Two trains leave different cities heading toward each other at different speeds. When and where do they meet?
Train A, traveling 70 miles per hour (mph), leaves Westford heading
toward Eastford, 260 miles away. At the same time Train B, traveling 60
mph, leaves Eastford heading toward Westford. When do the two trains
meet? How far from each city do they meet?
To solve this problem, we'll use the distance formula:
Distance = Rate x Time
Since an equation remains true as long as we perform the same operation on both sides, we can divide both sides by rate:
Distance
---------- = Time
Rate
or by time:
Distance
---------- = Rate
Time
So rate is defined as distance divided by time, which is a ratio. Speed is another word that is used for rate. When a
problem says that a train is moving at a speed of 40 mph, you can
understand this to mean that the train's rate is 40 mph, which means it
will travel 40 miles in one hour. Here are two different ways to approach this problem. Let's start by listing the information given:
Speed of Train A: 70 mph
Speed of Train B: 60 mph
Distance between Westford and Eastford: 260 miles
Method I: We'll use the notion of relative speed1 (or relative rate) in order to express the rates of the two trains in one number that can then be used in the distance formula.
Imagine you're on Train A. You're going 70 mph, so your speed relative
to the trees, houses, and other non-moving things outside the train is
70 mph. (All of those objects look as if they're going by at 70 mph.)
Now imagine you're the engineer and you can see Train B coming toward
you - not on the same track, of course! Since Train B is moving 60 mph,
it will look as if it's approaching faster than if it were sitting
still in the station - a lot faster than the trees and houses appear to
be moving. The relative speed of the two trains is the sum of the speeds they are
traveling. (If you're on either of the trains, this is the speed you
appear to be moving when you see the other train.) In our problem, the
relative speed of the two trains is 70 mph + 60 mph = 130 mph. What if
the trains were traveling in the same direction? Then we'd need to
subtract the speed of the slower train from the speed of the faster
train, and their relative speed would be 10 mph. At this point we know two of the three unknowns: rate and distance, so
we can solve the problem for time. Remember that time = distance/rate,
the distance traveled is 260 miles, and the relative speed is 130 mph:
t = 260 miles/130 mph
t = 2 hrs.
We find that the trains meet two hours after leaving their respective cities.
Method II: Here we'll begin by noting that the distance between
Westford and Eastford is 260 miles: this is the total distance the
trains will travel. Using the distance formula (Distance = rate x time,
or D = rt) we can express the distance traveled by each train:
Train A moving at 70 mph in t hours will cover 70t miles
Train B moving at 60 mph in t hours will cover 60t miles
Together the two trains will cover the distance 70t + 60t
Since we know that this distance is 260 miles, we can write the following
algebraic equation to represent this information.
70t + 60t = 260
Solving this equation we find that:
130t = 260
t = 2
which tells us that the trains will meet in 2 hours. Now, where do the trains meet? We again use the distance formula to find how far each train has traveled in two hours:
For Train A: 70 mph x 2 hrs = 140 miles
For Train B: 60 mph x 2 hrs = 120 miles
Thus the two trains meet at a point 140 miles from Westford and 120 miles from Eastford.
To check this, we can add 140 to 120: the answer is 260, which was the given distance between the two cities.
Let's look at a variation on this problem.
Train A, traveling 40 mph, leaves Westford heading toward Eastford, 260
miles away. One hour after Train A leaves Westford, Train B, traveling
70 mph, leaves Eastford heading toward Westford. When do the two trains
meet?
Notice that in this problem, the two trains do not leave their respective cities at the same time. Method I:
Let's move the starting point for Train A so we can
treat the problem as if the trains leave at the same time, which we
already know how to do. We know that Train A is moving 40 mph, and will therefore travel 40
miles in the hour before Train B leaves Eastford. This means that by
the time Train B starts moving, the two trains are only 260 - 40 = 220
miles apart. Now we can use the relative speed of the trains, which is
40 + 70 = 110 mph. Using the distance formula for time (time =
distance/rate), we write:
t = 220 miles/110 mph
t = 2 hrs.
Since t represents the time traveled by each train after Train A
has already traveled for one hour, Train B travels 2 hours before
meeting Train A. Adding the extra hour that Train A travels before
Train B starts moving, we see that Train A must travel 3 hours before
meeting Train B. Method II:
Let t represent the time that Train A travels. Since
Train B leaves one hour after Train A, let t -1 represent the time that
Train B travels. Again, the sum of the distances traveled by the two trains up until the
time they meet is 260 miles: between the two of them, they cover all 260
miles of track. Using the distance formula:
distance traveled by Train A = 40t
distance traveled by Train B = 70(t -1)
40t + 70(t-1) = 260 miles
t = 3 hrs.
Since t represents the time that Train A has been traveling, this means
that Train A travels 3 hours before meeting Train B. But how long has
Train B been traveling? Train B travels t -1 hours, which means that it
travels 2 hours before meeting Train A. Note:
To help us make sense of relative speed and how it relates to distance
traveled, let's think of it in terms of the distance the two trains
travel toward each other in one hour. We'll again use the distance
formula:
Rate x Time = Distance
Train A's rate is 70 mph, so: 70 mph x 1 hr = 70 miles
Train B's rate is 60 mph, so: 60 mph x 1 hr = 60 miles
If we add these two numbers together, we get 130 miles, which means that
130 miles of track will be covered in one hour. This is the same as
saying that the two trains are going 130 mph relative to each other, so:
130 mph x 1 hr = 130 miles
Either way, the numbers tell you that 130 miles of track will be covered in an hour.
1.Which company brought out the worlds first cell phone ? Answer - Motorola 2.Aam Aadmi Bima Yojana was launched on ? Answer - 2 October,2007 3.Founder of Swaraj Party are Chitharanjan Das &----------? Answer - Motilal Nehru 4.In which year Gandhiji attended second round table conference ? Answer - 1931 5.Who is known as the father of Indian budget ? Answer - P.C Mahalanobis 6.In which country Green Revolution started? Answer - Mexico 7.The ex officio chairman of National Development Council ? Answer - The Prime minister 8.India copied five year plans from ? Answer - Former Soviet Union 9.The final resting place of Nehru ? Answer - Santhi Van 10.Who was known as Desh Bandu? Answer - Chitharanjan Das
1.Green Revolution in India advanced during the period of ? Answer - Fourth Plan 2.Which type of clouds are implicated in the formation of Ozone hole? Answer - Nacreous 3.Which Indian State shares maximum number of states? Answer - Uttar Pradesh 4.Who was behind the 'Safety Valve' concept of congress? Answer - A.O Hume 5.Which Indian state first to introduce 'She Taxi' ? Answer - Kerala 6.World Radio day observed on? Answer - February 13 7.Who founded the Vidya Poshini Sabha? Answer - Sahodaran Ayyappan 8.Who was the first woman to address the congress session ?
Answer - Kadambini Ganguly
9.Which Prime minister of Britain lead the country to victory in second world war ? Answer - Winston Churchil
10.Which metal is stored in mineral oil ? Answer - Lithium
1.What is the time difference between Indian Standard time and Greenwich time ? Answer - 5 hours 30 minute 2.Sri Lanka is separated from India by a narrow sea called the ? Answer - Palk Strait 3.The boundary line between India and Pakistan is called ?
Plateau
Answer - Radcliffe Line 4.The boundary line between India and China is called ? Answer - McMahon Line 5-Srinagar is located along the banks of ?
Answer - Jhelum 6.The region acting as water divide between Indus and Ganga ? Answer - Ambala 7.The worlds highest Plateau? Answer - Tibet 8.The largest gap along the Western Ghat is ? Answer - Palghat gap 9.The river Narmada has its source at ? Answer - Amarkantak 10.The river which was known as the "Sorrow" of West Bengal ? Answer - Kosi
First telegraph line in India is operational between Calcutta and Diamond Harbour.
1853
First train in India runs from Bombay to Thane.
1857
First war of Indian independence also called the Sepoy Mutiny by the British.
1881
The first complete census of population conducted on a uniform basis throughout India.
1885
Formation of Indian National Congress by A.O. Hume, Dadabhai Naoroji, Dinshaw Wacha, W.C. Bonnerjee and others.
1905
Partition of Bengal by Lord Curzon. Swadeshi Movement launched.
1909
Indian Council Act popularly known as Minto Morley reforms allowing a limited increase in the involvement of Indians in governance passed.
1911
Visit of King George V to India, Shifting of capital from Calcutta to Delhi. Jana Gana Mana first sung at Calcutta session of INC. Launch of first air mail in India & World from Bumraulli to Allahabad.
1919
Government of India Act, 1919 introducing dyarchy, Rowlatt Act, Jallianwala Bagh tragedy.
1920
Khilafat movement, Launch of Non-cooperation movement.
1922
Chauri Chaura outrage in UP, Suspension of Non-cooperation movement.
1928
Visit of Simon Commission to India, Death of Lala Lajpat Rai
1929
Resolution of complete independence at Lahore session of Indian National Congress.
1930
Dandi March, Launch of the Civil Disobedience Movement.
1931
Gandhi Irwin pact, execution of Bhagat Singh, Sukhdev and Rajguru.
1935
Government of India Act.
1942
Quit India movement, Formation of Azad Hind Fauz.
1943
Visit of Cripps Commission to India.
1946
British Cabinet mission visited India.
1947
Independence and partition of India.
1948
Assassination of Mahatma Gandhi, 1st Pak aggression
1950
India became republic
1951
1st Five year plan and 1st Asian Games in Delhi
1952
1st General elections
1954
Panchasheel signed by India and China
1956
Reorganisation of Indian states on linguistic basis
1957
Introduction of decimal system in currency
1959
First television service started in India in New Delhi.
1961
Liberation of Goa from the Portuguese
1962
Chinese aggression
1964
Death of Jawaharlal Nehru
1965
Indo-Pak war
1966
Death of Lal Bahadur Shastri
1969
Split in Indian National Congress and nationalisation of 14 banks. India's first Atomic Power Station Tarapur commences commercial operation.
1971
Indo-Pak war
1972
Simla agreement signed between India and Pakistan
1974
First nuclear test codenamed Smiling Buddha carried out at Pokhran (Rajasthan)(May 18).
1.Who introduced rubber to Kerala?
- The Dutch 2.The largest public sector undertaking in Kerala ?
- KSRTC
3.Which river flows through the Silent Valley ?
- Kunthippuzha 4.Southern most river of Kerala ?
- Neyyar 5.Who called Pazhassi Raja as 'Kerala Simham'?
- Sardar K M Panicker 6.Pazhassi Raja was the ruler of ?
- Kingdom of Kottayam 7.Which temple is known as 'The Guruvayoor of South Kerala'?
- Sreekrishna Swami Temple,Ambalappuzha 8.In which language was Hortus Malabaricus was originally written?
- Latin 9.In which year was the Kulachal war?
- 1741 10.The first revolt against British in Kerala?
- Attingal Rebellion
