CPhill

Winter

H₃₂   means the element in the 3rd row, 2nd column of the matrix

This  =  23

pq - 4p - 2q =  2  

(p - 2)(q -4) =  pq -4p - 2q  + 8  ......so

pq - 4p - 2q = 2

pq - 4p - 2q + 8 - 8  = 2

(p - 2) (q - 4) - 8 = 2

(p - 2) (q - 4)  =  10

When

p = 1, q = -6

p = 3, q = 14

p = 4, q = 9

p = 7, q = 6

So....there are 4 p's  that satisfy the equation.....so.....the probability of choosing one of these  = 

4 / 10   =   2 / 5

Let A be the total amount of money

So   the amount that Dan has originally  =  (9/14)A

And the amount that   Paul has is (5/14)A

So.....when Dan gives Paul £39, they have an equal amount

So

(9/14)A - 39  =  (5/14)A + 39        simplify

[ 9/14 - 5/14] A = 78

[ 4/14]A = 78

[ 2/7 ]A = 78

A = 78 (7/2)  = £ 273

So ....Dan had   (273)(9/14)  = £ 175.5

Here's a possibility

Half of a person's age plus 8 years less than his age - 6 =  the person's age

Let the cost of a square inch of  paint = p

Let the cost of a cubic inch of concrete = c

The surface area  (in square inches) of a 1 inch cube = 6(1)^2 =  6

The volume of  1 inch cube (in cubic inches) =   1^3  =  1

The surface area of a two inch cube = 6(2)^2 = 24

The volume of a 2 inch cube = 2^3 = 8

So.....we have this system of equations

6p + 1c  = 1.30    ⇒  c = 1.30 - 6p      (1)

24p + 8c = 6.80      (2)

Sub (1) into (2)   and we have

24p + 8 ( 1.30  - 6p)  = 6.80       simplify

24p + 10.4 - 48p = 6.80

-24p  + 10. 4 = 6.80

-24p = -3.6   divide both sides by -24

p = .15

So  c =  1.30 - 6(.15)  =  1.30 - .90  =  .40

So......a 3 inch cube has  surface area of 6(3)^2 = 54 in^2

And the volume is (3)^3 = 27 in^3

So.....the cost of a 3 inch cube is

54(.15) +27(.40)  = $18.90

Angle  AOB   =  Angle COD

Angle ABO = Angle CDO

So...by AA congruency....triangle AOB  is similar to triangle COD

Since  AO / AC = 3/4 ......then CO must be (1/3)AO  ⇒  3CO = AO

So

OD/ CO = OB / AO

OD / CO = 6/ (3CO)

OD / 1  = 6/3

OD = 2

So  BD =  BO + OD   =  6 + 2   =   8

Very nice, EP......!!!!

This utility gives the result

https://www.easycalculation.com/statistics/quadratic-regression.php

0.155 X^2 -4.074 X +38.91

EP and AT look  to be correct.....!!!!

Thanks, EP.....!!!

Here's another way to see this :

Let A be the total amount received

Brianna's portion + 9.60  = Caleb's portion

Brianna portion = (3/15)A =  (1/5)A

Caleb's portion = (7/15)A

So

(1/5)A + 9.60 =  (7/15)A       

9.60  =  [ 7/15 -  3/15]A

9.60 = [ 4/15]A     multily both sides by 15/4

9.60 (15/4) = A = $36