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Get the total distance sum     then divide by the total time sum 

                                                          Rome time   3487.5 m / 465 m/hr = ........ hr 

total distance / total  time  = avg

3(8-4x)<6(x-5)   expand both sides

24-12x<6x-30

54 < 18x

x> 3                 the line which is greater than 3   BUT DOES NOT INCLUDE 3   is the answer

1/11  *  7/9 =.......

find the slope between the tw given points     (y1-y2) / (x1-x2)   =   6/(-12)    = m

  parallel is the same slope

    y = mx +b

    y = -1/2 x + b     sub in the  point to be on the new line

    9 = -1/2(7) + b     b = 12.5

y = - 1/2x + 12.5 

it turned out to be 5/9 but thank you for helping!!

See if you can discern how I found all of these angles leading up to x !

thanks :D

.55555555.......

1/3 + 2/9 =   5/9

What is 0.3... + 0.2...?     0.5

  !

yay! thanku

Yep...... You are correct ....(at least I found the same answer)    1 day later would be thursday    day two of her life

                 7days later would be wdnesday again  day 8 of her life

 994 days later would be a wednesday   day 995 of her life

995  days later = thursday       day  996 of her life

996= friday

997 = sat

998 = sunday  

999 = MONDAY       day 1000 of her life  

THANX , Guest !

True....leap years are irrev !    D'Oh!  

every 4 th digit of the result is a '9' ...all of the other digits are zeroes

ten 9's  =  90

This is a dome shaped parabola (due to the negative coefficient of t^2)

   the ma will ocur at a  't' value  = - b/2a    where b = 8   and a = -1           then t = 4

     put t= 4 into the equation       -(4)^2 + 8(4) -4 = max value = -16 + 32 -4  = 12

The equation of the circle is x2+y2=4x2+y2=4. Thus, the centre of the circle is (0,0)(0,0) and radius is 2.

Since the line y=x+n⇒x−y+n=0y=x+n⇒x−y+n=0 is a tangent, the perpendicular distance from the point (0,0) to the tangent will be 2.

Thus, we get:
∣∣x−y+n2√∣∣=2|x−y+n2|=2

⇒∣∣0−0+n2√∣∣=2⇒|0−0+n2|=2

⇒n=±22–√⇒n=±22

Thus the two tangent equations are y=x±22–√y=x±22.

https://artofproblemsolving.com/wiki/index.php/2016_AIME_II_Problems/Problem_3

Leap year days are irrelevant here. 

The question asks on what day of the week the celebration will occur after 1000 elapsed days, where first day is the day of her birth. If she was born on a Wednesday then the celebration will occur on a Monday, no matter what year or date she was born.  

let me try to read

Solution

The expression we want to find is .

Factor the given equations as  and , respectively. Dividing the latter by the former equation yields . Adding 3 to both sides and simplifying yields . Solving for  and substituting this expression into the first equation yields . Solving for , we find that , so . Substituting this into the second equation and solving for  yields . So, the expression to evaluate is equal to .        890

This is a bowl shaped parabola   and factors to 

(n-3)(n-6)        the zeroes are at   n = 3 and 6   it is negative in between....

                           the largest integer would be 5 where it is still negative

To ensure x has no real solutions. use the discriminant

b2 - 4ac < 0

n2 - 60 < 0

n2 < 60

Since n is a integer,

we now ask ourselves how many perfect squares that are below 60?

1, 4, 9, 16, 25, 36, 49.

There are 7 such perfect squares.

n can also be negative, so the answer should be 7 * 2 = 14

Find the slope, m ,  of the given line.....do you know how to do that?

PERPINDICULAR slope will be  - 1/m

then     use the line form    y = -1/m x + b      sub in the x,y point given to calulate 'b'

thank you :))

Neither the decimal ".727272..." nor the decimal ".722222..." can be turned into a fraction.
Also to respond to CPhill's comment I'm pretty sure that he was saying that he needed the answer based on ".72 repeating" not what you assumed he meant. Also the answer you gave can be simplified to

18/25

I guess this is the easiest way:

.727272727272/.272727272727 = 2.6666666666666667     which is   2 2/3

Here is the long way

let x = .72727272....

  100x = 72.72727272....

100x - x = 99x  = 72       x = 72/99     this is the numerator.....

           now do similar for the denominator....then do the division to get an exact answer.....

denominator = 27/99

72/99  /  27/99   =   72/27 = 24/9= 8/3 = 2 2/3