CPhill

Last digit pattern

Term            Last Digit

3                    2

4                    4

5                    8 

6                    2 

7                    6

8                    2

9                    2 

10                  4

11                  8 

12                  2

13                  6

14                  2          

The  pattern has a repeating length of  6

And  35  =   6n +  5

So.... the 35th term has the same last digit as the 5th term  =   8

Very nice, hectictar......!!!

Let x  be the number of pens  and let  y be the number of pencils

And we have these constraints

x ≤ y

3x +  1y ≤  10

The third graph is correct  [ both lines will be solid because of the  " ≤ "  ]

See  here  :  https://www.desmos.com/calculator/pd7tbftmpj

The number of creases separating the stamps on  a sheet of  n  stamps  is given by

(2n - 1) ( n - 1) + (n -1)   =

(n - 1) (2n - 1 + 1) =

(n - 1) (2n)

So.....for   3 stamps, we have that  that the length of each perforated crease is

18/ [ (n - 1)(2n) ]  =  18 / [ (2)(6)]  =  18 / 12   =   1.5 inches each

And the number of  perforated creases separating the stamp on a 6 X 6  sheet   =

(6 - 1)(2 * 6)  =  5 * 12  =  60

So....the total length of the creases  =  60 * 1.5   =  90  inches

Let the cost of one soft taco = s

Let the cost of one burrito  = b

So  we have this system

3s  + 3b  = 11.25 

4s +  2b   = 10.00   (1)

Divide  the first equation through by 3   and we have that

s +  b  =  3.75         subtract  b from both sides

s   =  3.75 - b   (2)

Sub  (2) into (1)  and we have

4 (3.75 - b)  +  2b  =  10.00      simplify

15 - 4b  + 2b   = 10.00

-2b + 15  =10.00      subtract 15 from both sides

-2b  =  - 5.00         divide both sides by -2

b  = 2.50

And using (2)

s  =  3.75 - 2.50   =   1.50

So...  a soft taco  is  $1.50  and a burrito is $2.50

Welcome, William....!!!

Colored  ones  =  Total  - White ones  =   8  - 5  =  3

So

Colored Socks / Total     =    3 / 8   =  .375  =     37.5 %

x ( x + 2) (4x  - 1)

Taking the first two terms  we have   x ( x + 2)  =  x^2 + 2x

So now, we are multiplying

(x^2 + 2x)  ( 4x  - 1)       

Distribute the terms in the first set of parentheses over (4x -1)

x^2  (4x - 1)      =               4x^3      -  1x^2

+

2x(4x  - 1)      =                               +  8x^2    -  2x

__________________________________________

                                        4x^3     +   7x^2    -   2x

x^2   +  ax  +  5a  = 0

Let m, n  be the integer roots...this implies that....

(x - m) (x - n)  =  0

x^2  - (m + n)x  + mn = 0

So....equating terms.....we have this

-m - n  = a   ⇒  -5m -5n   =  5a

mn  =  5a

So this implies that

-5m - 5n  =  mn

-5n - mn  =  5m

-n ( 5 + m)  =  5m

n  =   -5m / ( m + 5)

Note that, as m → ± inf,   n →  -5

Also.....m   must be  a multiple of 5  since  -5m  is a multiple of 5

So  we can  test some values

m       n         -m - n  =   a

0        0                          0

20     -4                       -16

All positive values of m > 20  means that -5 < n < -4   so no other integer values are possible for in this direction

-10   -10                       20

-30   -6                         36

All negative values of m < -30  means that  -6 < n < -5 so no other integer values are possible for n at all

So the possible values for "a" are :

  -16, 0, 20  and 36

Number of problems normally answered per day   is

p * t

So......when he drinks coffee, he solves

(3p + 7)  (t - 4)  problems....and this is twice the number he normally answers...so...

(3p + 7) (t - 4)    =   2p*t

3pt  +  7t   -  12p  -  28  =  2pt

pt  +  7t  -  12p  -  28  =  0

We have an equation with two unknowns....so....there will possibly be more than one solution

WolframAlpha shows that the (possible) feasible solutions are

p = 1     t  = 5   =    5 problems on a regular day

p = 7     t  = 8  =    56 problems on a normal day

So as a check

(3*1 + 7)(5 - 4)  = (2)(1*5)  ??

(10)(1)  = 10  is true

(3*7 + 7) (8 - 4)  = (2)(7*8)  ???

(28) (4)  = 112

112 =  112    is also true

So.......the number of problems that he answers when he drinks coffee is either

(3)(1) + 7    =  10  problems per day

or

(3)(7) + 7  =   112 problems per day