Great Mountain Ride shop has seen an increase in the number of people that have purchased a new snowboard over the last three days. In order to keep up with demand, the manager of the shop has recorded the number of people that have purchased a snowboard over a five-day period. His data was given in the following table:

\begin{tabular}{|c|c|c|c|c|c|}
\hline
& Day 1 & Day 2 & Day 3 & Day 4 & Day 5 \\
\hline
\begin{tabular}{c}
\% increase of the \\
number of people \\
who bought snow \\
boards
\end{tabular}
& [tex]$10 \%$[/tex] & [tex]$15 \%$[/tex] & [tex]$5 \%$[/tex] & [tex]$8 \%$[/tex] & [tex]$12 \%$[/tex] \\
\hline
\end{tabular}

Given that the total number of people that purchased a snowboard on day 5 was 250 people, determine how many people purchased snowboards the day before the manager started to collect her data. Round your answer to the nearest person.

A. 155 people
B. 156 people
C. 160 people
D. 161 people



Answer :

In order to determine the number of people who purchased snowboards the day before the manager started to collect her data, we need to work backwards through the given percentage increases from Day 5 to Day 0.

Given:
- Day 5: 250 purchases
- Day 4 experienced a 12% increase from Day 4 to Day 5.
- Day 3 experienced an 8% increase from Day 3 to Day 4.
- Day 2 experienced a 5% increase from Day 2 to Day 3.
- Day 1 experienced a 15% increase from Day 1 to Day 2.
- Day 0 experienced a 10% increase from Day 0 to Day 1.

Using this information, we can calculate the number of purchases for each preceding day.

1. Calculating Day 4 purchases:
- Day 5 purchases are 250 people.
- The number of people who purchased on Day 4 can be found by dividing the Day 5 purchases by 1 + the percentage increase.

[tex]\( \text{Day 4 purchases} = \frac{250}{1 + \frac{12}{100}} \)[/tex]

[tex]\( \text{Day 4 purchases} = \frac{250}{1.12} \approx 223.214 \)[/tex]

2. Calculating Day 3 purchases:
- Day 4 purchases are approximately 223.214 people.
- The number of people who purchased on Day 3 can be found by dividing the Day 4 purchases by 1 + the percentage increase.

[tex]\( \text{Day 3 purchases} = \frac{223.214}{1 + \frac{8}{100}} \)[/tex]

[tex]\( \text{Day 3 purchases} = \frac{223.214}{1.08} \approx 206.680 \)[/tex]

3. Calculating Day 2 purchases:
- Day 3 purchases are approximately 206.680 people.
- The number of people who purchased on Day 2 can be found by dividing the Day 3 purchases by 1 + the percentage increase.

[tex]\( \text{Day 2 purchases} = \frac{206.680}{1 + \frac{5}{100}} \)[/tex]

[tex]\( \text{Day 2 purchases} = \frac{206.680}{1.05} \approx 196.838 \)[/tex]

4. Calculating Day 1 purchases:
- Day 2 purchases are approximately 196.838 people.
- The number of people who purchased on Day 1 can be found by dividing the Day 2 purchases by 1 + the percentage increase.

[tex]\( \text{Day 1 purchases} = \frac{196.838}{1 + \frac{15}{100}} \)[/tex]

[tex]\( \text{Day 1 purchases} = \frac{196.838}{1.15} \approx 171.163 \)[/tex]

5. Calculating Day 0 purchases:
- Day 1 purchases are approximately 171.163 people.
- The number of people who purchased on Day 0 can be found by dividing the Day 1 purchases by 1 + the percentage increase.

[tex]\( \text{Day 0 purchases} = \frac{171.163}{1 + \frac{10}{100}} \)[/tex]

[tex]\( \text{Day 0 purchases} = \frac{171.163}{1.10} \approx 155.603 \)[/tex]

6. Rounding to the nearest person:

The calculated number of purchases on Day 0 is approximately 155.603 people, which rounds to 156 people.

Therefore, the number of people who purchased snowboards the day before the manager started to collect her data is approximately [tex]\( \boxed{156} \)[/tex] people. Thus, the correct answer is:
b. 156 people.