and time relationships. Grade 7 Mathematics: Rate Problems – Speed, Distance, and Time Subtopic Navigator Introduction Understanding Speed Finding Distance Finding Time Applications and Mixed Problems Cumulative Exercises Conclusion Lesson Objectives Understand the relationship between speed, distance, and time. Calculate speed when distance and time are given. Calculate distance when speed and time are given. Calculate time when speed and distance are given. Solve real-life problems involving rates such as travel and motion. Lesson Introduction Rate problems describe situations where something is happening over time. The most common type involves speed, distance, and time. The relationship is expressed as: [latex] text{Speed} = frac{text{Distance}}{text{Time}} [/latex] From this, we can also rearrange to find distance or time: [latex] text{Distance} = text{Speed} times text{Time} [/latex] [latex] text{Time} = frac{text{Distance}}{text{Speed}} [/latex] Understanding Speed Speed tells us how fast an object is moving, measured as distance per unit of time (e.g., kilometers per hour or meters per second). Example 1: A car travels 120 km in 3 hours. What is its average speed? Solution: To calculate speed, we divide the total distance by the total time taken. The car covered 120 km and used 3 hours, so: [latex] text{Speed} = frac{120}{3} = 40 text{ km/h} [/latex] This means the car moves 40 kilometers every hour on average. Example 2: A train moves 300 km in 5 hours. Find its average speed. Solution: The train covers 300 km in 5 hours. Dividing the distance by the time, we have: [latex] text{Speed} = frac{300}{5} = 60 text{ km/h} [/latex] So, the train travels 60 kilometers in each hour. Exercises (Speed) A cyclist travels 45 km in 3 hours. What is the average speed? A bus covers 150 km in 5 hours. Find its average speed. Finding Distance If the speed and time are known, the distance covered can be found by multiplying speed by time. Example 3: A car travels at 50 km/h for 4 hours. How far does it go? Solution: Distance is obtained by multiplying the given speed by the time. At 50 km/h, in one hour the car covers 50 km. In 4 hours, the car covers: [latex] text{Distance} = 50 times 4 = 200 text{ km} [/latex] Hence, the car travels 200 kilometers. Example 4: A ship sails at 25 km/h for 12 hours. Find the distance travelled. Solution: In each hour, the ship covers 25 km. Over 12 hours, the total distance is: [latex] text{Distance} = 25 times 12 = 300 text{ km} [/latex] Therefore, the ship sails 300 kilometers. Exercises (Distance) A bus moves at 60 km/h for 3 hours. Find the distance covered. A person runs at 8 km/h for 2 hours. How far do they run? Finding Time If the distance and speed are known, the time taken is found by dividing distance by speed. Example 5: A car covers 180 km at 60 km/h. How long does the journey take? Solution: Since the car moves 60 km each hour, to know how many hours it takes to reach 180 km, we divide the distance by the speed: [latex] text{Time} = frac{180}{60} = 3 text{ hours} [/latex] So, the journey lasts 3 hours. Example 6: A cyclist rides 90 km at 30 km/h. Find the time taken. Solution: The cyclist covers 30 km each hour. To travel 90 km, we divide 90 by 30: [latex] text{Time} = frac{90}{30} = 3 text{ hours} [/latex] Thus, it takes 3 hours. Exercises (Time) A truck moves 200 km at 50 km/h. How long does it take? A bus travels 240 km at 60 km/h. Find the time required. Applications and Mixed Problems Example 7: If a person runs 10 km in 50 minutes, what is the average speed in km/h? Solution: First, the time must be expressed in hours since speed is usually measured in km/h. 50 minutes equals [latex] tfrac{50}{60} = tfrac{5}{6} [/latex] hours. Dividing the distance by this time, the speed is: [latex] text{Speed} = frac{10}{5/6} = 10 times frac{6}{5} = 12 text{ km/h} [/latex] Therefore, the average running speed is 12 km/h. Example 8: A train travels 200 km at 100 km/h and then another 150 km at 75 km/h. Find the total time taken. Solution: For the first part, the time is distance divided by speed: [latex] frac{200}{100} = 2 text{ hours} [/latex]. For the second part, the time is [latex] frac{150}{75} = 2 text{ hours} [/latex]. Adding both, the total time is 2 + 2 = 4 hours. Thus, the whole trip takes 4 hours. Example 9: A bus leaves a town at 9:00 am and travels 180 km at 60 km/h. At what time does it arrive? Solution: The time of travel is the distance divided by the speed, which is [latex] frac{180}{60} = 3 text{ hours} [/latex]. Adding this to the departure time of 9:00 am, the arrival time is 12:00 noon. Exercises (Applications) A boy cycles 24 km in 2 hours. What is his average speed? A plane flies 600 km at 200 km/h. How long does the flight last? Cumulative Exercises A car travels 160 km in 4 hours. What is the average speed? A train moves at 90 km/h for 5 hours. Find the distance covered. A cyclist covers 75 km at 25 km/h. How long does it take? A truck travels 240 km at 60 km/h. Find the time taken. A motorbike goes 80 km in 2 hours. Find the speed. A bus runs 300 km at 100 km/h. Find the time. A runner completes 21 km in 2 hours. Find the average speed. A train covers 400 km at 80 km/h. Find the travel time. A car travels 60 km in 1.5 hours. What is the speed? A person walks 6 km in 1 hour 30 minutes. Find the speed in km/h. Conclusion/Recap Rate problems help us understand motion and speed in everyday life. By using the relationships between speed, distance, and time, we can solve travel problems, plan journeys, and estimate arrival times. Mastery of these concepts is useful both in mathematics and in real-world applications. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c