Question:A can run 1 km in 3 min. 10 sec. and B can cover the same distance can A beat B ?
Clearly, A beats B by 10 sec. Distance covered by B in 10 sec.` = (1000/200 xx 10) m` = 50 m. :. A beats B by 50 metres.
Question:In a 100 m race, A runs at 8 km per hour. If A gives B a start of 4 m and still beats him by 15 seconds, what is the speed of B ?
Time taken by A to cover 100 m` = ((60 xx 60)/8000 xx 100) sec.`
= 45 sec.
:. B covers (100 - 4) m = 96 m in (45 + 15) sec
= 60 sec.
:. B's speed` = ((96 xx 60 xx 60)/(60 xx 1000))` km/hr = 5.76 km/hr.Question:A, B and C are three contestants in a km race. If A can give B a start of 40 m and A can give C a start of 64 m, how many metre's start can B give C ?
While A covers 1000 m, B covers (1000 - 40)m = 960 m and C covers (1000 - 64) m or 936 m. When B Covers 960 m, C covers 936 m When B covers 1000 m, C covers` (936/960 xx 1000) m` = 975 m. :. B can give C a start of (1000 - 975) or 25 m.
Question:In a game of 80 points, A can give B 5 points and C 15 points. Then how many points B can give C in a game 60 ?
A : B = 80 : 75, A : C = 80 : 65 `B/C = (B/A xx A/C)` `= (75/80 xx 80/65)` `= 15/13 = 60/52 = 60 : 52` :. In a game of 60, B can give C 8 points.
Question:Find the day of the week on 6th July, 1776.
(i) 16th july, 1776 means
= (1775 years + period from 1st January to 16th July)
Now, 1600 years have 0 odd day.
100 years = (18 leap years + 57 ordinary years)
= (36 + 57) odd days = 93 odd days
= (13 weeks + 2 days) of odd days
= 2 odd days.
:. 1775 years have (0 + 5 + 2) odd days i,e 0 odd days.
Now, days from 1st Jan, 16th July ; 1776
Jan Feb March April May June July
31 + 29 + 31 + 30 + 31 + 30 + 16 = 198 days
= (28 weeks + 2 days) = 2 odd days.
:. Total number of odd days = (0 + 2) = 2.
:. The day of the week was ' Tuesday'Question:What was the day of the week on 12th January, 1979 ?
Number of odd days in (1600 + 300) years = (0 + 1) = 1.
78 years = (19 leap years + 59 ordinary years)
= (38 + 59) odd days = 97 odd days = 6 odd days.
12 days of January have 5 odd days.
:. Total number of odd days = (1 + 6 + 5) = 5 odd days.
:. The desired day was 'Friday'.Question:Find the day of the week on 25th December, 1995.
1600 years have 0 odd day. 300 years have 1 odd day. 94 years = (23 leap years + 71 ordinary years) = (46 + 71) odd days = 117 odd days = 5 odd days. Number of days from 1st Jan, 1995 to 25 th Dec, 1995 = 359 days. = (51 weeks + 2 days) = 2 odd days. :. Total number of odd days = (0 + 1 + 5 + 2) odd days = 1 odd day. :. Required day is 'Monday'.
Question:On what dates of October, 1994 did Monday fall ?
Find the day on 1st October, 1994. 1600 years contain 0 odd day. 300 years contain 1 odd day. 93 years = (23 leap years + 70 ordinary years) = (46 + 70) odd years = 116 odd days = 4 odd days. Days from 1st Jan, 1994 to 1st Oct, 1994 Jan Feb March April May June July Aug Sept Oct 31 + 28 + 31 + 30 + 31 + 30 + 31 + 31 + 30 + 1 = 274 days = 39 weeks + 1 days = 1 odd day. :. Total number of odd days = (0 + 1 + 4 + 1) = 6. :. 1st Oct, 1994 was 'Saturday' :. Monday fell on 3rd October, 1994. During October 1994, Monday fell on 3rd, 17th and 24 th.
Question:Prove that the calendar for 1995 will for 2009.
In order that the calendar for 1995 and 2006 be the same, 1st January of both the years must be on the same days week. For this, the total number of odd days between 3 1st Dec, 1994 and 31 st Dec, 2005 must be zero. We know that an ordinary year has 1 odd day and a leap year has 2 odd days. During this period there are 3 leap years, namely 1996 , 2000 and 2004 and 8 ordinary years. :. Total number of odd days during this period (6 + 8) + 14 = 0 i.e. 0 odd day. Hence, the calendar for 1995 will serve for 2006.
Question:Prove that any date in March is the same day of the week as the corresponding date in November of that year.
In order to prove the required result, we have to show that the number of odd days between last day of February and last day of October is Zero. Number of days between these dates are. March April May June July Aug Sep Oct 31 + 30 + 31 + 30 + 31 + 31 + 30 + 31 `= 241 days = 35 weeks :. Number of odd days during this period = 0. Hence, the result follows.