Exercises

1. Consider the equation \(P=2x+2y\). If \(x=4\) and \(y=9\), find the value of \(P\).
2. The equation \(V=5n+25q\) describes the total value, \(V\), of \(n\) nickels and \(q\) quarters.
   1. If you have \(4\) nickels and \(7\) quarters in your hand, what is your total amount?
   2. If \(V=200\), then \(200=5n+25q\). Find one solution to the equation.
   3. Show that there is more than one solution to the equation \(200=5n+25q\).
3. Amusement park admission prices are listed as:
   • Adult: \($48\) 
   • Student: \($30\)
   • Child under 10: \($24\)
   1. Write an equation to represent the total cost, \(T\), for a family of \(a\) adults, \(s\) students, and \(c\) children.
   2. If a group spent \($150\) on admissions to the amusement park, how many adults, students, and children might have gone?
4. 1. \(100\) can be written as the sum of two numbers, one of which is a multiple of \(2\) and the other a multiple of \(3\). Find two numbers that fit this description.
   2. Explain how your solution from part a) gives you a solution to the equation \(100=2x+3y\).
5. Exactly \(120\) tickets were sold for a concert. The tickets cost \($12\) each for adults, \($10\) each for seniors, and \($6\) each for children. The number of adult tickets sold was equal to the number of child tickets sold. Given that the total revenue from the ticket sales was \($1100\), how many senior tickets were sold?
6. One soccer ball and one soccer jersey together cost \($100\). They each cost a whole dollar amount.
   1. Write an equation to represent this problem.
   2. Describe the possible costs for the soccer ball and soccer jersey.
7. A dodecahedron is a convex polygon with \(12\) pentagonal faces. How many edges and vertices does a dodecahedron have?
8. A game starts with the integer \(0\) and produces a sequence of integers. On each turn, you may perform one of the following four operations on the current number to produce the next number in the sequence:
   • Add \(5\)
   • Subtract \(5\)
   • Add \(17\)
   • Subtract \(17\)
   Therefore, the first term in the sequence is \(0\) and the second term will be \(5\), \(-5\), \(17\), or \(-17\), depending on what operation is performed on \(0\). Here is an example of the beginning of one possible sequence in this game:\[0 \xrightarrow{+17} 17 \xrightarrow{-5} 12 \rightarrow \ldots\] In this game, is it possible to produce a sequence that contains the number \(1\)? If so, explain how.