MathProblemSolver101

Let $x = 0.\overline{6}$

Thus, $10x = 6.\overline{6}$

Subtract these equations to get: $9x = 6.$

Thus, $x = 0.\overline{6} = \frac{2}{3}.$

We have $3 \sqrt{\frac{2}{3}} = 3 \frac{\sqrt{2}}{\sqrt{3}} = \frac{3 \sqrt{2}}{\sqrt{3}}.$

Then, if we multiply both the numerator and denominator by $\sqrt{3}$ we have 

$\frac{3 \sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}} = \frac{3 \sqrt{6}}{3} = \sqrt{6}$.

Thus, the answer is $\sqrt{6}$

Thank you, CPhill !

Mary and Mark are saving to buy new bikes. They both start with 20 dollars. If Mary saves 15 dollars a week and Mark saves 20 dollars a week, how much will each have saved in week 6? Make a table and graph to solve.

Let Mary = a

Let Mark = b

Let w = number of weeks

We know that: a=20+15w

b=20+20w

In 6 weeks, just plug in w=6.

A = 20 + 15*6 = 110

B = 20 + 20*6 = 140

Mary: 110 dollars

Mark: 140 dollars

The width of the rectangles is 50% its length. Given that the perimeter of 3 such rectangles is 90 cm, find the length of 1 rectangle.

Let l be the length

Let w be the width

w=1/2 l (1)

2(l+w) * 3 = 90 (2)

Simplifying (2): 2l+2w=30

l+w=15 (3)

(1) & (2) : 3/2 l = 15

1/2 l = 5

L = 10

Length is 10.

Enjoy you ungrateful guest.

Honestly, I think @mathisopandcool's solution is nice and clear enough. @guest, it is not his/her fault if YOU can't understand a thing. Why are you being rude?

Consider the geometric series 4 + 20/a + 100/a^2 + .... If the sum is a perfect square, what is the smallest possible value of a where a is a positive integer?

Using the geometric series formula, we have: 4/(1 - 5/a)

Simplifying we have: 4a/(a-5)

4a/(a-5) = k^2

4a = ak^2 - 5k^2

4a/k^2 = a-5

k^2 will always stay positive, so a-5 = 1 (least square)

a=6

Evaluate the infinite geometric series: 1 + 2/7 + 4/49 + 8/343 + ...

Sum: (first term)/(1 - rate of increase)

First term = 1

Rate of increase = 2/7

1/(1-2/7)

1/(5/7)

7/5

Find the greatest integer value of b for which the expression (9x^3+4x^2+11x+7)/(x^2+bx+8) has a domain of all real numbers.

Domain is all real numbers iff x^2+bx+8 ≠ 0.

Let x^2 + bx + 8 = 0 and then we will find a discriminant that will result in no real solutions if x^2+bx+8 = 0 because the denominator ≠ 0.

A equation is unreal iff b^2 - 4ac < 0

So b^2 - 32 < 0

b^2 < 32

floor(sqrt(32)) = 5

5 is the answer I think

Thank you!

The consecutive angles of a particular trapezoid form an arithmetic sequence. If the largest angle measures 120 degrees, what is the measure of the smallest angle?

Let a be the smallest angle

Let n be the difference between two consecutive angles

a+(a+n)+(a+2n)+(a+3n)=360 (1)

a+3n=120 (2)

Substituting, we have: a+(a+n)+(a+2n)+120=360

a+a+n+a+2n=240

(a+a+a)+(n+2n)=240

3a+3n=240

a+n=80 (3)

(3) * 3 = 3a + 3n = 240 (4)

(4) - (2) => 2a = 120

a = 60 degrees

60 degrees