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P==Pens,       c==pencils

P / c == 3/5.....................(1)

[P + 3] / [c  - 9]==1.........(2), solve for P, c

P ==18 - original number of Pens

c ==30 - original number of pencils.

18 + 3 ==21 Pens

30 - 9 ==21 pencils

That's actually a really cool formula!

[A  +  n] / [B  -  n] ==3,     

[A  - n] / [B  + n]==2, solve for A, B 

A==17n   and   B==7n

A / B ==17n / 7n ==17 / 7 - original ratio of their money.

n == 1

Best way of doing this sort of thing is to use Newton's method.

The formula  for finding the square root of the number A is,

\(\displaystyle x_{n+1}=\frac{1}{2}\left(x_{n} + \frac{A}{x_{n}}\right)\)

where \(\displaystyle x_{n}\)

is your current approximation

and \(\displaystyle x_{n+1}\)

is your new approximation.

So, if your first approximation for the square root of 12 is 3.5, your next approximation will be

(3.5 + 12/3.5)/2 = 3.46 (2dp).

Next would be

(3.46 + 12/3.46)/2 = 3.4641 (4dp).

And the next

(3.4641 + 12/3.4641)/2 = 3.4641016 (7dp).

The number of correct decimal places roughly doubles at each step.

(So if 3.5 is correct to 1dp, then 3.46 is probably correct to 2 dp, and so on.)

Easiest is to put each current approximation into memory and then recall it when needed, (i.e you wouldn't enter 3.46 at the second step, you'd put the result of the calculation (3.5 + 12/3.5)/2 into memory and then use that).

Square root of 12

3^2=9   4^2=16  so it is about half way between between  3 and 4.  If I estimated at this point I'd say 3.5

3.5^2=12.25  that is too big but very close

3.45^2=11.9025  to small

So to one decimal place it is 3.5

You can do the others yourself.

Let the distance between matthew and the school be \(x\)

Then, the distance between Berthe and the school is 7 km more than that of matthew; so: \(x+7\) is the distance between Berthe and the school.

"They live a total of 31 km from school"
So, the sum of the distances we wrote above is 31:

\(x+(x+7)=31 \\ \iff 2x+7=31\\ \iff 2x=24\\ x=12\)

So, mathew lives 12 km far, and Berthe lives (12+7)=19 km far.

I hope this helps!

I got a different answer, Melody.

If a number is both a square and a 5th power, it must also be a perfect 10th power.

Note that \(1024^2 = 4^{10}\), so there are 4 numbers that work: \(1^{10}\), \(2^{10}\), \(3^{10}\), and \(4^{10}\)

There are \({4 \choose 2} = 6\) successes and \({1024 \choose 2} = 523776\) total outcomes.

So, the probability is \({6 \over 523776} = \color{brown}\boxed{1 \over 87296}\)

1024^2 = 1048576

1048576^(1/5) = 16

4/1024  * 3/1023 =  12/1047552 = 3/261888     

EDIT:  It is only my last disision that is incorrect    12 /1047552 =  1/87296  NOW our answers agree

Thanks Builderboi.    Oh, and your answer is a lot more elegent than mine is    

This is a post I was referring to.  It is rude to not respond and then just repost.

b = binder

n = notebooks

p = pencils

b + n + p = 14

p = 2b

n = b + 2

Substitute => b + (b + 2) + (2b) = 14

4b + 2 = 14 => b = 3, n = 5, p = 6

binders = 3$

pack of notebooks = 5$

box of pencils = 6$

10!/8! * (8)! = 10!

10! = 3,628,800

You did it again Melody. Called someone out for something not true, and provided a non-workable solution. All solutions on google require for you to pay. 

f(x) = (x-9)/(5x+4)    Find an equation for the tangent line of f at x=8

\(f(8)= \frac{-1}{44}\)

\( f(x) = \frac{(x-9)}{(5x+4)}\\ f'(x)=\frac{(5x+4)(1)-5(x-9)}{(5x+4)^2} \\ f'(x)=\frac{5x+4-5x+45}{(5x+4)^2} \\ f'(x)=\frac{49}{(5x+4)^2} \\ f'(8)=\frac{49}{(44)^2} \\ f'(x)=\frac{49}{1936}\)

Equ of tangent

\(\frac{49}{1936}=\frac{y+\frac{1}{44}}{x-8}\\ 49(x-8)=1936(y+\frac{1}{44})\\ 49x-392=1936y+44\\ 49x-1936y=436\)

I suggest you check this carefully

Thanks for responding and you are very welcome.

Proyaop,

Please do not answer over other people's teaching answers. 

You are just making their answer worthless and facilitating cheating.

I think Guest (the one who asked the question) means ___ * ___ = 56. I believe he means that the same number times itself is 56.

If that is so, then we can set the blank as a variable "x". x * x = 56. The star means "times/multiply". \(x^2 = 56\). Square root both sides, you get "blank" \( = \pm 2\sqrt{14}\).

Just to clarify @mrinank, the angle bisector theorem is thus:

BC/AB = CY/AY

Then you get BC.

Then you do angle bisector theorem again, use BZ as x, and AZ as 8-x. Apply angle bisector theorem, AC/BC = AZ/BZ = (8-x)/x.

To solve: BC = 4. 9/4 = (8-x)/x => 9x = 32 - 4x, 13x = 32, x = 32/13 = BZ

I don't know what the semi-colons mean, but try using a calculator.

35% can be written as 0.35. So 35% of 80 is 0.35 * 80 = 28

Remember that converting a percentage to a decimal is taking the number in the percentage and dividing by 100! :)

Use stars and bars like proyaop said: 

There are 4 stars (balls) and 3 - 1 = 2 bars.

So, there are \({4 + 2 \choose 2} = {6 \choose 2} = \color{brown}\boxed{15}\) ways.