Answer

Verified

453k+ views

Hint: Distance in both cases will be equal. So, find equations of distance and both the cases and after that we can get the length of journey.

\[ \Rightarrow \]Let the actual speed of the train be \[x{\text{ km/hr}}\]

\[ \Rightarrow \]And the actual time taken be \[{\text{y}}\] hours.

\[ \Rightarrow \]As distance covered \[ = \] speed\[*\]time.

\[ \Rightarrow \]So distance covered \[ = (xy){\text{km}}\] ………………………...(1)

When the speed is increased by 6 km/hr,

Then time of journey is reduced by 4 hours.

So, speed will become,

Speed is \[(x + 6){\text{km/hr}}\] and time of journey is \[(y - 4)\]hours.

So, distance covered will be,

\[ \Rightarrow \]Distance covered \[ = {\text{ }}(x + 6)(y - 4).\] ………………………...(2)

As we know that the distance covered will always be the same.

So, comparing equation 1 and 2. We get,

\[ \Rightarrow - 2x + 3y - 12 = 0\] ………………………………...(3)

When the speed is reduced by 6 km/hr,

Then time of journey is increased by 6 hours.

So, speed will become,

Speed is \[(x - 6){\text{km/hr}}\] and time of journey is \[(y + 6)\] hours.

So, distance covered will be,

\[ \Rightarrow \]Distance covered \[ = {\text{ }}(x - 6)(y + 6).\] ……………………………..(4)

As we know that distance covered will always be the same.

So, comparing equation 1 and 4. We get,

\[ \Rightarrow xy = (x - 6)(y + 6) = xy - 6y + 6x - 36\]

From the above equation. We will get,

\[ \Rightarrow x - y - 6 = 0\] ………………………………….(5)

Thus, we obtain following system of equations:

\[

\Rightarrow - 2x + 3y - 12 = 0 \\

\Rightarrow x - y - 6 = 0 \\

\]

By using cross-multiplication, we have,

\[ \Rightarrow \dfrac{x}{{(3)*( - 6) - ( - 1)*( - 12)}} = \dfrac{{ - y}}{{( - 2)*( - 6) - (1)*( - 12)}} = \dfrac{1}{{( - 2)*( - 1) - (1)*(3)}}\]

On solving the above equation. It becomes,

\[ \Rightarrow \dfrac{x}{{ - 30}} = \dfrac{{ - y}}{{24}} = \dfrac{1}{{ - 1}}\]

\[ \Rightarrow \]So, \[x = 30\] and \[y = 24\]

Putting the value of x and y in equation 1. We get,

Distance \[ = (30*24)km{\text{ }} = {\text{ }}720{\text{km}}\]

\[ \Rightarrow \]Hence, the length of the journey is 720km

Note: Whenever we came up with this type of problem then easiest and efficient way to find the length of journey is first, assume speed as a variable x and time as variable y and then make equations using formula, distance \[ = \]speed\[*\]time. And then solve the equations by using a cross multiplication method to get the required length of journey.

\[ \Rightarrow \]Let the actual speed of the train be \[x{\text{ km/hr}}\]

\[ \Rightarrow \]And the actual time taken be \[{\text{y}}\] hours.

\[ \Rightarrow \]As distance covered \[ = \] speed\[*\]time.

\[ \Rightarrow \]So distance covered \[ = (xy){\text{km}}\] ………………………...(1)

When the speed is increased by 6 km/hr,

Then time of journey is reduced by 4 hours.

So, speed will become,

Speed is \[(x + 6){\text{km/hr}}\] and time of journey is \[(y - 4)\]hours.

So, distance covered will be,

\[ \Rightarrow \]Distance covered \[ = {\text{ }}(x + 6)(y - 4).\] ………………………...(2)

As we know that the distance covered will always be the same.

So, comparing equation 1 and 2. We get,

\[ \Rightarrow - 2x + 3y - 12 = 0\] ………………………………...(3)

When the speed is reduced by 6 km/hr,

Then time of journey is increased by 6 hours.

So, speed will become,

Speed is \[(x - 6){\text{km/hr}}\] and time of journey is \[(y + 6)\] hours.

So, distance covered will be,

\[ \Rightarrow \]Distance covered \[ = {\text{ }}(x - 6)(y + 6).\] ……………………………..(4)

As we know that distance covered will always be the same.

So, comparing equation 1 and 4. We get,

\[ \Rightarrow xy = (x - 6)(y + 6) = xy - 6y + 6x - 36\]

From the above equation. We will get,

\[ \Rightarrow x - y - 6 = 0\] ………………………………….(5)

Thus, we obtain following system of equations:

\[

\Rightarrow - 2x + 3y - 12 = 0 \\

\Rightarrow x - y - 6 = 0 \\

\]

By using cross-multiplication, we have,

\[ \Rightarrow \dfrac{x}{{(3)*( - 6) - ( - 1)*( - 12)}} = \dfrac{{ - y}}{{( - 2)*( - 6) - (1)*( - 12)}} = \dfrac{1}{{( - 2)*( - 1) - (1)*(3)}}\]

On solving the above equation. It becomes,

\[ \Rightarrow \dfrac{x}{{ - 30}} = \dfrac{{ - y}}{{24}} = \dfrac{1}{{ - 1}}\]

\[ \Rightarrow \]So, \[x = 30\] and \[y = 24\]

Putting the value of x and y in equation 1. We get,

Distance \[ = (30*24)km{\text{ }} = {\text{ }}720{\text{km}}\]

\[ \Rightarrow \]Hence, the length of the journey is 720km

Note: Whenever we came up with this type of problem then easiest and efficient way to find the length of journey is first, assume speed as a variable x and time as variable y and then make equations using formula, distance \[ = \]speed\[*\]time. And then solve the equations by using a cross multiplication method to get the required length of journey.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Establish a relation between electric current and drift class 12 physics CBSE

Guru Purnima speech in English in 100 words class 7 english CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Copper is not used as potentiometer wire because class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE