1. In a stationery store, pencils have one price and pens have another price. Two pencils and three pens cost 78 cents. But three pencils and two pens cost 72 cents. How much does one pencil cost?
2. What number multiplied by itself is equal to the product of 32 and 162?
3. Julius Caesar wrote the Roman Numerals I, II, III, IV, and V in a certain order from left to right. He wrote I before III but after IV. He wrote II after IV but before I. He wrote V after II but before III. If V was not the third numeral, in what order did Caesar write the five numerals from left to right?
4. Thirteen plums weigh as much as two apples and one pear. Four plums and one apple have the same weight as one pear. How many plums have the weight of one pear?
5. A train can hold 78 passengers. The train starts out empty and picks up 1 passenger at the first stop, 2 passengers at the second stop, 3 passengers at the third stop, and so forth. After how many stops will the train be full?
6. If a kindergarten teacher places her children 4 on each bench, there will be 3 children who will not have a place. However, if 5 children are placed on each bench, there will be 2 empty places. What is the smallest number of children the class could have?
7. When I open my mathematics book, there are two pages which face me and the product of the two page numbers is 1806. What are the two page numbers?
8. Alice and Betty each want to buy the same kind of ruler. But Alice is 22 cents short and Betty is 3 cents short. When they combine their money, they still do not have enough money. What is the most the ruler could cost?
9. The owner of a bicycle store had a sale on bicycles (two-wheelers) and tricycles (three-wheelers). Each cycle had two pedals. When he counted the total number of pedals of the cycles, he got 50. When he counted the total number of wheels of the cycles, he got 64. How many tricycles were offered in the sale?
10. A jar filled with water weighs 10 pounds. When one-half of the water is poured out, the jar and remaining water weigh 5¾ pounds. How much does the jar weigh?
11. A certain natural number is divisible by 3 and also by 5. When the number is divided by 7, the remainder is 4. What is the smallest number that satisfies these conditions?
12. Alice needs 1 hour to do a certain job. Betty, her older sister, can do the same job in 1/2 hour. How many minutes will it take them to do the job if they work together at the given rates?
13. 30! represents the product of all natural numbers from 1 through 30 inclusive: 1 x 2 x 3 x 4 x 5 x ... x 28 x 29 x 30. If the product is factored into primes, how many 5s will the factorization contain?
14. Peter has one of each of the following coins in his pocket: a penny, a nickel, a dime, a quarter, and a half-dollar. Four of these coins are taken out of the pocket and the sum of their values is calculated. How many different sums are possible?
15. When the order of the digits of 2552 is reversed, the number remains the same. How many counting numbers between 100 and 1000 remain the same when the order of the number's digits is reversed?
16. Tom multiplied a number by 2½ and got 50 as an answer. However, he should have divided the number by 2½ to get the correct answer. What is the correct answer?
17. Said Anne to Betty: "If you give me one marble, we will each have the same number of marbles. Said Betty to Anne: "If you give me one marble, I will have twice as many marbles as you will then have." How many marbles did Anne have before any exchange was made?
18. If a is divided by b, the result is 3/4. If b is divided by c, the result is 5/6. What is the result when a is divided by c?
19. In the following sequence of numbers, each number has one more 1 than the preceding number: 1, 11, 111, 1111, 11111, ... . What is the tens digit of the sum of the first 30 numbers of the sequence?
20. A study of 50 high school students showed that exactly 25 of them took Biology, exactly 20 of them took Chemistry, and exactly 12 of them took both subjects. How many of the 50 students took neither Biology nor Chemistry?
21. An Olympiad team is made up of students from the 4th, 5th, and 6th grades only. Seven students are 5th graders, eleven students are 6th graders, and one-third of the entire team are 4th graders. How many students are on the team?
22. I have a drawer which contains 40 socks in the following numbers and colors: 12 tan, 9 brown, 11 gray, and 8 blue. Suppose I am blindfolded. What is the fewest number of socks I must pick from the drawer to be absolutely certain that I have two socks of the same color among those I have picked?