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.
