Question

Problem Solving in Santa's Workshop

Clifford works in the gift-wrapping department of Santa's
workshop. His job is to finish all the packages with a bow.
He is running low on bows and only has 100 bows left. He
begins working at 5:00 a.m. During the first half-hour, he
uses 12 bows. In the second half-hour, he uses 9 bows.
During the third half-hour, he uses 14 bows and in the
fourth half-hour, he uses 11 bows. Clifford uses 16 bows
during the fifth half-hour. If the pattern continues, during
which half-hour will Clifford run out of bows?

Answer 1

Eighth half-hour

Explanation:

Given pattern:

• 1st half hour = 12 bows used
• 2nd half hour = 9 bows used
• 3rd half hour = 14 bows used
• 4th half hour = 11 bows used
• 5th half hour = 16 bows used

Calculate the differences between the number of bows used:

`12\underbrace{}_(-3)9\underbrace{}_(+5)14\underbrace{}_(-3)11\underbrace{}_(+5)16`

Therefore, the pattern for the bows used is:

• -3, +5, -3, +5, etc...

If the pattern continues:

`12\underbrace{}_(-3)9\underbrace{}_(+5)14\underbrace{}_(-3)11\underbrace{}_(+5)16\underbrace{}_(-3)13\underbrace{}_(+5)18\underbrace{}_(-3)15\underbrace{}_(+5)20`

• 6th half hour = 16 - 3 = 13 bows used
• 7th half hour = 13 + 5 = 18 bows used
• 8th half hour = 18 - 3 = 15 bows used
• 9th half hour = 15 + 5 = 20 bows used

Create a table with the given and found data, and add a cumulative total column:

`\begin{array}c\vphantom{\frac12} \textsf{Half Hour} & \textsf{Bows used per half hour} & \sf Cumulative\; total\\\cline{1-3} \vphantom{\frac12} \sf 1st & 12 & 12\\\vphantom{\frac12} \sf 2nd &9 &21 \\\vphantom{\frac12} \sf 3rd & 14&35 \\\vphantom{\frac12} \sf 4th & 11& 46\\\vphantom{\frac12} \sf 5th & 16& 62\\\vphantom{\frac12} \sf 6th & 13& 75\\\vphantom{\frac12} \sf 7th &18 & 93\\\vphantom{\frac12} \sf 8th & 15& 108\\\vphantom{\frac12} \sf 9th & 20& 128\\\end{array}`

If Clifford has a total of 100 bows, he will run out of bows during the eighth half-hour.