Word Problems

Anderson's Pumpkin Patch

Ms. Juncos is the lead teacher for the fourth grade team at Newton Elementary School. One week before Halloween, the team teachers assigned all of the fourth graders a new project. They thought that the students might enjoy making geometric jack-o-lanterns, so they planned a field trip to Anderson's Pumpkin Patch to hunt, clean, and create the jack-o-lanterns.

Looking at the class lists the night before the trip, Ms. Juncos decided to put the students into groups so no one would have to work in the pumpkin patch alone. Her first attempt using pairs did not work - one person was left out.

Next she tried groups of five, but that didn't work either.

Then she tried groups of three and again had someone left over. Still no luck when she tried groups of four. Each attempt ended with one person left out.

If each grade at Ms. Juncos's school has fewer than 80 students, how many pupils are on her team?

Bonus: What number of group members should Ms. Juncos use to make this work?

The number of students is 61.

The number is less than 80 and odd since division by 2 leaves a remainder.

Dividing the number by 3, 4, or 5 also leaves a remainder of one.

Therefore a number that is one less than the number of students has 2, 3, 4, and 5 as factors.

So 3 X 4 X 5 = 60,

which also has 2 as a factor.

So the number of students is one greater which is 61. There does not seem to be a factor that can reduce this number, so it is prime, which means its only factors are itself and one.

So the smallest group she can use in the pumpkin patch in which all group(s) are the same is a single group of 61.

Coding Conundrums

In a certain code, each of the 26 letters of the English alphabet is represented by a number (A=1, B=2, C=3,... Z=26).

A word is then encoded by multiplying the numbers that represent its letters.

For example, CAT is encoded by 3* 1* 20 = 60,

MATH is encoded by 13*1*20*8 = 2080.

Find a word that would be encoded as 7560 and explain how you found it.

Could there be other words? Explain why or why not.

Bonus: Find an encoded number that can be deciphered as exactly one word.

Explain how this works.

Many answers are possible. Here are two highlighted solutions.

BORN (2*15*18*14)
First I made a list of all the letters of the alphabet with each letterÕs corresponding number. Next, I eliminated all of the letters that were not a factor of 7560, leaving only the possible letters. I then tried to put together words using guess and check. I finally ended up with the word ÒTORN,Ó or (20*15*18*14).
This product was 75600. This number is ten times more than 7560 (the right answer). I then found a letter to replace T (20). I realized that the letter B (2) was perfect. I put the letters together and found the solution. BORN (2*15*18*14) the product of the word is 7560.

Rude or "18, 21, 4, 5."
We began to solve the equation by making a factor tree. We factored the number 7560 until we recieved four numbers that translated into letters, which we formed into a real word. This is an example of our factor tree: 7560 / \ 20 378 /\ / \ Numbers used: 4 5 18 21 Translation of numbers: D E R U
Letters unscrambled: R U D E

Big number

I am looking for the largest six-digit number that can be written following two simple rules.

* Rule 1: Each digit must be different.

* Rule 2: No digit may be prime.

986,410

The method was a chart of the numbers that I could and couldn't use:
9 - yes, not prime
8 - yes, not prime
7 - no, prime
6 - yes, not prime
5 - no, prime
4 - yes, prime
3 - no, prime
2 - no, prime
1 - yes, not prime
0 - yes, not prime

The chart left me with 986,410. Since there are 6 digits left in the highest combination possible, and there are no repeating numbers nor prime, this is the answer.

Ages

The Ages of Sigma Sigma, the latest and hottest singing group to ever hit the charts, is made up of three talented musicians. Slinky Sue is the lead vocalist, and she's backed up by Stunning Steve and Super Sean.

Strange as it may sound, they are also quite stuck on math. One day when a reporter for a fan magazine asked them for their ages, Slinky said, "Well, that's getting a bit too personal for my taste, but I will tell you this.

If you take our three ages and add them up two-by-two, you will get these sums: 50, 51, and 55. If you can unravel that, my age is neither the largest nor the smallest."

How old is Slinky Sue?

Bonus: Each of the three ages is a number that can be described by its own p-word (i.e. a word beginning with the letter p.) What are those three words?

Sue is 27.

Let the ages be x, y, z

x + y = 55 (1)
x + z = 51 (2)
y + z = 50 (3)

From (1)-(2)
y - z = 4 (4)

From (3)+(4)
2y = 54
y = 27

From (3)-(4)
2z = 46
z = 23

Substitute z into (2)
x + 23 = 51
x = 28

Ages are 23, 27, 28.

Since Sue's age is not the largest or smallest, she is 27 years old. Now, 23 is prime, 27 is powerful, and 28 is perfect. A prime number is one whose factor other than 1 is itself.

A perfect number is one which is the sum of all its factors (not including itself)

A number (n) is powerful if for any positive integer p, p|n and p^2|n