geno3141

If you go, the buffet will cost you $6.00.

If you want your total cost to be $11.00, the bus must only cost you $5.00.  ($11.00 - $6.00  =  $5.00)

How many persons must go so that your portion is only $5.00?

So:  How many persons times $5.00 each will have a total cost of $95.00?

$95.00 ÷ $5.00  =  the answer

First, you will need to factor:             10x² + 11x - 6

Because the last term (-6) is a negative, the parentheses will include one plus sign and one minus sign.

Find the factors of  10; they will be either  10  and  1    or    5  and  2.

Find the factors of 6; they will be either  6  and  1    or    3  and  2.

So, the possibilities are:

    (10x + 6)(1x - 1)              (10x - 6)(1x + 1)              (10x + 1)(1x - 6)              (10x - 1)(1x + 6)

    (5x + 6)(2x - 1)                (5x - 6)(2x + 1)                (5x + 3)(2x - 2)               (5x - 3)(2x + 2)

    (5x + 2)(2x - 3)                (5x - 2)(2x + 3)

Check each one!  The middle term must become +11x.

Once you have determined which one is correct, set each factor equal to zero to find the two answers.

5/5  =  1

1 ÷ 1/2   =   1 x 2/1   =   2

To divide by a common fraction, multiply by the reciprocal.

An appropriate equation is:

     rate x time (of the first person) +  rate x time (of the second person)  =   amount completed

Alex's rate is 1 wall painted in 6 hours, which is:  1 wall / 6 hrs

David's rate is 1 wall painted in 4 hours, which is:  1 wall / 4 hrs

Amount completed is 1 wall.

The time is the same for both, so let's use the variable  t.

    (1/6)·(t)  +  (1/4)·(t)   =   1

         t / 6   +   t / 4        =   1

Simplify the equation by multiplying both sides by a common denominator; I'll use 24:

     (24) · (1/6)·(t)  +  (24) · (1/4)·(t)    =  (24) · 1

            4t   +   6t   =  24

                      10t   =   24

                          t   =   2.4 hours

In geometry and trigonometry, 'r' usually represents 'radius'.

In statistics, it can represent rank, rank correlation, random number.

CPill's answer is far more complete than mine, with exactly the correct procedure. But, in the problem, he transposed 2014 into 2104, so check the last two numbers. 

My way of looking at this is:

every number that ends in a 2 multiplied by a number ending in a 5 adds one more zero to the number of zeroes;

also multiplying by a number that ends in a zero adds another zero.

This means that 10! has two zeroes because it has 2 x 5 x 10; one zero for the 2 x 5 and another zero for the 10.

But this means: 1 through 10 adds 2 zeroes; so does 11 through 20; 21 through 30; up through 91 through 100 

for a total of 20 zeros

But I forgot about 25, since it contains 2 5's as factors, it will add 2 zeros not just one (there are a lot of extra 2's in the other numbers); so here is an extra zero.

Then, there is also, 50, 75, and 100; here are three more zeros.

OK, that's how you can get 24 zeros in 100!.

Can you use this to finish the analysis for 2014! ?

(2/3) to the second power is  (2/3)²  =  (2/3)·(2/3)  =  4/9

2 1/3 as a common fraction is  (2 x 3 + 1) / 3  =  7/3

3/4 x (7/3)  =  (3 x 7) / (4 x 3)  =  21 / 12  =  -7/4     [reduced]

Putting all these together:  4/9 - 7/4   =   4/9 + 7/4

The common denominator is 36            =  16/36 - 63/36   =  -47/36

2x - 2/x² - 2x + 1   =   x + 2/x² + 4x + 4

Simplifying the left side:  2x - 2x  = 0.

Simplifying the right side:  x + 4x  =  5x

- 2/x² + 1  =  2/x² + 5x + 4

Further simplifying:

             0  =  4/x² + 5x + 3

Multiplying both sides by x²:     0  =  4 + 5x³ + 3x²

This will have a solution, you can approximate it by graphing:  y  =  5x³ + 3x² + 4.

[And, like the quadratic equation, there is a cubic equation, but it's not simple.]