1. Each of eighteen students was given a different integer from 1 through 18, inclusive. When asked to select a dance partner, the sum of the numbers for each of the nine couples was a perfect square. Who was dancing with whom?

2. In a 10-foot-wide alley, two ladders lean against opposite walls. One ladder reaches 30 feet up the wall, the other ladder reaches 20 feet high. The bases of the ladders rest against opposite walls. How high above the ground will the ladders cross?

3. A 100-pound watermellon is 95% water. It is dehydrated until it is 90% water. What is its weight after dehydration?

4. The flag of the United States has 13 stripes, alternating red and white, with a blue canton containing 50 stars. An American flag that is 78" by 136" with a canton that is 42" by 51" is wat percent red?

Any help would be much appreciated! Thanks!

2. #1

Given the relation $\displaystyle R=\{aRb$ $\displaystyle |$ $\displaystyle a+b=perfect$ $\displaystyle square\leq{25}\}$, the couples are $\displaystyle \{(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,18) ,$$\displaystyle (8,17),(9,16)\}$

3. Originally Posted by epetrik
...

2. In a 10-foot-wide alley, two ladders lean against opposite walls. One ladder reaches 30 feet up the wall, the other ladder reaches 20 feet high. The bases of the ladders rest against opposite walls. How high above the ground will the ladders cross?

...
1. Draw a sketch.

2. You get 2 proportions.
Left right triangle:

$\displaystyle \dfrac hx=\dfrac{20}{10}~\implies~x=\dfrac12 h$

Right right triangle:

$\displaystyle \dfrac h{10-x}=\dfrac{30}{10}$

3. Substitute x by the term from the first equation and solve for h.

$\displaystyle \dfrac h{10-\dfrac12 h}=\dfrac{30}{10}$

Spoiler:
For your confirmation only: h = 12

4. Originally Posted by epetrik
...

3. A 100-pound watermellon is 95% water. It is dehydrated until it is 90% water. What is its weight after dehydration?

...
1. The melon contains 5% dry material = 5 pounds.

2. Let x denote the amount of water which has been taken from the melon. Then you know:

$\displaystyle \dfrac{\overbrace{100-5-x}^{water\ left}}{\underbrace{100-x}_{actual\ weight}}=0.9$

3. Solve for x.

Spoiler: