RATE ( SPEED , DISTANCE AND TIME, WORK PROBLEM, AND WORK-RATE PROBLEM)

Summary

This video covers essential concepts of calculating speed, distance, and time, along with average speed, through practical examples. It explains how to compute average speed for different time intervals and speeds, and applies this to real-life situations like vehicle travel and running. The video also delves into solving work-rate problems, demonstrating how to calculate how long multiple people take to complete tasks together. Using formulas for speed, distance, time, work, and rate, the video provides step-by-step solutions to problems, making the concepts easier to grasp.

Takeaways

• 📏 Distance, speed, and time are interconnected through formulas.
• 🚌 To find speed, use the formula: speed = distance / time. For example, a bus traveling 200 km in 4 hours has a speed of 50 km/h.
• 🏍️ To find the average speed of a vehicle over different time periods, first calculate the distance for each period, then find the total distance and divide by the total time.
• 🏎️ For a motorcycle traveling 36 km in 3 hours and 40 km in 2 hours, the average speed is calculated by adding both distances and dividing by the total time: 188 km / 5 hours = 37.6 km/h.
• 🏃 To find the average speed of a runner who changes speeds, calculate the distance covered at each speed and then the total distance divided by the total time.
• 🔨 Work problems can be solved using the formula: work = rate x time. The rate is often given as 1/time.
• 👷 For work problems involving two people, calculate the individual work rates and then the combined work rate.
• 🧮 To find the time remaining to complete a task when one person starts and another finishes, use the formula: time remaining = 1 - work done.
• 🧹 When solving how long it takes for two people to complete a task together, add their rates and find the combined rate.
• 📊 Use calculators for precise calculations when dealing with fractions and rates to ensure accuracy.

Q & A

What is the formula for calculating speed?

The formula for calculating speed is: Speed = Distance ÷ Time.

In the example of a bus traveling 200 km in 4 hours, what is its speed?

Using the formula Speed = Distance ÷ Time, the bus's speed is 200 km ÷ 4 hours = 50 km per hour.

How do you calculate the average speed when there are two different rates for a trip?

To calculate average speed, first find the total distance by adding the distances traveled at each rate, then divide by the total time. The formula is: Average Speed = (Distance 1 + Distance 2) ÷ (Time 1 + Time 2).

In the motorcycle problem, how do you find the total distance and total time?

For the motorcycle problem, first find the distance for each rate: 36 km/h for 3 hours gives 108 km, and 40 km/h for 2 hours gives 80 km. The total distance is 108 km + 80 km = 188 km, and the total time is 3 hours + 2 hours = 5 hours.

What is the motorcycle's average speed over the two time periods?

The average speed is calculated by dividing the total distance by the total time: 188 km ÷ 5 hours = 37.6 km per hour.

In the problem where Katy runs at two different speeds, how is the average speed calculated?

First, calculate the distance for each speed: 15 km/h for 2 hours gives 30 km, and 8 km/h for 1 hour gives 8 km. The total distance is 38 km, and the total time is 3 hours. The average speed is 38 km ÷ 3 hours = 12.6 km per hour.

What is the formula for calculating work when rate and time are given?

The formula for work is: Work = Rate × Time.

In the chair-assembling problem, how long did it take Look to finish assembling the chair after Phil worked on it for 20 minutes?

Phil worked for 20 minutes, completing 2/3 of the work. The remaining work was 1/3. Look took 20 minutes to complete the remaining 1/3 of the work.

How do you calculate the time remaining to complete a task if part of it has been completed?

The time remaining is calculated by the formula: Time Remaining = 1 - Work Done.

How long did it take Alex to finish cleaning the room after Haley cleaned for 1 hour?

Haley worked for 1 hour, completing 1/2 of the work. The remaining 1/2 of the work took Alex 1.5 hours to complete.

Outlines

00:00

🚍 Calculating Speed and Average Rate in Motion Problems

This paragraph discusses how to calculate speed using the formula speed = distance ÷ time. A problem is solved where a bus travels 200 kilometers in 4 hours, resulting in a speed of 50 km/h. Another example follows where the average speed of a motorcycle is computed. The motorcycle covers 36 km in 3 hours and 40 km in 2 hours. The process involves calculating the total distance (108 km + 80 km = 188 km) and dividing by the total time (5 hours) to find the average speed, which is 37.6 km/h.

05:01

🪑 Work Rate Problems: Chair Assembly Example

This paragraph focuses on work rate problems, using the formula Work = Rate × Time. A scenario is provided where Phil assembles a chair in 30 minutes and his son takes 60 minutes. Phil works for 20 minutes, and the problem is to determine how long his son will take to finish the task. Using work rate formulas, it is determined that Phil completes 2/3 of the task, leaving 1/3 for his son. His son, with a rate of 1/60 chairs per minute, finishes the remaining work in 20 minutes.

10:05

🧹 Cleaning the Room: A Shared Task Problem

This paragraph describes a shared work problem where Haley and Alex clean a room. Haley can clean the room in 2 hours, and Alex in 3 hours. Haley works for 1 hour, and the task is to find how long Alex will take to finish the job. After calculating the work done by Haley (1/2), the time remaining is found to be 1/2. Alex’s rate is 1/3 rooms per hour, so the time required for Alex to finish the job is calculated as 1.5 hours.

15:05

🏠 House Painting: Joint Effort Problem

This paragraph explains a scenario where Betty and Jan are painting a house together. Betty can paint the house in 6 days, and Jan can paint it in 8 days. The problem is to calculate how long it will take them to paint the house if they work together. Using the formula for combined rates, 1/6 + 1/8, the total rate is found to be 7/24 houses per day. The time required to paint the house is then calculated as 3.42 days.

Keywords

💡Speed

Speed refers to the rate at which an object covers a distance. In the video, the formula for speed is given as distance divided by time, and it is used in multiple examples, such as calculating the speed of a bus that travels 200 km in 4 hours, resulting in a speed of 50 km/h.

💡Distance

Distance is the total length of the path traveled by an object. In the video, distance is a crucial variable in calculating speed and is highlighted in examples like the bus traveling 200 km or the motorcycle covering 108 km in the first part of its journey.

💡Time

Time refers to the duration taken to cover a certain distance. It is used alongside distance to compute speed. In the script, time is a key factor when solving problems like how long it takes for a bus to travel 200 km or how long it takes to complete tasks, such as assembling a chair.

💡Average Speed

Average speed is the total distance traveled divided by the total time taken. In the video, the concept is explained with examples like the BMW motorcycle traveling different distances at different speeds, leading to an average speed calculation of 37.6 km/h.

💡Work

Work in this context refers to a task completed over time, such as assembling a chair or cleaning a room. The work formula is introduced in the video as work = rate × time. It is used in examples like how long it takes for Phil and his son to assemble a chair.

💡Rate

Rate is the speed at which work is done or a task is completed. In the video, rate is used in problems involving tasks like assembling chairs or cleaning rooms. It is calculated as 1 over time (rate = 1/time) and helps in determining how long it takes to complete shared tasks.

💡Distance-Time Relationship

The distance-time relationship is a core concept in the video, where distance is derived by multiplying speed and time. This is seen in examples such as calculating the distance traveled by a BMW motorcycle at a rate of 36 km/h for 3 hours, giving a total distance of 108 km.

💡Task Sharing

Task sharing refers to the division of labor between two or more individuals to complete a task faster. This concept is explored in examples like Phil and his son assembling a chair, where each person works for a different duration, and the combined effort reduces the total time needed.

💡Fractional Time

Fractional time is the remaining time needed to complete a task after part of it has been completed. In the video, this is demonstrated in problems involving task-sharing, such as how long Alex needs to finish cleaning a room after Haley has cleaned for one hour.

💡Problem Solving

Problem solving is the process of applying mathematical formulas and logic to find solutions. The video covers various problem-solving scenarios, such as calculating speed, average rates, and work-time for tasks. Each example walks through step-by-step solutions using formulas like speed = distance/time or work = rate × time.

Highlights

Introduction to distance, speed, and time formulas with a bus travel example.

Speed formula: speed equals distance divided by time. Example: 200 km in 4 hours equals 50 km/h.

Average speed calculation using multiple speeds for different time periods. Example of a motorcycle traveling 36 km for 3 hours and 40 km for 2 hours.

Step-by-step breakdown of finding distance for two different speeds: 108 km for 36 km/h over 3 hours and 80 km for 40 km/h over 2 hours.

Calculating average speed by summing the distances (108 km + 80 km = 188 km) and dividing by total time (5 hours), yielding an average speed of 37.6 km/h.

Another average speed example with a runner: 15 km/h for 2 hours and 8 km/h for 1 hour, resulting in an average speed of 12.6 km/h.

Introduction to work rate problems and formulas, including work equals rate times time.

Rate formula: rate equals 1 divided by time. Demonstrating with Phil and his son assembling a chair.

Phil works for 20 minutes assembling the chair, completing 2/3 of the task.

Using time remaining formula: time remaining equals 1 minus the completed work, resulting in 1/3 of the work left.

Calculating the son's work rate, where the remaining work takes him 20 minutes to finish assembling the chair.

Work rate problem example with Haley and Alex cleaning a room together, calculating how long Alex needs to finish the task after Haley starts.

Introduction to another work rate problem: Betty and Jan painting a house together.

Calculating the combined work rate of two workers using the formula 1/A + 1/B, demonstrating with Betty and Jan's task.

Final example results in Betty and Jan finishing the painting task in 3.42 days.

Transcripts

rate of objects and rational

numbers so first is we going to discuss

distance speed and time so these are the

formula for this distance speed and

time so for example here we have a

problem a bus travel 200 kilm in 4 hours

what is its

speed so using the speed formula we have

speed is equals to distance divide time

and speed is equals to your distance is

200 kilm in time is 4 hours so speed is

equals to 50 kilm per hour so that's how

you compute the

speed uh distance and time you use the

formula

so okay so we have here another problem

a BMW gs1 1260 can travel at a rate of

36 kilm for 3 hours and 40 km for the

next 2 hours what is the average rate of

the

motorcycle so the question here is what

is the average rate so we have the First

Rate which is 36 kilm for 3 hours and

the next rate which is 40 kilm for the

next 2

hours so to solve this one we begin with

step one Sol for 36 kilm uh for per 3

hours 36 kilm per hour for 3 hours so

this problem need uh uh needs to find

the distance so the distance is equals

to speed time time the distance is

equals to 36 km/ hour * 3 and the answer

is 108

kilm okay next is for the step two solve

for 40 km/ hour for the next 2

hours so we have here the distance is

equals speed time time and the distance

is equals to 40

kilm per hour time 2 hours and 40 * 2

that is 80 okay for step three okay we

need to find the uh average okay so

solve for the average rate so we have

here so da stands for distance average

is equal to distance 1 plus distance 2

ta n is equals to our time average is

equals to T1 + T2 and we have the speed

is equals to distance uh average divide

time

average so D1 class is step one distance

and step two is the D2 okay so we have

108 km plus 80 km

so the distance average is equals to

188

kilm okay so that's the da now solving

for ta okay ta Nam is the RS RS step one

RS n step two so Step One is 2 hours

while step two is 3 hours so that is T1

and

T2 so ta is equals to 5 hours now

solving for the speed so this last uh

column here is the speed so distance

average divide time average and that is

188 kilm y then 5 hours that is iton so

the speed is equals to 37.6 km per

hour okay next problem so we have

another problem which is the same

um Kina Katy runs at a rate of 15 kilm

per hour for two hours after that she

slow down to 8 kilm for the next hour

what is her average

speed so to compute for the speed

first step one step

one 15 km per hour for 2 hours so that

is distance that is 30 and the distance

next problem which is 8 km per hour for

the next hour next hour is 1 hour okay

equivalent to 1 hour so the distance is

equals to 8 kilm now solving for the

average rate or your speed we have da is

equal to

38 and ta is equal to 3 hours Sil okay

we have da a TA then we have the speed

okay 38 km divide 3 hours that is 12.6

km per hour okay the answer is 12.6 km

per

hour okay for the next topic we have

word problem a work problem so we have

here formula tatong formula to remember

work is equals to rate time time and the

rate this rate s formula rate is

equivalent to one over time okay we'll

discuss it later

to further understand and the last time

and the last formula is work time is

equals to time remaining divide rate

okay let's use this to an actual

problem okay fi okay field can assemble

a chair in 30 minutes his sonl can

assemble the chair in 60 minutes Bill

work on the task for 20 minutes and let

his son work on the rest how long will

it take look to finish assembling the

chair so so we have here a problem no so

the question here is how long will it

take look to finish assembling the chair

so let's go on with the solution so step

one so we have here the given so

remember that phield can assemble a

chair in 30 minute so in 30 minutes no

all individual so this is the individual

rate okay so field only field can finish

assembling a chair in 30 minute so the

rate the rate here means the individual

rate

Kip look okay so we have here uh rate

equals to one over time and the time

used in rate is rate time individual so

K is 30 30 so we have rate is equals to

1 over 30 okay so now we have to find

the work okay work is equals to rate

time time so the work here is equals to

1 over 30 rate while the time is used

for okay work is to

solve remember Phil okay Phil and uh his

sonl work together to finish the same

task okay

yeah okay so fi work on the task for 20

minutes so we have the time is 20 so

multiply L uh 1 / 30 * 20 you can use a

calculator

here okay 1

over3 divide 20 and the answer is and

multiply

Pala the answer is 2/3 so the answer is

2/3 okay next step

two note we have a note here formula for

time remaining is 1 minus work so this

is a constant

formula so always remember this formula

for the time

remaining so to solve for the time

remaining time remaining equals 1us 2/3

remember the 2/3 is work work

so we have 1 - 2/3 okay we use the

calculator okay

1 1

-

2/3 will equals to 1/3 so the time

remaining is

1/3 okay now solving for work time of

look so remember that that the work time

is equals to time remaining divide rate

and time remaining is equals to

1/3 okay so individual Work N okay so

individual remember that field can

finish in 30

minutes while look can finish in 60

minutes okay so look can assemble the

same chair in 60 minutes so the rate of

look is 1 / 60 remember 1/ 60 is the

rate

of minutes

so we have here work time is equals to

remember the time remaining is 1/3

divides 1 / 60 which is the rate of look

so to calculate in

calculator 1/

3

divides 1 /

60 the answer is 20 so we have here the

work time for look is 20

minutes this means that Phil finish the

work for 20 minutes and then look also

finish it within 20

minutes okay for another problem the

same problem Hy and Alex are sharing a

room Haley can clean their room in two

hours while Alex can clean it in 3 hours

haly cleaned the room for 1 hour and

Alex continued the rest how long did it

take Alex to clean the room so

individual rate n is for Alex Haley

first is 2 hours K Alexan is 3 hours now

uh what did what they do is they

combined their work and heey started for

1 hour and Alex continued the rest so

how long did Alex finish the

job so again solve for the rate so given

Haley can clean the room in 2 hours so

the rate of Haley is 1/2 now for the

time is 1 since Haley finished the room

uh cleaning the room in 1 hour so work

is equals 1 12 * 1 and that is equals to

1

12 okay we're going to use the

calculator the answer is

1/2 okay next solve for the time

remaining okay remember the time

remaining is one minus work again the

work okay you work Haley okay so 1 - 12

in

calculator and the answer is 1 12 so

this is the time remaining now solving

for the work time of Alex so again we

have the formula for work time work time

is equals to time remaining divide rate

and this is the time remaining and Alex

now can clean it in 3 hours so the rate

of Alex is 1/3 so work time is equals to

12 IDE 1/3 use the calculator to solve

for the work

time we have 1/2 divides 1/3 and it will

equal to 3 over two or 1 or 1/2 or 1.5

so always the answer is in a decimal or

whole number which is and the answer

here is 1.5 hours so this means that

Alex took 1.5 hours that Hal finish in 1

hour so start Alex one hour I mean Haley

then Alex into uh 1.5

hours so last for the rate is work rate

problem so this is very

easy so we have an example Betty can

paint a house in 6 days and JN can paint

the same house in 8 days how long will

it take them to paint the house so this

time both uh both

the uh both the given or Betty and Jan

okay P work time n or job n in just um

just a one scenario

okay okay one at um one time P talaga so

how do you solve this one so solving

for okay 1 a + 1 B so Betty can paint a

house in six days while Jan can paint

the same house in 8 days so B Betty is a

so

first a

Anda B so we have one six and 1 over8

y six J or a is days nil or whatever the

number it is hours maybe days maybe time

or weeks okay yeah so we have here six

and 8 so let's add it 16 +

18 1 / 6

+ 1 over 8 equals to 7

over 24

now step two solving for one / t Okay

the one over T here class

is the total of 1 a + 1 B which is 1 6

+8 = to 7 / 24 and our T is equals to

7/ 24 so in

fraction okay in

fraction one okay and time remaining mag

another fraction click Naman 7 over

24 and click the equals click the SD and

the answer is

3.42 so we're going to get the two

decimal places so the answer is 3.42

days okay so s Hindi Cas okay use this

away 1

divide okay 7 divide

24 and the answer is

3.42 okay still the same so use this one

use

parenthesis so this

is okay this is the calculator technique

okay so 24/ 7 or 3 + okay this uh this

one is an example of a mixed number so

this one three and 3 over 7 so 3

+ 3/

7al equal

to okay

3.24 okay so that is all for rate

problem