The average of 16 consecutive positive integers is 30.5. what is the average of the first 8 integers of this set?

consecutive integers are 1 apart

16 of them can be represented as

x, x+1, x+2, x+3, x+4, x+5, x+6, x+7, x+8, x+9, x+10, x+11, x+12, x+13, x+14, x+15

the average is

there might be some trick to get the average of the first 8 integers only

the average of the first 8 integers would be

how can we get from

first, let's match the denomenators (make them both 1 for ease)

multiply the first eqution by 8 to get

x+x+1+x+2+x+3+x+4+x+5+x+6+x+7=8?

multiply the 2nd equation by 16 to get

x+x+1+x+2+x+3+x+4+x+5+x+6+x+7+x+8+x+9+x+10 x+11+x+12+x+13+x+14+x+15=488

notice that we can try and force the 1st equation into the 2nd equation by adding some numbers

we can already do this:

x+x+1+x+2+x+3+x+4+x+5+x+6+x+7+x+8+x+9+x+10 x+11+x+12+x+13+x+14+x+15=488

[x+x+1+x+2+x+3+x+4+x+5+x+6+x+7]+x+8+x+9+x+10 x+11+x+12+x+13+x+14+x+15=488

[8?]+x+8+x+9+x+10 x+11+x+12+x+13+x+14+x+15=488

we can try to force another 8? into it

[8?]+x+7+1+x+7+2+x+7+3+x+7+4+x+7+5+x+7+6+x+7+7+x+7+8=488

[8?]+(x)+7+(1+x)+7+(2+x)+7+(3+x)+7+(4+x)+7+(5+x)+7+(6+x)+7+(7+x)+7+8=488

[8?]+(x)+7+(x+1)+7+(x+2)+7+(x+3)+7+(x+4)+7+(x+5)+7+(x+6)+7+(x+7)+7+8=488

group

[8?]+[x+x+1+x+2+x+3+x+4+x+5+x+6+x+7]+7+7+7+7+7+7+7+7+8=488

[8?]+[8?]+8(7)+8=488

16?+64=488

minus 64 from both sides

16?=424

divide both sides by 16

?=26.5

the average of the first 8 integers is 26.5

I'm not sure if there is a simpler way to do it or not

maximum area = xy = (16)(32) = 512 ft2

to maximize the area of the garden, the side of the fence parallel to the side of the house should be 32 ft long, and the other two sides should both have length 16 ft.'

source

tube or goog would be a great resource to with that question