Word Problem and Magic Square Problem

[geekily]

I'm not sure if this is the right forum to post this in because I don't quite see the algebra, but I'm in mathematics for elementary teachers (MTH 130), which is supposed to be on the same level as college algebra.

My questions:

1. Mr. Morgan has five daughters. They were all born the number of years apart as the youngest daughter is old. The oldest daughter is 16 years older than the youngest. What are the ages of Mr. Morgan's daughters?

The back of the book says the ages are 4, 8, 12, 16, and 20, which makes sense, but I don't understand how to arrive at this conclusion myself. Is there a formula I can use? I've checked the chapter and it doesn't cover this type of problem or anything similar.

The other problem is "complete the following magic square in which the sum of each row, each column, and each diagonal is the same. When completed, the magic square should contain each of the numbers 10 through 25 exactly once."

It then gives a 4x4 table with a few numbers filled in
Row 1: 25, blank, blank, blank
Row 2: blank, blank, 19, 17
Row 3: 18, 16, blank, blank
Row 4: blank, 23, blank, 10

Again, I can just get the answer from the back of the book, but I want to understand it for myself. I tried adding up the numbers in each row and column and then guessing and placing the smallest numbers (11, 12, 13) with the rows or columns with the higher ones, but this was taking me forever and getting me no where. Is there a better way?

Thank you so much for your help, I really appreciate it!

 

[galactus]

1. Mr. Morgan has five daughters. They were all born the number of years apart as the youngest daughter is old. The oldest daughter is 16 years older than the youngest. What are the ages of Mr. Morgan's daughters?

The gap betwen their ages is the same, call that x. It's also the youngest daughters age.

Their respective ages would be x, 2x, 3x, 4x, 5x.

The oldest is 16 years older than the youngest. There are 5 daughters.

5x-x=16. Solve for x. That's the youngest's age.

The back of the book says the ages are 4, 8, 12, 16, and 20, which makes sense, but I don't understand how to arrive at this conclusion myself. Is there a formula I can use? I've checked the chapter and it doesn't cover this type of problem or anything similar.

 

[Denis]

The other problem is "complete the following magic square in which the sum of each row, each column, and each diagonal is the same. When completed, the magic square should contain each of the numbers 10 through 25 exactly once."
It then gives a 4x4 table with a few numbers filled in
Row 1: 25, blank, blank, blank
Row 2: blank, blank, 19, 17
Row 3: 18, 16, blank, blank
Row 4: blank, 23, blank, 10

numbers 10 to 25 add up to 280; so common sum = 280 / 4 = 70

take 1st column; you have 25 + 18 = 43; so other 2 = 70 - 43 = 27;
what 2 numbers add up to 27? Well:
10 + 17 : out, since 10 already used
11 + 16 : out, since 16 already used
12 + 15 : possible
13 + 14 : possible

Get my drift ?

 

[geekily]

galactus and Denis, thank you both so much for your help! Denis, your clues were just the start I needed to help me figure out the problem. Once I knew what to do, I had it solved in minutes. Galactus, your response really helped me make sense of the other problem. Just one question to clarify, though, the 5x-x: The 5x is the 5 daughters, take away the youngest daughter?

Thanks again, guys! I really appreciate it. I like to get things out of the way on my day off and there's nothing more frustrating to me than not being able to get this figured out and finished.

 

[Denis]

Just one question to clarify, though, the 5x-x: The 5x is the 5 daughters, take away the youngest daughter?

Not quite: 5x is the age of the oldest, x is the age of the youngest;
so age of oldest less age of youngest = 16 : kapish ?

 

[geekily]

Thanks, Denis, that definitely makes sense now.

 

[soroban]

Hello, geekily!

I solved #2 quickly, because I'm familiar with the 4-by-4 Magic Square.

2) Complete the following magic square in which the sum
of each row, each column, and each diagonal is the same.
When completed, the magic square should contain
each of the numbers 10 through 25 exactly once.

. \(\displaystyle \begin{array}{ccccc}\hline \\| & 25 & | & & | & &| & & |\\ \hline \\
| & & | & & | & 19 & | & 17 & | \\ \hline \\
| & 18 & | & 16 & | & & | & & |\\ \hline \\
| & & | & 23 & | & & | & 10 & | \\ \hline
\end{array}\)

There are several ways to construct a 4-by-4 Magic Square with 16 consecutive integers.
Here's an easy way to remember how to construct one of the forms.

Starting at the upper-left, write the numbers 1 to 16 in the cells
. . but only in the cells on the main diagonal.

. \(\displaystyle \begin{array}{ccccc}\hline \\| & 1 & | & & | & &| & 4 & |\\ \hline \\
| & & | & 6 & | & 7 & | & & | \\ \hline \\
| & & | & 10 & | & 11 & | & & |\\ \hline \\
| & 13 & | & & | & & | & 16 & | \\ \hline
\end{array}\)

Starting at the lower-right, write the numbers 1 to 16 in the cells,
. . moving left and up, but only in the unoccupied cells.

. \(\displaystyle \begin{array}{ccccc}\hline \\| & 1 & | & 15 & | & 14 &| & 4 & |\\ \hline \\
| &12 & | & 6 & | & 7 & | & 9 & | \\ \hline \\
| & 8 & | & 10 & | & 11 & | & 5 & |\\ \hline \\
| & 13 & | & 3 & | & 2 & | & 16 & | \\ \hline
\end{array}\)

This is a Magic Square with a magic sum of 34.

This square starts with the smallest number in the upper-left
. . and ends with the largtest in the lower-right.
The square we are given is just the opposite.
. . Invert our magic square:

. \(\displaystyle \begin{array}{ccccc}\hline \\| & 16 & | & 2 & | & 3 &| & 13 & |\\ \hline \\
| &5 & | & 11 & | & 10 & | & 8 & | \\ \hline \\
| & 9 & | & 7 & | & 6 & | & 12 & |\\ \hline \\
| & 4 & | & 14 & | & 15 & | & 1 & | \\ \hline
\end{array}\)

Then add $\displaystyle 9$ to every number:

. \(\displaystyle \begin{array}{ccccc}\hline \\| & 25 & | & 11 & | & 12 &| & 22 & |\\ \hline \\
| & 17 & | & 20 & | & 19 & | & 17 & | \\ \hline \\
| & 18 & | & 16 & | & 15 & | & 21 & |\\ \hline \\
| & 13 & | & 23 & | & 24 & | & 10 & | \\ \hline
\end{array}\;\;\) . . . There!

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Otherwise, Denis had the best approach.