Word Problems
(Speed, Distance & Time)
As most of the students are familiar with the equation used in physics, if you ever read, that gives relationship between Distance, Speed and Time. The more the speed, the lesser is the time it take to cover some distance and vice versa; provided that the distance remains the same. Below is the famous relationship between these three variables:
Distance = Speed × Time
This implies that:
This relationship would help us to solve problems throughout this topic. Notice that in the last equation, which states that Speed equals to distance divided by time. Similarly, its unit would also be the ratio of units of Distance and that of Time. So generally, unit of Speed is in ‘miles per hour’ (m/h) or ‘kilo-meters per hour’ (km/h); but occasionally, the unit of speed may be given in ‘kilo-meters per minute'(km/min) or ‘kilo-meter per second’ (km/s) and so on. You must be good in conversion of these units to each others, because sometimes its tricky to solve a problem having such different units. So you have to be careful while solving the problem if you don’t see same unit in question stem (i.e statement given in question) and in answer choices.
Units Conversion:
120 km/h is how much speed in ‘meters per second‘?
First we need to convert km to meters. For that purpose, we have to multiply (remember don’t divide) 120 by 1000. As there are 1000 meters in a kilo-meter; so, in order to convert larger unit to smaller unit, we always multiply by the unit conversion factor (i.e 1km = 1000m, so 1000 is unit conversion factor).
Important Note: When a big unit needs to be converted to small unit, always multiply by the number of pieces that makes up big unit. And divide in reverse case, i.e converting small unit into big unit (e.g converting meters to kilo-meter).
For those who still unable to understand this, just use simple sense. When large unit is broken down to smaller units, the number of units increases, so this makes a sense that we must multiply. On other hand, when small unit is required to express in term of large unit, it always mentioned in fraction or in part, which suggests that we must divide by the unit conversion factor.
After conversion of 120 km into meters, we’ll have 120 × 1000 = 120000 meters.
Now we have speed in ‘meters per hour‘ but we need to convert into ‘meters per second’, so we need to convert ‘hour’ into seconds. As we know there are 60 minutes in an hour, while 60 seconds in a minute. Therefore, there are 60 × 60 which equals to 3600 seconds in an hour. We multiplied 60 twice because 1 hour has 60 minutes, and each minute has 60 seconds. For those who are unable to understand this see the simple method explain below:
1 minute = 60 seconds
By multiplying both sides by 60,
⇒ 60 minutes = 60 × 60 seconds = 3600 sec
⇒ 1 hour = 60 min = 3600 sec
⇒ 1 hour = 3600 sec
So when conversion from hour to seconds is require, we need to multiply, rather than divide, by unit conversion factor (i.e 3600) as we have to convert big unit into small units. As 3600 seconds combine to make an hour. In the required question, notice that 1 hour is in denominator, as the time is in denominator. So we need to place 3600 as well in denominator. i.e
So, Speed in meters per second = 100⁄3 ≈ 33.33 meters per second Answer
Let’s proceed to this topic and cover different scenarios.
As you know that:
Similarly,
Where Average Speed is the average of all different speeds applied in the complete journey.
This basic equation should be kept in mind, because whole topic of distance, speed & time will revolve around this important equation. Secondly for remainder all equations that are used in quantitative section will be merged within a one last lecture of quantitative that later on you’ll find after completion of your course.
Now, let’s begin with considering different scenarios from simple and basic to complex and advanced.
Simple Scenarios:
One Person scenario:
Let’s suppose a person starts jogging form point A to another point B at 40 miles/hr, and then come back at 60 miles/hr as shown below:
Let’s find out the average speed of the entire journey. Do you think the average speed of the entire journey would be the average of the two speeds which would be (40 + 60)⁄2 = 100⁄2 = 50 miles/hr?
If yes, then you are unfortunately thinking in wrong way. As the lower speed (40 miles/hr) requires more time as compared to higher speed (60 miles/hr). So the avg speed must be more close to the slower speed (40 miles/hr) and hence avg speed must be less than 50 miles/hr. If still not understood, then tell me what you think about the average weights of 5 boxes having each weight 10kg, 10kg, 10kg, 20kg and 20kg. Does the avg weight of these 5 boxes equals to avg of 10 and 20 (i.e. 15)? Your answer would be never, which is right. Because number of 10kg boxes are more than that of 20kg boxes. So the avg boxes must be more close to 10 kg boxes from the mid point of 10 & 20. In other words the avg weight must be less than 15kg (but would obviously remain greater than 10kg). Similarly, in above scenario, the time required with speed 40 miles/hr is more than the time required with speed 60 miles/hr, because 40 miles/hr being slower in speed than 60 miles/hr hence requiring more time at 40 miles/hr than 60 miles/hr. If you make a finger from point A and move it slowly to point B, and then move same finger from point B and move it quickly to point A, you would come to realize that the more repetition/time required by the finger is on slower speed than as compared to faster speed. So in this scenario, always the average speed would be more closer to slower speed rather than at middle of the two speeds.
So in this way you may eliminate all those options in the answer choices that are above or at the average of the two speeds (i.e the middle of the two speeds).
Now let’s find the exact value of the average speed in this scenario. For that purpose you have two methods to solve (2nd method is recommended being short-cut).
Weighted Average Method:
As we know,
Here t1 and t2 are the times of traveling from point A to point B and from point B to point A respectively.
Average Speed = 2 × 120⁄5 = 2 × 24
Average Speed = 48 Answer
Did you notice that the answer is not average of 40 and 60 (i.e. 50), rather it’s less than the average because 40 speed takes more time/repetition.
Alternate Method (Recommended):
When you see the one person scenario, where a person travel from certain point to another point and then come back from the same path, always use the following equation to solve for the average speed of the entire trip.
Avg. Speed = 2 S1 S2⁄(S1 + S2)
Where S1 is the speed of moving from start point to another point, while S2 is the speed of moving back from another point to starting point.
So in this method, if the two speeds of going and coming back is given, we can find the average speed.
Important Note: The above equation/formula is only applicable in one person scenario, where the two parts of the journey are same in distance, but different in time. On other hand when the time of the two parts of the journey is same, than we’ll just take simple average of the two speeds.
Now let’s find the average speed of the same question as mentioned in one person scenario by using 2nd method.
Average Speed = 2 (40) (60)⁄(40 + 60)
= 2 (40) (60)⁄100
= 2 (4) (6) {By cancelling the two zero’s of numerator and denominator}
= 2 × 24
Average Speed = 48 Answer
Important Note: When you see the form of expression above, avoid multiplication first, and do division first. It’s the best method to simplify any expression as you have learnt in Arithmetic section of Study Plan for Beginners Level.
Now let’s consider a tough scenario in same question as below:
You may find the answer through the same way as above in 1st method, but here there’s no alternate method. This type of question may come in high difficulty level questions. Let’s solve it by weighted average method.
So Average Speed ≈ 55.4 miles/hr Answer
Now let’s discuss second category of one person scenario. In this category a person move from point A to point B at some certain speed. And then move from point B to point C at different speed. Look at the figure below:
How to find the distance covered by the person while moving at 15 miles/hr?
To solve this, let’s begin with the diagram above. What you consider as the starting point from where we can go to solve this? The answer of this most important question lies on how quickly can you figure out the desired way. The quickness is hugely depends on your level of practice. To help you in practice, you will be bombarded with plenty of medium to hard difficulty level question in form of assignments and quizzes. This would make you expert in quickly identifying first step to proceed to the answer of the problem.
To solve the above problem we should start from the avg. speed equation. As the value of average speed of entire trip is given, so you can feel quite handy to find other helpful variable needed to solve the problem.
As we know,
So,
⇒ Total Distance = 18 × 6 = 108
Now, we have found the distance of the entire trip. Now we can find the distance of first part of the trip as it was the main question. For that purpose, we need to initiate with total distance as follows:
Total Distance = d1 + d2
⇒ 108 = S1 T1 + S2 T2 {As Distance = Speed × Time}
⇒ 108 = (15) T1 + (24) T2 ————— (eq. 1)
As it’s given that,
Total time = 6
⇒ Total time = T1 + T2 = 6
⇒ T2 = 6 – T1 ——————————(eq. 2)
By putting value of (equation 2) in (equation 1), we’ll get
⇒ 108 = 15 (T1) + 24 (6 – T1)
⇒ 15 (T1) + 144 – 24 (T1) = 108
⇒ –9 (T1) = –36
⇒ T1 = 4 ————————————(eq. 3)
Now, as the first part of the trip is d1 = 15 (T1)
By putting value of eq. 3 we’ll get
⇒ d1 = 15 (4)
⇒ d1 = 60 Answer
A question may come in your mind, that how to make these long calculation in such short duration of time? Well I would like you to just focus on your accuracy more than your speed of calculation at this stage. I’ll guide you accordingly when you start focus on speed after you’ll have enough practice on accuracy in calculation. So at start never miss any step during calculation.
Two Person Scenario:
Two person scenario is split into three categories. 1) Both persons moving in same direction; 2) Both persons moving in opposite direction towards each other; 3) Both persons moving in opposite direction against each other. Let’s discuss first category:
1). Both persons moving in same direction
Let’s consider the following figure where Mr. A starts moving from a certain point and towards certain direction at 15 miles/hr. After 2 hrs, Mr. B started moving from the same point and towards the same direction (i.e. along the same path) at 20 miles/hr. How much time would it take Mr. B to overtake Mr. A?
From the figure above, we see that the two people have same distance to cover as shown below:
So, according to the given condition,
Distance covered by Mr. A = Distance covered by Mr. B
⇒ DA = DB
⇒ SA TA = SB TB (As Distance = Speed × Time)
⇒ (15) TA = (20) TB
By dividing both side by 5 to simplify, we’ll get
⇒ 3 TA = 4 TB ——————– (eq. 4)
Now think about another equation that may be made regarding the time taken by the two people.
If you see the figure above, you’ll come to understand that Mr. B started his journey 2 hours late. So We can say that the traveling time required by Mr. B is 2 hours less than that of Mr. A. Here many of my students get confuse and rise question why Mr. B would take 2 hours less time if he already was 2 hours late. The answer is simple because Mr. B was at rest during first 2 hours, while Mr. A was keep moving. The visual form of this scene is shown below:
Here you can see that Mr. A took 2 hours to cover some distance indicated by straight line, while the remaining distance is denoted by doted line. After that Mr. B also started his journey shown by doted line. Now at this instance the time taken by the two people to cover their distance of doted line is same. Because starting time to travel in doted line and the ending time to overtake is same, so time during the trip to cover the doted line is same.
Hence, 2 hours is extra time which Mr. A has take more. In other words, we can say that Mr. A took 2 hours more than Mr. B to reach the meeting point. I’m sure your question is now cleared on why Mr. A will take more time even if Mr. B start his journey late.
So accordingly to the relationship of time, we can write,
TA = TB + 2 (As Mr. A took 2 hrs more time than Mr. B)
As we need to find time required by Mr. B that is TB, so By putting this value of TA in (eq. 4), we’ll get
⇒ 3 (TB + 2) = 4 TB
⇒ 3 TB + 6 = 4 TB
⇒ TB = 6 hrs Answer
2). Both persons moving towards each other:
Now, let’s consider the following figure in which 2 persons standing a some distance and then started moving towards each other as shown below:
How to find the distance covered be Mr. A after he meet with Mr. B?
In this scenario, as you can see that the total distance is given, so let’s begin with this variable lead towards the answer.
According to the given condition,
DA + DB = total distance (DT)
⇒ SA TA + SB TB = DT
⇒ 15 TA + 20 TB = 120
By taking 5 common from left hand side (L.H.S) of the above equation,
⇒ 5 (3 TA + 4 TB) = 120
Now dividing by 5 on both sides, we’ll get,
⇒ 3 TA + 4 TB = 24 ————————— (eq. 5)
Now, think about another equation that may be made regarding the time taken by the two people.
You see that both people started moving towards each other, so it means both have started at the same time. Now, the starting time of both people is same (for instance 9:00 am), also the meeting time of both people is same.
In other words,
TA = TB
As we need to find the distance covered by Mr. A, that is DA which is equals to SA × TA = (15) TA, By putting this value of TB in (eq. 5) to find value of TA, we’ll get
⇒ 3 TA + 4 TA = 24
⇒ 7 TA = 24
⇒ TA = 24⁄7 Answer
Some tricky questions may asked from this concept. For instance, from previous question, what is the distance between Mr. A and Mr. B 1 hr before they met?
Take a moment and Try yourself!
If you can’t solve yet, let’s solve such easy question.
Always remember a distance, speed and time question always require a diagram. So you need to make your own quick diagram such as shown below:
You can see that the speed of Mr. A is 15 miles/hr which means he travels a distance of 15 miles in 1 hour. Similarly speed of Mr. B is 20 miles/hr, suggesting that Mr. B can travel a distance of 20 miles in 1 hour. Now according to the above diagram, Mr. A & Mr. B were (15 + 20 = 35) miles apart 1 hour before they met.
For those who still didn’t understood whats happening, you see the figure that we suppose the points 1 hour before both persons are standing. Right!
Now, Mr. A can travel a distance of 15 miles in 1 hour as indicated by his speed. While Mr. B can travel a distance of 20 miles in 1 hour as indicated by his speed. So Mr. A will take 15 miles of distance to reach to the meeting point, while Mr. B will take a distance of 20 miles to reach to the meeting point. So total distance between both persons was (15 + 20 = 35) miles 1 hour before they met.
3). Both persons moving against each other:
Similarly, when the two persons moving from the same point but in opposite direction, the total distance between them would be the sum of their individual distances. So we’ll do the same calculation and with the method used in scenario of ‘2). Both person moving towards each other’, we’ll get the same answer here. I’m not writing example here, as you’ll see plenty of questions in assignments and quizzes for your practice.