Question

cular field has a circumference of 360 km. Three cyclists start together and can cycle 48, 60 and 72 km a day, round the field. When will they meet again

Answer 1

$\underline{\underline{\huge{\textbf{Question:}}}}$

A circular field has a circumference of 360 km. Three cyclists start together and can cycle 48, 60 and 72 km a day, round the field. When will they meet again

$\underline{\underline{\huge{\textbf{Solution:}}}}$

Given,
Circumference of Circular field $C = 360\: km$

Three Cyclists start cycling at the same time,

Cyclist 1 can cycle 48 km a day round the field.

Cyclist 2 can cycle 60 km a day round the field.

Cyclist 3 can cycle 72 km a day round the field.

They will meet again after ?

$\underline{\large{\textbf{Step-1:}}}$
Find the time taken by each cyclists to cover the total Circumference of the field.

We have,
$\bigstar \mathbf{time\: taken = \dfrac{Distance\: covered}{Speed}}$

To cover the Circumference of field:

time taken by Cyclist -1, $t_{1}= \frac{360}{48}$

$\implies t_{1} = \frac{15}{2} = 7\, \dfrac{1}{2} = 7.5\: days$

time taken by Cyclist -2, $t_{1}= \dfrac{360}{60}$

$\implies t_{2} = 6\: days$

time taken by Cyclist -3, $t_{1}= \dfrac{360}{72}$

$\implies t_{3} = 5\: days$

$\underline{\large{\textbf{Step-2:}}}$
Express the time taken by each cyclists to cover the total Circumference of the field in hours

We have,
$\bigstar \textsf{1 Day = 24 Hours}$

$\implies t_{1} = 7.5 \times 24 = 180\: Hrs$

$\implies t_{2} = 6 \times 24 = 144\: Hrs$

$\implies t_{3} = 5 \times 24 = 120\: Hrs$

$\underline{\large{\textbf{Step-3:}}}$
Find the Interval of meeting of all the three Cyclists.

$\textit{Interval of meeting is equal to the LCM of time taken by all three Cyclists}$

Interval of meeting = LCM($t_{1}, t_{2},t_{3})$

Substituting Values

Interval of meeting = LCM(180, 144, 120) ------(1)

$\underline{\large{\textbf{Step-4:}}}$
Find LCM(180, 144, 120)

- Express 180, 144, 120 as product of Prime factors

$180 = 2^{2} \times 3^{2} \times 5$

$144 = 2^{4} \times 3^{2}$

$120 = 2^{3} \times 3 \times 5$

- $\textsf{LCM is the product of highest powers of Prime factors.}$

$\implies LCM(180, 144, 120) = 2^{4} \times 3^{2} \times 5 = 720$

$\underline{\large{\textbf{Step-5:}}}$
Substituting value of LCM in (1)

Interval of meeting = 720 Hours

The three Cyclists will meet after 720 Hours after start time

$\underline{\large{\textbf{Step-6:}}}$
Express Interval of Meeting in days

Interval of meeting = $\dfrac{720}{24}$

Interval of meeting = 30 days.

$\therefore$
$\bigstar \textsf{The three Cyclists meet again after \underline{\textbf{30\: days}}}$

Answer 2

Answer:

after 720 hrs

Step-by-step explanation:

no need to do all that big steps just find the LCM of 40,60 and 72 which is 720