**Problem of the Week**

__3/27/2017__

**A New Mathematical Operation**

If *a***b*
= *a*^{b}*
– b*, find (2*3)*4

__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
80*h* + 14*f* + 10*c* + 200 < 800.
Using the information in the problem we can write 80(4) + 14*f* + 10(6) + 200 < 800. This implies
that 14*f* < 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 10^{6} = 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 1.3^{5P }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.

__1/16/17__

**What Is the Price?**

Matthew
and Matilda want to buy a set of DVDs. Matthew has $47 less than the purchase
price, and Matilda has $2 less. If they pool their money, they still do not
have enough to buy the DVDs. If the set costs a whole number of dollars, what
is its price?

**Solution**

$48.

If *P* is the price of the DVDs, Matthew has *P* - 47 dollars and Matilda has *P* - 2 dollars.

We
know that (*P* - 47) + (*P* - 2) < *P*, since the amount of their combined savings is still less than
the price of the DVDs.

Solving for *P*, we have *P* < 49; but *P* > 47,
since Matthew had some money. Therefore, *P*
is a whole number such that 47 < *P*
< 49, so the price of the DVDs is $48.

__1/9/17__

**Tom and Bill**

Tom
is standing in a hole that is 4 feet deep. Bill asks him how much deeper he is
going to dig the hole. Tom replies that he will dig 4 feet 2 inches deeper and
that the top of his head will then be the same distance below ground level that
it is now above ground level. How tall is Tom?

**Solution**

6 feet 1
inch. The top of his head will go down 4 feet 2 inches with the additional
digging. Half that distance was above the hole before the additional digging,
so he is 4 feet + 2 feet 1 inch, or 6 feet 1 inch, tall.

__1/2/17__

**Wire Length**

In the diagram, the rectangular wire grid contains
15 identical squares. The length of the grid is 10. What is the length of wire
needed to construct the grid?

**Solution**

76. Since the length of the rectangular grid is 10, the side of each square in the grid is 10 ÷ 5 = 2. The height of the grid is therefore 6 (3 squares). There are four horizontal wires, each of length 10, and six vertical wires, each of length 6, for a total length of wire of 4 • 10 + 6 • 6 = 40 + 36 = 76.

__12/19/16__

**How Tall Was the Tree?**

Lightning hit a tree
one-fourth of the distance up the trunk from the ground and broke the tree so
that its top landed 60 feet from its base, thus creating a triangle. How tall
was the tree before it broke?

**Solution**

Approximately 84.85 feet tall.

Let *h* be the height, in feet, of the tree
before it fell. Then (*h*/4)^{2} + 60^{2} = (3*h*/4)^{2},
so *h*^{2}/16 + 3600 = 9*h*^{2}/16.

Solving for *h* reveals that h^{2}/2 =
3600, so *h* = , and the original height of the
tree equals , or approximately 84.85 feet.

__12/12/16__

**Facetious**

In how many possible
arrangements of the letters in the word FACETIOUS are the vowels in
alphabetical order?

**Solution**

3024 ways.

A total of 9! (9x8x7x6x5x4x3x2x1) ways exist to
arrange the letters in the word FACETIOUS (nine ways of picking the first
letter exist, eight ways to pick the second exist, and so on).

For each arrangement of the consonants in the word,
the vowels can be arranged in 5! (5x4x3x2x1) ways. In only one of them are the
vowels in alphabetical order.

Hence, 1/5! of the arrangements have the vowels in
alphabetical order, for a total of (9!/5!), or 3024.

__12/5/16__

**Townhouses**

A contractor is asked to build a new set of
townhouses in attached clusters of different sizes. He created plans for one-,
two-, and three-house clusters, as shown in the diagram below. The builder used
computer software to draw the line segments used to represent the houses. How
many line segments are needed to draw 4 houses? 10 houses? 47 houses? *n*
houses? Explain your strategy.

**Solution**

21; 51; 236; 5*n* + 1. One house takes 6
segments. Each additional house only requires 5 additional segments. Therefore,
the sequence for the number of segments is
6, 11, 16, 21, 26, 31, 36, . . . .
If n is the number of houses, then the *n*th term for this sequence is 5*n* + 1,
or 5 times the number of houses plus 1. The 5th house would require
5(5) + 1 = 26 segments; the 10th house would require
5(10) + 1 = 51 segments; and the 47th house would require
5(47) + 1 = 236 segments. Another method is to begin with 6
segments for the first house and notice that each additional house adds 5
segments to the total. Therefore, the total number of segments is
6 + 5(*n* – 1) = 5*n* + 1.

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