Sophia has been making two types of necklaces: small necklaces that contain four beads each and large necklaces that contain seven beads each.
After creating a certain number of small and large necklaces, a total of \(99\) beads have been used. Determine all possibilities for how many of each type of necklace Sophia has made.
Let \(S\) represent the number of small necklaces and \(L\) represent the number of large necklaces she has made. Since \(S\) and \(L\) represent numbers of necklaces, both must be integers greater than or equal to \(0.\) Since the small necklaces use \(4\) beads each, \(S\) necklaces would use \(4\times S\) or \(4S\) beads in total. Since the large necklaces use \(7\) beads each, \(L\) necklaces would use \(7\times L\) or \(7L\) beads in total. Since a total of \(99\) beads have been used, \(4S+7L=99.\)
We can also determine a maximum value for \(L\). Since each large necklace uses \(7\) beads, \(99\div 7 \approx 14.1\), and \(L\) must be an integer, we know that \(L\) must be less than or equal to \(14.\) Thus, \(L\) is an integer greater than or equal to \(0\) and less than or equal to \(14\). We could at this point check all of the possible integer values for \(L\) from \(0\) to \(14.\) However, we can narrow down the possibilities even more.
In the equation, \(4S+7L=99\), \(4S\) will always be an even integer since \(4\) times any integer is always even. We have the even integer \(4S\) plus \(7L\) is equal to the odd integer \(99\). This means that \(7L\) must be an odd integer. (The sum of an even integer and an even integer is an even integer, not an odd integer.) For \(7L\) to be an odd integer, \(L\) must be odd. (If \(L\) is even, \(7L\) would be even.) This observation reduces the possible values for \(L\) to the odd positive integers between \(0\) and \(14\), namely \(1\), \(3\), \(5\), \(7\), \(9\), \(11\), \(13\). For each possible value of \(L\), we now determine \(7L\), the total number of large beads used, \(4S = 99-7L\), the total number of small beads used, and finally \(S = (99-7L)\div 4\), the number of small necklaces for that value of \(L.\) If \(S\) is a non-negative integer, then we have found a valid possibility.
| \(\boldsymbol{L}\) | \(\boldsymbol{7L}\) | \(\boldsymbol{4S = 99-7L}\) | \(\boldsymbol{S = (99-7L)\div 4}\) | Valid Possibility? |
|---|---|---|---|---|
| \(1\) | \(7\times 1=7\) | \(99-7=92\) | \(92\div 4=23\) | Yes, \(S\) is an integer |
| \(3\) | \(7\times 3=21\) | \(99-21=78\) | \(78\div 4=19.5\) | No, \(S\) is not an integer |
| \(5\) | \(7\times 5=35\) | \(99-35=64\) | \(64\div 4=16\) | Yes, \(S\) is an integer |
| \(7\) | \(7\times 7=49\) | \(99-49=50\) | \(50\div 4=12.5\) | No, \(S\) is not an integer |
| \(9\) | \(7\times 9=63\) | \(99-63=36\) | \(36\div 4=9\) | Yes, \(S\) is an integer |
| \(11\) | \(7\times 11=77\) | \(99-77=22\) | \(22\div 4=5.5\) | No, \(S\) is not an integer |
| \(13\) | \(7\times 13=91\) | \(99-91=8\) | \(8\div 4=2\) | Yes, \(S\) is an integer |
Therefore, Sophia has either made \(1\) large necklace and \(23\) small necklaces, or \(5\) large necklaces and \(16\) small necklaces, or \(9\) large necklaces and \(9\) small necklaces, or \(13\) large necklaces and \(2\) small necklaces.