1) The last Friday of a particular month is on the 25th day of the month. What day of the week is the first day of the month?
2) Three water pipes are used to fill a swimming pool. The first pipe alone takes 8 hours to fill the pool, the second pipe alone takes 12 hours to fill the pool, and the third pipe alone takes 24 hours to fill the pool. If all three pipes are opened at the same time, how long will it take to fill the pool?
3) Suppose two days ago was Sunday. What day of the week will 365 days from today then be?
1) Three weeks or 21 days before Friday the 25th is Friday the 4th. Count back to the first day. The first day of the month is Tuesday.
2) In one hour, the first pipe will fill 1/8 of the pool, the second pipe 1/12, and the third pipe 1/24. Together, in one hour, they will fill 1/8 + 1/12 + 1/24 or 3/24 + 2/24 + 1/24 = 6/24 = 1/4 of the pool. Therefore, the three pipes will fill the entire pool in 4 hours.
3) Today is Tuesday. 365 days from now is equivalent to 52 weeks and 1 day. 52 weeks from today is also Tuesday. The 365th day is Wednesday.
1) If a class of children is separated into groups of 5 children, 2 children will be left over. If the class is separated into groups of 6 children, 3 children will be left over. What is the smallest number of children the class could have?
2) There is an even number between 200 and 300 that is divisible by 5 and also by 9. What is the number?
3) A book has 500 pages numbered 1, 2, 3, and so on. How many times does the digit 1 appear in the page numbers?
1) Method 1 –
must be an odd number. Then the only possibilities for the class size are: 7, 17, 27, 37, . . Divide each by
6. The first number that has a remainder of 3 is the required number. That number is 27.
Method 3 - Algebra: Let A and B be numbers such that 5A + 2 = 6B + 3. Subtract 2 from each member of
the equation. Then 5A = 6B + 1. Since 6B + 1 is an odd number, 5A is also odd. 5A may be 5, 15, 25, 35,
45, . . . Divide each of these numbers by 6. The first number that has a remainder of 1 is 25: 25/6 = 4 R1. So B = 4. Since 6B + 3 is the class size, and B = 4, the class size is 6 x 4 + 3 = 2.
2) Since the number is divisible by 2, 5, and 9, it is also divisible by the product 2 x 5 x 9 = 90. The multiple of 90 between 200 and 300 is 270.
3) Consider the frequency of appearance of the digit "1" in each of the places.
1) Suppose today is Tuesday. What day of the week will it be 100 days from now?
2) The four-digit numeral 3AA1 is divisible by 9. What does A represent?
3) A purse contains 4 pennies, 2 nickels, 1 dime, and 1 quarter. Different values can be obtained by using one or more coins in the same purse. How many different values can be obtained?
1) Every 7 days from "today" will be Tuesday. Since 98 is a multiple of 7, the 98th day from today will be Tuesday. Then the 100th day from today will be Thursday.
2) If the number is divisible by 9, then the sum of its digits is divisible by 9. The digit sum is 3 + A + A + 1 = 4 + 2A. The digit sum cannot by 9, otherwise A = 2 1/2. So 4 + 2A = 18 which produces A = 7.
3) The largest amount that can be made is $0.49. Using the given set of coins, any amount from $0.01 to $0.49 can be made. Therefore there are 49 different amounts that can be made.