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as shown where?

why pin in sticky topics?

1/a  - 1/b  = 1/12

(a)   Note that....if this is true, then   0 < a < 12     where a is an integer

[ b - a ] / ab  =  1/12

12 [ b - a]  =  ab

12b - 12a = ab

12b - ab  = 12a

b  [ 12 - a ]  =  12a

b =    12 a / [ 12 - a  ]

a = 3     b  =  4

a = 4     b =   6

a = 6     b =  12

a = 8     b = 24

a = 9     b =  36

a  = 10  b =  60

a = 11   b =  132

We can stop here because of  the provision of  (a)  

Find the total number of digits then divide by half.

do some experimenting with numbers, come on.

whats 1+1?       

To solve this, we must count on our fingers.

Hold up a closed hand.

Say "one..."         and raise your index (first) finger.  

Say "plus one..."  and raise your middle (second) finger.

Examine your hand which has the first two fingers raised.

In a voice full of joyous discovery, shout "Bunny!"    

_.

BD  is a diagonal  of the base....so.....since the base is square.....one side of the base = 6/√2

The height of the equilatreal triangle  =  height of the pyramid .....so....

tan 60°  =  height of pyramid / 3

3 tan 60°  =  height of pyramid

3√3  = height of pyramid

So....the volume of the pyramid  = 

(1/3)  base area *  height  =

(1/3) (6/√2)^2 * 3√3  =

18√3   units^3

I think the easiest way would be to count the cubes that do have paint on them and then subtract.

Each of the 8 vertices has paint on three sides of one cube*           =    8 cubes

Each of the 6 faces has paint on one side of eight "center" cubes   =  48 cubes 

Total cubes with paint are 56 subtracted from 125 cubes in the block  =  69 cubes with no paint  

The fraction of cubes with no paint is 69 / 125

* Although each corner cube has paint on three sides, it's still just one cube. 

_.

See the graph here :  https://www.desmos.com/calculator/s42iuonkkh

The overlap of these areas  what we want.....  this is  the area of a square with a side of 4  less the area of a square with a side of 3

So....the area of interest  is 4^2  - 3^2   =  16 - 9   =    7 units^2

Thanks

Yes maybe I could do it in spreadsheet if I wanted to but I was really trying to extract the logic.

When you answer using a program it is nice if you say that so everyone knows where your answers comes from. 

I like people doing this. I am not trying to discourage you at all. 

a=1;b=1;p=0; cycle:d=a*10+b;if(d%a==0 and d%b==0, goto loop, goto next); loop:printd," ",;p=p+1; next:b++;if(b<10, goto cycle, 0);b=1;a++;if(a<10, goto cycle, 0);print"Total = ",p

11  12  15  22  24  33  36  44  48  55  66  77  88  99  Total =  14 such numbers

Hi Melody: There is this formula that is programmed into my computer that gives you instantaneous  answer when  you enter the total under "S" and the number of dice under "N" :

S=6; N=4; sumfor(k, 0, ((S - N)/6), ((-1)^k * (N nCr k) * (S - 6*k - 1) nCr (N - 1))

 /* This formula calculates the number of permutations for "N" dice */
 /* given a total in "S" */

P.S. If you don't use a programming language, I believe you could programm it as a "Macro" into a Spreadsheet.

I'll give you an outline:

Use similar triangles to show that TY is congruent to TW

Now you have three congruent sides and one pair is parallel.

Now draw in the diagonal YW 

You now have enough info to show that triangles  TYW  and XYW are congruent.

Now you can prove the rest.

Hi guest.

How did you work all these totals out?

Thanks Heureka,

Am I right in thinking you just used a formula.

I mean can you see the logic behind the forumula - I can't.

Which quadratic equation fits the data in the table?

y=−x2−4x+1

y=x2−4x+1

y = x² + 4x + 1

y=x2−4x−1

Hello Guest!

The second equation fits your table.

\(y=x^2−4x+1\)

x |  0    1    2    3    4    5    6

y |  1   -2   -3   -2   1    6   13

  !

The above is true for the first 100 NOT 1000 numbers. This formula applies to any number of Fibonacci Numbers: Floor(N/3). So, the first 100 F. Numbers have:Floor(100/3) =33. The first 1,000 F. Numbers should have: Floor(1,000 / 3) =333 Even Numbers.

index=1000; i=0; n=1;i++;p=1;i++;printn,",",p,", ", ;loop1:n=n+p;i++;printn,", ", ;if(i>=index, goto loop3,0);loop2:p=p+n; i++;printp,", ", ;if(i>=index, goto loop3,0); if(i<=index and index%2==0, goto loop1, goto loop1); loop3:print"= Fibonacci Nos =",i

only 33 integers found: 0 | 2 | 8 | 34 | 144 | 610 | 2584 | 10946 | 46368 | 196418 | 832040 | 3524578 | 14930352 | 63245986 | 267914296 | 1134903170 | 4807526976 | 20365011074 | 86267571272 | 365435296162 | 1548008755920 | 6557470319842 | 27777890035288 | 117669030460994 | 498454011879264 | 2111485077978050 | 8944394323791464 | 37889062373143906 | 160500643816367088 | 679891637638612258 | 2880067194370816120 | 12200160415121876738 | 51680708854858323072
(assuming only positive integers)

The answer is adventurous.

The point is (18,11), and the sum of the coordinates is 18 + 11 = 29.

The domain of g(x) is [-32,1].

a=125; b=100;c=1/a - 1/b; if(c==1/12, goto4, goto5);printa, b; a--;if(a>1, goto2, 0);a=125;b--;if(b>1, goto2, discard=0;

a             b
10          60
9            36
8            24
6            12
4             6
3             4

Total of 6 = 10 permutations

Total of 9 = 56 permutations

Total of 12=125 permuations

Total of 15=140 permutations

Total of 18 = 80 permutations

Total of 21 = 20 permutations

Total of 24 = 1   permuation

GRAND TOTAL =432 / 6^4 = 1 / 3

We have the equation :

V_t  =  V₀k^t   where V₀  = 3000 = the original volume  and t is the number of hours

So....when t  = 2 we have

2881.2  =  3000k²         divide both sides by   3000

2881.2 / 3000  = k²        take the square root of both sides

0.98  =  k

This means  that every hour after V₀, the tanks loses   1 - .98   = .02   = 2% of its water from the previous hour

So

k = 0.98    and  x  = 2%

(c)  the radius of the circle will meet the tangent at  a right angle

Thus...points MNP  from a right triangle

We know the distance fom M to P  = the radius = 7

The distance from M to N  is the hypotenuse  =    sqrt  [ (25 - 10)^2 + (32 - 12)^2 ]  =

sqrt [ 15^2  + 20^2 ]   =

sqrt [ 225 + 400 ]  =

sqrt [ 625]  =

25 

So....using the Pythagorean Theorem  MP^2 + NP^2  = MN^2...so....

NP  =  sqrt [ MN^2 - MP^2 ]     =  sqrt [ 625 - 49 ]   =  sqrt [ 576]   =   24