# Quantitative Reasoning: Average Speed YouTube Lecture Handouts for GRE

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## Average Speed

All speed questions are also based on our standard formula:

Average Speed

So, given here is total it means that for a trip the average speed would be his total distance that was covered in the trip divided by the total time taken and total time also includes all the time during which let՚s say of cars for this talk it is the total trip time it does not matter whether your car was the stop because you had a flat tire or you went to a picnic and you were enjoying that it does not matter all the time of the trip would have to be taken into account in this total time.

Let us now look at two cars now this is a fast car a racing car probably driven by a young driver and this is a slow car. Now this fast car racing car would go like this and then it stops it would raise and then frequently slow down whereas this other car this is a kind of event probably driven by a man with a family and these would travel at a constant speed so this is similar to of a tortoise race where these would be it is running for a while and then it stopped and this car is travelling at a constant rate not so happens that the distance travelled by this car which frequently stops and races.

The distance travelled by this car and this car is the same so they both reach their destination they start from the same place they both reach their destination so they cover the same distance not only that but they start and stop at the same time and this has been traveling with a constant velocity. So the velocity with the speed with which this car was traveling is the average speed of this car.

It is the average speed although this car has been taking frequent stops and then racing forward but the constant velocity with which you travel the same distance in same amount of time would be this average speed.

All the questions related to average speed would be grouped into one of the following categories:

Types of questions

### 1. Distances Given

- Absolute
- Proportion (Fraction)
- Equal (Harmonic Mean)

### 2. Times Given

- Absolute
- Proportion (Fraction)
- Equal (Average of Arithmetic Mean)

## Types of Questions

First in the distances are given.

Distances for each segment during which the velocity the speed was constant and in the second kind of question times are given. So the time during which the speed were constant so if you take our car example.

This car let՚s say it was first driving slowly then it became it then it started travelling faster than its people travelling slowly again then this slow-fast-slow. These are the different segments in which this car journey could be divided and in each of these segments this speed remains constant.

So let us say that this speed , and in this segment and so on and each of these in the first kind of problem the distances for each of these segments are given so, let՚s say the distances are a, b and c so, these are the distances which are given not the times.

Now given this kind of information our average speed would come out to be total distance which is divided by the total times what is the time required for each of these segments time is equal to distance by speed so, the time required

Notice that are coming in denominator of the denominator of this form. There are three ways in which a question it specifies the distances:

The distances in kilometers there is the first kind of question , the question can also give you these distances in some kind of fraction or some kind of ratio so it could say that the different distances but in the ratio is a: b: c and these distances were covering the speed .

If these are the ratios a, b and c then our absolute distance becomes .

So, our absolute distance becomes

Third kind of question which comes around the ratio a: b: c is the same so now let՚s see what happens this a, b and c are same so they all cancel out and what are the left with

If so,

This formula that means distances are the same each of these segments has same distance the average speed that we call this term harmonic mean.

Harmonic Mean is the reciprocals of the name of the reciprocal so first we take the reciprocal then we take its main mean of the reciprocal and this restrict reciprocal of that mean of the reciprocals would be

Our journey is divided into various segments the speed for these both segments is and the time required absolute hour are a, b and c instead of giving the distances this time.

If you are given that information you can quickly find out what the distances would be distance is equal to speed into

Similarly distance and

*Figure 1 Average Speed*

Now average speed

Sometimes they can be given in ratio so, let us now see what happens if this were in ratio and absolute becomes and the distances becomes

Average speed for the second case

Third took equal time you say if then what happens this are same quantity cancel out then you are left at the top is

It is the arithmetic means of the individual segments.

## Harry Takes His Family Out for a Picnic. They Travel 80 Km to Coast and Arrived After 1 Hr. Their Picnic Lasts for 2 Hr. Afterward They Travel Home in 1 Hr. What Was Their Average Speed?

There is a family man Harry who went for a picnic the total distance they traveled was comes home to the picnic spot 80 km they went for the entire distance for the trip was 80 plus 80 the picnic last part to us so they took 1 hr to go and 1 hr to comeback and they spent 2 hrs in the picnic spot.

*Figure 2 Harry Speed*

The average speed for the trip would have to consider these two acts that is the average speed.

It means that if there was its slower car which was travelling constantly and then it returned back immediately. This the speed this car needs to have to make the same distance in same time and the other car has been travelling constantly.

Average Speed

## Jim Travels the First 3 Hours of His Journey at 60 Mph Speed and the Remaining 6 Hours at 24 Mph Speed. What is the Average Speed of Jim՚s Travel in Mph?

In this question instead of time is being given.

So the Jim travels first 3 hours of his journey at 60 mph and remaining 6 hrs at 24 mph speed.

So, 1^{st} segment 2^{nd} segment

The distance of 3 hrs equal to 3 hrs at 60 mph and 6 hrs at 24 mph divided by total time i.e..

Distance of the 3 hrs

In this question,

1 ⁄ 3^{rd} of the time at 60 mph speed and the time left would be

Total time

The distance covered in the 1^{st} segment is

Time for the 2^{nd} segment is

## Jim Travels the First 180 Miles of His Journey at 60 Mph Speed and the Remaining 144 Miles at 24 Mph Speed. What is the Average Speed of Jim՚s Travel in Mph?

It takes miles at 60 mph

So this is 180 total distance is with some of these times of each of those segments there were two segments.

What is the time required for the 1^{st} segment it is , 2^{nd} segment it is

Average speed

Jim travels the first 5 ⁄ 9^{th} of his journey at 60 mph speed and the remaining at 24 mph speed. What is the average speed of Jim՚s travel in mph?

The absolute value of the distances we were given the relative value or the fraction of distances for each of the segments.

In given that 5 ⁄ 9^{th} of its journey at 60 mph, so 1^{st} segment

2^{nd} segment

Average of speed

## Tim Drove at an Average Speed of 30 Miles Per Hour for the First 30 Miles of a Trip, at an Average Speed of 60 Miles Per Hour for the Next 30 Miles and at an Average Speed of 90 Miles ⁄ Hr for the Remaining 30 Miles of the Trip. If He Made No Stops During the Trip, Tim՚s Average Speed in Miles ⁄ Hr for the Entire Trip Was?

Tim who is driving at a speed of 30 mph for 30 miles,

60 mph for 30 miles and

90 mph for 30 miles.

Average speed

The answer is the harmonic means of these individuals speed 30,60 and 90 and not the arithmetic mean.

You can simplify that the answer would be .

## Priya Drove at an Average Speed of 30 Miles Per Hour for Some Time and then at an Average Speed of 60 Miles/Hr for the Rest of the Journey. If She Made No Stops During the Trip and Her Average Speed for the Entire Journey Was 50 Miles Per Hour, for What Fraction of the Total Time Did She Drive at 30 Miles/Hour?

Priya drove at an average speed of 30 mph

So, the keyword here is sometimes not for some distance

Let՚s say that total time required for the journey, I am assuming in terms of time because of the keyword sometimes.

So, assume that the total tile

Fraction of the times t that 30 miles

An average was expended that 60 mph

So, the average speed

Average speed

drive at 30 mph

## The Speed of a Bus Increases by 4 Km/Hr Every 2 Hours. If the Distance Travelled in First 2 Hours Was 70 Km. What is the Average Speed in 6 Hours?

In this question, the speed of a bus increased by 4 km/hr

One segment for the first segment we are given this to be 70 km and the speed was 35 km/hr

A bus increased by 4 km/hr so, the speed

Each of this is 2 hrs segments, so we are being asked what the average speed is for this 6 hrs duration. The key thing here is to understand that when this term is equal in the distances are times of each of the segments is equal.

The entire journey is just an average of arithmetic mean of the individual speed, arithmetic means is 39.

Therefore the answer will be 39 mph.

## There is a Road Beside a River. 2 Friends Started from a Place a and B and then Back. One of Them on a Cycle at a Speed of 12 Km/Hr, the Other on a Boat at a Speed of 10 Km/Hr. If the River Flows at the Speed of 4 Km/Hr, Which of the Two Friends Will Return to a First?

There are 2 friends, one of them is travelling in a boat in a river and the other one is cycling on the bank.

You use the concept of boats and streams, so the average speed for the friend who is cycling hen went and he came back and his speed

2^{nd} friend distance was for the second friends when going

Downstream speed would be

Let՚s calculate that total distance covered it

## If a Car Must Achieve an Average an Average Speed 250 Km/Hr to Qualify in a Racing Event on a 1600 M Track and Covers an Average Speed 200 Km/Hr on the First Half, What Minimum Average Speed Must It Have in Second Half to Qualify?

So, the total distance is 1600 km

The first half is covered at a speed of 200 km ⁄ hr

The entire 1600 km has to be covered at the speed of 250 km ⁄ hr

Average speed

We want to find the value of .

Next class we will start a very important topic is percent, profit and loss, simple compound interest

✍ Mayank