Joe sent out 100 emails in 5 days. Each day he sent out 6 more than the day before. Seventy five percent of all emails sent on the first day were not sent to his director. How many emails did he send to his director on the first day?
Let D be the number of emails sent to the director on the first day.
He sent out D + 0.75 * D = 1.75 * D emails on the first day
The number of emails sent each day increased by 6 so the number of emails sent on the second day was 1.75 * D + 6
On the third day, he sent out 1.75 * D + 6 + 6 = 1.75 * D + 12 emails
On the fourth day, he sent out 1.75 * D + 12 + 6 = 1.75 * D + 18 emails
On the fifth day, he sent out 1.75 * D + 18 + 6 = 1.75 * D + 24 emails
The sum of the number of emails sent each day is 1.75 * D + 1.75 * D + 6 + 1.75 * D + 12 + 1.75 * D + 18 + 1.75 * D + 24 = 5 * 1.75 * D + 60 emails.
He sent out 100 emails in total so 5 * 1.75 * D + 60 = 100.
Therefore 5 * 1.75 * D = 100 - 60 = <<5*1.75=8.75>>8.75
Therefore D = 8.75 / (5 * 1.75) = <<8.75/(5*1.75)=1>>1. Answer: \boxed{1}.
Let's break down the problem step-by-step:
Step 1: Find the total number of emails sent on the first day.
Let's assume that Joe sent x emails on the first day. Since 75% of all emails sent on the first day were not sent to his director, 25% of the emails were sent to his director. Therefore, the number of emails sent to his director on the first day is 0.25x.
Step 2: Calculate the total number of emails sent on the second day.
On the second day, Joe sent 6 more emails than on the first day. So, the total number of emails sent on the second day is x + 6.
Step 3: Calculate the total number of emails sent on the third day.
On the third day, Joe sent 6 more emails than on the second day. So, the total number of emails sent on the third day is (x + 6) + 6.
Step 4: Calculate the total number of emails sent on the fourth day.
On the fourth day, Joe sent 6 more emails than on the third day. So, the total number of emails sent on the fourth day is ((x + 6) + 6) + 6.
Step 5: Calculate the total number of emails sent on the fifth day.
On the fifth day, Joe sent 6 more emails than on the fourth day. So, the total number of emails sent on the fifth day is (((x + 6) + 6) + 6) + 6.
Step 6: Calculate the total number of emails sent in 5 days.
Since Joe sent 100 emails in 5 days, we can write the following equation:
x + (x + 6) + ((x + 6) + 6) + (((x + 6) + 6) + 6) + ((((x + 6) + 6) + 6) + 6) = 100.
Simplifying this equation gives us:
5x + 60 = 100.
Step 7: Solve the equation to find the value of x.
Subtracting 60 from both sides of the equation gives us:
5x = 40.
Dividing both sides of the equation by 5 gives us:
x = 8.
Therefore, Joe sent 8 emails to his director on the first day.