A circular field has a circumference of 360 km. Three cyclists start together and can cycle 48, 60 and 72 km per day, round the field. When will they meet again?
(a) 7 days
(b) 15 days
(c) 23 days
(d) 30 days
Answer
381k+ views
Hint: One of the multiples of the LCM of the velocities will be the distance travelled by them to meet again. As the path is circular, it is certain that they will meet again otherwise, they will never meet.
Complete step-by-step answer:
Let us assume that the three cyclists meet again after x days from starting.
Therefore, the first cyclist travelled \[48x\] km before the meeting. Similarly, the second and third cyclist travelled \[60x\] and \[72x\] km respectively.
Since the path is circular with circumference 360 km, after travelling 360 km they cross the starting point again.
Therefore, we can write \[48x-360{{k}_{1}}=60x-360{{k}_{2}}=72x-360{{k}_{3}}\] where \[{{k}_{1}},{{k}_{2}},{{k}_{3}}\] integers are as they meet finally and after 360km, they travel a whole circle completely.
Now, from first and second equations, we get,
\[12x=360({{k}_{2}}-{{k}_{1}})\]
Or,\[x=30({{k}_{2}}-{{k}_{1}})\]
Similarly, taking second and third equation or third and first equation we get,
\[x=30({{k}_{3}}-{{k}_{2}})\] and \[x=15({{k}_{3}}-{{k}_{1}})\] respectively.
So, \[30({{k}_{2}}-{{k}_{1}})=30({{k}_{3}}-{{k}_{2}})\]
Or, \[{{k}_{2}}-{{k}_{1}}={{k}_{3}}-{{k}_{2}}\]
So, we can see \[{{k}_{1}},{{k}_{2}},{{k}_{3}}\] are in Arithmetic Progression.
Hence, we can write \[{{k}_{1}}={{k}_{2}}-d\] and \[{{k}_{3}}={{k}_{2}}+d\] where d is the common difference of the A.P.
So, \[x=30d\]
Now d cannot be 0 or negative as x is positive.
So, the minimum integral value of d is one. Hence for d=1, we get x = 30
Therefore, they meet after 30 days.
Hence, the correct option for the given question is option (d) 30 days.
Note: After completing each complete cycle, they are just passing the starting point. If we consider the LCM of the velocities, we will get it as 720, which can be individually covered by them in 10, 12 and 15 days respectively. Hence, the distance they meet again must be a multiple of 720. If we multiply 30 with the velocities of all of them, we can clearly see that they are all multiples of 720.
Complete step-by-step answer:
Let us assume that the three cyclists meet again after x days from starting.
Therefore, the first cyclist travelled \[48x\] km before the meeting. Similarly, the second and third cyclist travelled \[60x\] and \[72x\] km respectively.
Since the path is circular with circumference 360 km, after travelling 360 km they cross the starting point again.
Therefore, we can write \[48x-360{{k}_{1}}=60x-360{{k}_{2}}=72x-360{{k}_{3}}\] where \[{{k}_{1}},{{k}_{2}},{{k}_{3}}\] integers are as they meet finally and after 360km, they travel a whole circle completely.
Now, from first and second equations, we get,
\[12x=360({{k}_{2}}-{{k}_{1}})\]
Or,\[x=30({{k}_{2}}-{{k}_{1}})\]
Similarly, taking second and third equation or third and first equation we get,
\[x=30({{k}_{3}}-{{k}_{2}})\] and \[x=15({{k}_{3}}-{{k}_{1}})\] respectively.
So, \[30({{k}_{2}}-{{k}_{1}})=30({{k}_{3}}-{{k}_{2}})\]
Or, \[{{k}_{2}}-{{k}_{1}}={{k}_{3}}-{{k}_{2}}\]
So, we can see \[{{k}_{1}},{{k}_{2}},{{k}_{3}}\] are in Arithmetic Progression.
Hence, we can write \[{{k}_{1}}={{k}_{2}}-d\] and \[{{k}_{3}}={{k}_{2}}+d\] where d is the common difference of the A.P.
So, \[x=30d\]
Now d cannot be 0 or negative as x is positive.
So, the minimum integral value of d is one. Hence for d=1, we get x = 30
Therefore, they meet after 30 days.
Hence, the correct option for the given question is option (d) 30 days.
Note: After completing each complete cycle, they are just passing the starting point. If we consider the LCM of the velocities, we will get it as 720, which can be individually covered by them in 10, 12 and 15 days respectively. Hence, the distance they meet again must be a multiple of 720. If we multiply 30 with the velocities of all of them, we can clearly see that they are all multiples of 720.
Recently Updated Pages
Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Find the values of other five trigonometric ratios class 10 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts
What is 1 divided by 0 class 8 maths CBSE

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

What is the past tense of read class 10 english CBSE

What is pollution? How many types of pollution? Define it

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

How many crores make 10 million class 7 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE

Students Also Read