There are 12 points are put at equal intervals on the circumference of a circular truck as shown in Fig.1.
Boy runs clockwise and girl runs counterclockwise with a fixed speed, respectively.
Girl goes around a truck in 24 minutes and boy runs by double speed of the girl.
For example when one boy and two girls begin to run from the location of Fig.2 at the same time, boy and girls pass each other once at each of P point and Q point before the boy reaches spot R for the first time.
Thus boy and girls will pass each other twice all together.
Answer the following questions.
(1) One boy and two girls begin to run from the location of Fig.2 at the same time.
How many times do they pass each other all together before girls go around a truck?
(2) Two boys and three girls begin to run from the points of Fig.3 at the same time.
How many times do boys and girls pass each other at the point P all together before girls goes around a truck?
(3) Two boys and three girls begin to run from the points of Fig.3 at the same time.
How many hours and minutes later do they become to be the position of Fig.4 after started running?
Noted that boys and girls passed each other 9 times at the point P all together after they begin to run until they become to the position of Fig.4.
Answer
(1) 6 times
(2) 2 times
(3) 1 hour 48 minutes
Solution
(1) Boy moves one piece per one minute according to 12/12 = 1.
Girl moves 0.5 piece per one minute according to 12 / 24 = 0.5.
According to 3/ (1 + 0.5)= 2, 6/(1 + 0.5) = 4 and 12 / (1 + 0.5) = 8, there are six times of 2 minutes later, 4 minutes later, 10 minutes later, 12 minutes later, 18 minutes later and 20 minutes later when boy and girls meet in 24 minutes.
(2) In the figure below the time when girls come to P in 24 minutes are 18 minutes later for G1, 24 minutes later for G2 and 3 minutes later for G3.
The time when boys come to P in 24 minutes are 10 minutes 22 minutes later for B1 and 6 minutes and 18 minutes later for B2.
Therefore it's twice when a man and a girl meet at P.
(3) Explore time when boys and girls come to P.
B1 comes 10, 22, 34, 46, 58, 70, 82, 94 ---- minutes.
B2 comes 6, 18, 30, 42, 54, 66, 78, 90, 102 ----- minutes.
G1 comes 18, 42, 66, 90, 114 ---- minutes, and G2 comes 24, 48, 72, 96, 130 ------ minutes, and G3 comes 6, 30, 54, 78, 102 ---- minutes. Therefore the position when a boy and a girl passed each other 9th times is as shown in the Fig. 8 and this will be 102 minutes later.
When checking the time boys come to S,U and time girls come to T,U,V thereafter, it is as following.
B1 comes to S in 108 minutes and comes to U in 112 minutes.
B2 come to S in 104 minutes and comes to U in 108 minutes.
G1 come to T in 108 minutes, U in 102 minutes and V in 120 minutes.
G2 comes to T in 114 minutes, U in 108 minutes and V in 102 minutes.
G3 comes to T in 124 minutes, U in 114 minutes and V in 108 minutes.
Therefore it'll be 108 minutes later = 1 hour and 48 minutes later when the position of boys and girls become as Fig.4 after 102 minutes.
