In 7 days, Mario cooked 98 pounds of spaghetti. Each day after the rst, he cooked 2 more pounds than he cooked than the day before. What is the difference between the average (arithmetic mean) number of pounds of spaghetti he cooked per day and the median number of pounds he cooked during 7 days ?
98/7 = 14 lbs/day average
7/2 (2a+6*2) = 98
so, a = 8
The median is the 4th day, so he cooked 8+3*2 = 14 lbs on that day.
So, the mean and median are the same. In fact, that is true for all arithmetic sequences.
Sn = n/2 (a1 + an) the mean is thus Sn/n = (a1+an)/2
The median is the n/2 term, which is a + (n-1)/2 * d = (2a + (n-1)d)/2 = (a1+an)/2
To find the average number of pounds of spaghetti cooked per day, we need to divide the total pounds of spaghetti cooked by the number of days.
Total pounds of spaghetti cooked = 98 pounds
Number of days = 7 days
Average number of pounds of spaghetti cooked per day = Total pounds of spaghetti cooked / Number of days = 98 pounds / 7 days
Average number of pounds of spaghetti cooked per day = 14 pounds
To find the median number of pounds of spaghetti cooked during 7 days, we need to arrange the amounts in ascending order and find the middle value.
First day: x pounds (unknown)
Second day: x + 2 pounds
Third day: x + 4 pounds
Fourth day: x + 6 pounds
Fifth day: x + 8 pounds
Sixth day: x + 10 pounds
Seventh day: x + 12 pounds
To find the median, we need to arrange these values in ascending order: x, x + 2, x + 4, x + 6, x + 8, x + 10, x + 12.
The middle value is the fourth value when arranged in ascending order. Since there are an odd number of values (7), the median will be the fourth value.
Median number of pounds of spaghetti cooked = x + 6 pounds
Now we need to find the difference between the average and the median.
Difference = Average number of pounds of spaghetti cooked per day - Median number of pounds of spaghetti cooked
Difference = 14 pounds - (x + 6 pounds)
Unfortunately, without any additional information or constraints on the value of x, we cannot determine the exact difference between the average and the median.
To find the average and median, we first need to calculate how many pounds of spaghetti Mario cooked each day.
Let's denote the number of pounds cooked on the first day as "x". Since each day after the first, Mario cooked 2 more pounds than the day before, we can represent the number of pounds cooked on each subsequent day as x+2, x+4, x+6, x+8, x+10, and x+12.
Now we can set up an equation to find the total number of pounds Mario cooked during the 7 days:
x + (x + 2) + (x + 4) + (x + 6) + (x + 8) + (x + 10) + (x + 12) = 98
Simplifying the equation, we have:
7x + 42 = 98
7x = 98 - 42
7x = 56
x = 56 / 7
x = 8
So, Mario cooked 8 pounds of spaghetti on the first day.
Now, we can calculate the number of pounds cooked on each subsequent day:
First day: 8 pounds
Second day: 8 + 2 = 10 pounds
Third day: 10 + 2 = 12 pounds
Fourth day: 12 + 2 = 14 pounds
Fifth day: 14 + 2 = 16 pounds
Sixth day: 16 + 2 = 18 pounds
Seventh day: 18 + 2 = 20 pounds
Now we can find the average number of pounds cooked per day:
Average = (8 + 10 + 12 + 14 + 16 + 18 + 20) / 7
Average = 98 / 7
Average = 14 pounds
To find the median number of pounds cooked, we need to arrange the numbers in ascending order:
8, 10, 12, 14, 16, 18, 20
Since there are an odd number of values, the median is the middle value, which is 14 pounds.
Finally, to find the difference between the average and the median:
Difference = Average - Median
Difference = 14 - 14
Difference = 0 pounds
Therefore, the difference between the average and median number of pounds of spaghetti Mario cooked during the 7 days is 0 pounds.