Problem of the Week

5/1/2017

Less than 2005°

The sum of the interior angles of a polygon is less than 2005°. What is the largest possible number of sides of the polygon?


4/24/2017

One or Ten? 

Given that they are made of the same material, which is heavier: a ball with a radius of 10 inches or 10 balls each with a radius of 1 inch?

 

Solution

The ball with a radius of 10 inches is heavier. Since the volume of a sphere is (4/3)πr3, one ball with a radius of 10 inches would have a volume of (4/3)π103 = (4000/3)π in3. Thus 10 balls of radius 1 inch would have an accumulated volume of only

10[(4/3)π13] = (40/3)π in3.


4/3/2017

Tamika > Carlos?

Tamika selects two different numbers at random from the set {8, 9, 10} and adds them. Carlos takes two different numbers at random from the set {3, 5, 6} and multiplies them. What is the probability that Tamika's result is greater than Carlos's result?

Solution

4/9. Tamika can get the numbers 8 + 9 = 17, 8 + 10 = 18, or 9 + 10 = 19. Carlos can get 3 × 5 = 15, 3 × 6 = 18, or 5 × 6 = 30. The possible ways to pair these numbers are (17, 15), (17, 18), (17, 30), (18, 15), (18, 18), (18, 30), (19, 15), (19, 18), and (19, 30). Four of these nine pairs show Tamika with a higher result, so the probability is 4/9.


3/27/2017

A New Mathematical Operation 

If a*b = ab – b, find (2*3)*4

Solution

621 

(2*3)*4 = (23 – 3)*4 = (8 – 3)*4 = 5*4 = 54 – 4 = 625 – 4 = 621.


3/20/2017

A Quarter Pounder

A quarter-pound hamburger contains approximately 80 calories per ounce of meat, an average french fry contains about 14 calories, a cola contains about 10 calories per ounce, and a bun contains 200 calories. Suppose you have a quarter-pound hamburger with a bun and six ounces of cola. How many french fries can you eat and still keep your meal below 800 calories?

Solution

15. We have 80h + 14f + 10c + 200 < 800. Using the information in the problem we can write 80(4) + 14f + 10(6) + 200 < 800. This implies that 14f < 220 and f < 16. Therefore, you can eat 15 french fries.


3/13/17

Perfect Squares and Cubes...NOT

How many numbers from 1 to 1 million, inclusive, are not perfect squares or perfect cubes?

Solution

998,910. There are 1000 perfect squares between 1 and 1 million; these are the squares of the first 1000 integers. Similarly, there are 100 perfect cubes—the cubes of the numbers from 1 to 100. Subtract the squares and the cubes from 1 million to get 998,900. However, every number that is a perfect sixth power has been subtracted twice (the largest of these is 106 = 1,000,000). Adding these back in gives 998,910.


3/6/17

999 Coins

Starting with a single pile of 999 coins, a person does the following in a series of steps: In step one, he splits the pile into two nonempty piles. Thereafter, at each step, he chooses a pile with 3 or more coins and splits this pile into two piles. What is the largest number of steps that is possible?

Solution

997. The number of steps is one less than the number of piles, and 998 is the largest number of piles, 997 with 1 coin and 1 with two coins.


2/27/17

House Prices

The average house price in Boomtown rose 30 percent each year for the last five years. If the average house price is currently $250,000, what was the average house price five years ago?

Solution

Approximately $67,332. If P was the average house price five years ago, then the current average price is 1.35P or 3.71293P. Thus, 250,000 = 3.71293P, so P=250,000/3.71293, which equals $67,332.27


2/6/17

Upright Integers 

An integer is defined as upright if the sum of its first two digits equals its third digit. For example, 145 is an upright integer since 1 + 4 = 5. How many positive three-digit integers are upright?

Solution

45. From the definition, the first and second digits of an upright integer automatically determine the third digit, which is the sum of the first two digits. Consider those upright integers beginning with 1: 101, 112, 123, 134, 145, 156, 167, 178, and 189; there is a total of 9 such numbers. (Note that the second digit may not be 9; otherwise, the last “digit” would be 1 + 9 = 10.) Beginning with 2, the upright integers are 202, 213, 224, 235, 246, 257, 268, and 279; there is a total of 8 such numbers. We may continue this pattern of analysis to show that the numbers of upright integers beginning with a digit of 3, 4, 5, 6, 7, 8, or 9 are 7, 6, 5, 4, 3, 2, and 1, respectively. Therefore, there is a total of 9+8+7+6+5+4+3+2+1 = 45 three-digit upright integers.

 

1/30/17

e-traders 

Three students open e-trade accounts and become day traders. Although they all work hard, they achieve the following steady rates of losing money: The first student loses $1000 in one hour, the second student loses $1000 in two hours, and the third student loses $1000 in three hours. Find the number of minutes it takes for the three students together to lose a total of $2000.

Solution

About 65 minutes (to the nearest minute). The losing rates are $1000/hr., $500/hr., and $333.33/hr. The combined losing rate is $1833.33/hr. Thus,


1833/1 = 2000/x  so x = 1 and 1/11 hours or 65 minutes



1/23/17

Investments

A man has $10,000 to invest. He invests $4,000 at 5 percent and $3,500 at 4 percent. To have a yearly income of $500 from the investment, at what rate must he invest the remainder of the money? 

Solution

At 6.4 percent or higher.

The man wants to earn $500 in interest each year. Investing $4000 at 5 percent yields yearly interest of 0.05 x $4000, or $200; whereas $3500 invested at 4 percent yields 0.04 x $3500, or $140; so the remaining $2500 needs to be invested at a rate that yields interest of $160 per year (to have a total interest of $500 each year). If r is the annual percentage rate for the $2500, then

 

so the man needs to invest the remaining $2500 at 6.4 percent or higher to guarantee an interest income of at least $500 a year.