# Jan 2019 (CML)

I will post seven new problems every weekend, one for each day of the week. The idea is that you work on your own during the week and I will post answers the following Saturday. For those of you who practiced with me last year, the process is similar.

For instructions on submitting weekly answers, click on the “Logistics” tab.
For previously posted problems, click on the “Archive” tab.
For previous prizes, click on the “News” tab.

Week of Jan 27 – Feb 2.

Jan 27: Consecutive integers are numbers that follow one another in order. For example, 29, 30, 31, 32 are consecutive integers. The sum of any three consecutive integers will always be divisible by which one-digit number other than 1?

Answer: 3. (One of the three numbers is a multiple of 3, one has remainder 2 when divided by 3, and on has remainder 1.)

Jan 28: The sum of any five consecutive integers will always be divisible by which one-digit number other than 1?

Answer: 5. (When the numbers are divided by 5, the 5 remainders are 0, 1, 2, 3, 4.)

Jan 29: The sum of the prime factors of 60 is 2+2+3+5=12. Find the sum of the prime factors of 630?

Answer: 630 = 2*3*3*5*7.  The sum is 20.

Jan 30: One leaky faucet drips at the rate of 1 drip every 4 seconds. Assume 240 drips fills an 8-ounce cup. One gallon is 128 ounces. How many gallons of water will drip in 24 hours?

Answer: In 24 hours, there are 24*60*15 drips which fill up 24*60*15/240 = 90 cups. Since 16 cups fill up one gallon, 90/16=5.625 gallons will be filled.

Jan 31: A set of positive numbers has a sum of 15 and a product of 84. If the smallest of the three numbers is 2, what is the middle number?

Answer: x+y=13, xy=42. The three numbers are 2, 6 and 7. The middle one is 6.

Feb 1: If x is 25% of 80 and y is 20% of 60, what is x+y?

Feb 2: The sale price at a department store is 25% off the regular price. The closeout price is 40% off the sale price. If Mrs. L bought a dress at the closeout price of $45, what is the regular price of the dress? Answer:$100.

Week of Jan 20 – Jan 26.

Jan 20: In the multiplication problem $3B3 \times 15 = 5745$, what is $B$?

Jan 21: A quart bottle has all three of its dimensions doubled. How many quarts will the new bottle hold?

Jan 22: If $4 \times A >20$ and $4 \times A<35$, then $A$ represents which integer?

Jan 23: The surface area of a cube is 54 square inches. What is the volume of the cube?

Answer: The side length of the cube is 3. The volume is 27.

Jan 24: Mrs. M bought 4 new tires for a total of $360. If she replaces them at 80000 miles, the cost of the 4 tires will be ___ cents for every 100 miles. Answer:$360/800 = \$0.45 = 45cents.

Jan 25: Willy rides his bike at 4 miles in 15 minutes. He is traveling at ___ miles per hour.

Jan 26: A piece of wire is bent to a square of an area of 225 square inches. If the wire is bent to form a regular pentagon, what is the side length the pentagon?

Answer: The wire is 15*4=60 inches long. If the wire is bent to form a pentagon, the side length is 12 inches.

Week of Jan 13 – Jan 19.

Jan 13: In Mr. D’s class, 1/5 of the students failed the math test. Of the remaining students, 1/4 scored an A. What fraction of the students passed the test, but did not score an A?

Answer: 3/5 (1/5 failed, 1/5 scored A)

Jan 14: John is 14 and his mom is 41. Their ages have the same 2 digits. The next time that their ages have the same 2 digits, how old is John?

Answer: John is 25 and mom is 52.

Jan 15: If 8 less than k is 6 more than m, how much bigger is k than m?

Jan 16: Six people can be seated comfortably in a room 6′ by 8′. How many people can be seated comfortably in a room 12′ by 16′?

Jan 17: If 1/4 is greater than m/16 where m is an integer, what is the biggest possible value of m?

Jan 18: Two numbers are in the ratio of 4:5. If the smaller of the two numbers is 16, what is the bigger number?

Jan 19: A local pizza shop has 10 different choices of toppings. What is the number of combinations for 2 toppings?

Happy New Year!

Week of Jan 6 – Jan 12.

Jan 6: Two sides of a triangle have lengths 7” and 15”. If the third side also has an integral length, what is the longest possible length of the third side?

Jan 7: If 3/4 of a job can be done in 15 days, how many days does it take to finish the whole job?

Jan 8: There are 3 contestants in a hot dog eating contest. By the time Al finishes one hot dog, Ben finishes 3 and Carl finishes 2. The contest is over when Ben finishes his 27th hot dog. How many hot dogs do the three people eat altogether?

Answer: Al: 9, Ben: 27, C: 18. So the total is 54.

Jan 9: If 3 people can make 3 widgets in 3 hours, then how many widgets can 25 people make in 3 hours?

Jan 10: John gives 1/2 of his marbles to Marty, 1/4 of his collection to Norm, 1/6 to Otto and the remaining 30 marbles to Pete. How many marbles does Marty receive?

Answer: Pete gets 1/12 of the marbles, or equivalently 30 marbles. So Marty gets 90 marbles.

Jan 11: A rectangular kitchen floor is 11 ft by 15 ft. Each square tile has perimeter 24 inches. How many tiles are needed to cover the kitchen floor?

Answer: The square tile is 1/2 ft by 1/2 ft. So the kitchen floor needs 4*11*15 = 660 square tiles.

Jan 12:  During a track practice, A runs 3/5 of a mile, B runs 4/7 of a mile, C runs 1/2 a mile and D runs 2/3 of a mile. Who runs the farthest?