Problem Solving

If x^4 + y^4 = 100, then the greatest possible value of x is between

For any positive integer n, n>1, the "length" of n is the number of positive primes (not necessary distinct) whose product is n. For ex, the length of 50 is 3, since 50=2x5x5. What is the greatest possible length of a positive integer less than 1000.

Machines A and B always operate independently and at their respective constant rates. When working alone, Machine A can fill a production lot in 5 hours, and Machine B can fill the same lot in x hours. When the two machines operate simultaneously to fill the production lot, it takes them 2 hours to complete the job. What is the value of x ?

Which of the following must be equal to zero for all real numbers x ? I. − 1 x II. x + (-x) III. x^0

A soccer team played 25% more matches year-on-year in the second year of its formation. It won 84% of the matches it played in the second year, which was an improvement of 5% over the percentage of matches won in the first year. In the third year, the team played 4% fewer matches year-on-year but lost only 12.5% of the matches it played, which was one less than the number of matches lost by the team in the second year. Which of the following statements must be correct? I. The team lost the same number of matches in the first yea and the second year II. The team won the same number of matches in the second year and the third year III. The team played five more matches in the third year than the first year

The table above gives the gasoline costs and consumption rates for a certain car driven at 50 miles per hour, using each of two brands of gasoline. How many miles farther can the car be driven at this speed on $12worthofbrandXgasolinethanon$12 worth of brand Y gasoline?

Which of the following CANNOT yield an integer when divided by 10 ?

If s, u, and v are positive integers and 2s = 2u + 2v, which of the following must be true? I. s = u II. u ≠ v III. s > v

The value of − 89 − − √ 3 − 89 3 is:

In a certain class consisting of 36 students, some boys and some girls, exactly 1/3 of the boys and exactly 1/4 of the girls walk to school. What is the greatest possible number of students in this class who walk to school?

Four hours from now, the population of a colony of bacteria will reach 1.28 * 10 6 10 6 . If the population of the colony doubles every 4 hours, what was the population 12 hours ago?

38 69 22 73 31 47 13 82 Which of the following numbers is greater than three fourths of the numbers but less than one fourth of the numbers in the list above?

If Jack had twice the amount of money that he has, he would have exactly the amount necessary to buy 3 hamburgers at $0.96apieceand2milkshakesat$1.28 apiece. How much money does Jack have?

A bank customer borrowed $10,000,butreceivedydollarslessthanthisduetodiscounting.Iftherewasaseparate$25 service charge, then, in terms of y, the service charge was what fraction of the amount that the customer received?

If r and s are integers and rs + r is odd, which of the following must be even ?

If (t- 8) is a factor of t^2 - kt - 48, then k=

An empty swimming pool with a capacity of 5,760 gallons is filled at the rate of 12 gallons per minute. How many hours does it take to fill the pool to capacity' ?

A hiker walked for two days. On the second day the hiker walked 2 hours longer and at an average speed 1 mile per hour faster than he walked on the first day. If during the two days he walked a total of 64 miles and spent a total of 18 hours walking, what was his average speed on the first day?

Coins are dropped into a toll box so that the box is being filled at the rate of approximately 2 cubic feet per hour. If the empty rectangular box is 4 feet long, 4 feet wide, and 3 feet deep, approximately how many hours does it take to fill the box?

The organizers of a fair projected a 25 percent increase in attendance this year over that of last year, but attendance this year actually decreased by 20 percent. What percent of the projected attendance was the actual attendance?