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Yes  thanks,

I found that when I was looking at what you did.    

Thanks for bringing it to my attention.  

Melody, your diagram has a mistake...

\(\overline{AB} = 44\), not  \(\overline{CB}\)

Deleted. Diagram was wrong.

It always helps if you read and inprete the question properly.

Thanks Builderboi.

I have only half checked what you have done it but it looks like a really nice solution.   

Alright, I'll take a shot...

Reflect \(\triangle AME\) over angle bisector \(\overline{AD}\).

You now have 2 triangles, and the sum of the areas equals \(\triangle ACE\)

The first triangle has a height of 10, and a base of 22 (Note: It has the same side lengths as \(\triangle AME\) because it is a reflection)

The second triangle has a height of 10, and a base of 8 (both bases sum to 30). 

Thus, the area is \(110 + 40 = \color {brown} \boxed {150}\)

Here is the messy diagram I made: 

\(j=24.\)

Inverse proportion means that if

\(x\cdot{a\over b}, y\cdot{b\over a}.\)

Write as a system of 3 equations: 

\(3a = b+c\)  (Equation 1)

\(5b = a+c\)  (Equation 2)

\(c=80+b\)  (Equation 3)

Sub in (Equation 3) into (Equation 2):  \(5b = a+b+80\)

Simplify:  \(b = {a \over 4}+20\)

Now, sub what we found for b, and (Equation 3) into (Equation 1): \(3a= {a \over 4} +20+80+ {a \over 4} +20\)

Solving, we find \(a = 48\)

Subbing the value of \(a\) we just found and (Equation 3) into (Equation 2), we get: \(5b = 48+80+b\)

Solving, we find \(b = 32\)

Plugging in what we found for the value of \(b\) into (Equation 3), we find that \(c = 112\).

Now do \(a + b+c\) to find your answer. 

Ans: \(90 \text{ minutes.}\)

\(1-{1\over5}={4\over5}\)

\({4\over5}\cdot{1\over3}={4\over15}\)

\({1\over5}+{4\over{15}}={7\over{15}}\)

\(1-{7\over15}={8\over15}\)

\({8\over15}\cdot\text{total time of the test}=48\)

\(\text{total time}=48\cdot{15\over8}=6\cdot15=90\text{ minutes, or } 1 \text{hr. }30\text{ minutes.}\)

Thanks! It helped a lot!(I forgot about using theorems literally)

~ PikachuLovesKetchup

4/9 of the large hose will take  9/4 of the time    9/4 * 4 hrs = 9 hours/pool   = 1/9  pool /hrs

8 small hoses will take 6/8 of the time                  6/8 * 8 hrs = 6 hrs/pool = 1/6 pool /hrs

1 pool/ (1/9 + 1/6) = 3.6 hrs

Let the time it takes you to catch up to your friend==T

5T ==3 [T  +  5],  solve for T

5T ==3T  + 15

5T - 3T ==15

2T == 15

T ==15 / 2 ==7.5 minutes -for you to catch up to her

7.5  +  5 ==12.50 minutes - your friend had been walking when you caught up to her.

2022 / 2  +  1 ==1012

We cannot pick two numbers from that range with ratio greater than 2022/1012<2.

Any number less then 1012 is a factor of some number between 1012 and 2022

The answer is 1011+1=1012.

sum of roots is 121/24

\(a+b+c=\frac{121}{24}\\ ab+ac+bc=\frac{87}{24}=\frac{29}{8}\\ abc=\frac{8}{24}=\frac{1}{3}\)

\(\log_3(a)+\log_3(b)+\log_3(c)\\ =log_3(abc)\\ =log_3(\frac{1}{3})\\ =log_3(3^{-1})\\ =-1log_3(3)\\ =-1\)

If I understand the question correctly then the smallest 47 numbers will be

0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8, 9,9,10,10,11,11,12,12, ...... 22,22,23

average of 0,1,2, ... 22 =  [23/2(0+22) ]/23= 11

average of  0,1,2,3, ... 23  =  [24/2(0+23]/24= 11.5

avverage of 11 and  11.5  =  11.25

x = 5

y = 135

To be 5 units away from a point means that it is on a circle with radius = 5 and center = (-1,0).

The equation is:  (y - -1)² + (x - 0)²  =  5²

                                    (y + 1)² + x²  =  25

Placing (2, n) into the equation:  (2 + 1)² + n²  =  25

                                                              9 + n²  =  25

                                                                    n²  =  16

So, either  n = 4  or  n = -4. 

If the first term of an arithmetic sequence is a and the common difference is d,

then the fifth term is:  a + 4d  and the 32nd term is:  a + 31 d.

We have these equations:  a +    4d  =  9

                                            a + 31d  =  84

Solving, we have:                       27d  =  75     --->    d  =  75/27  =  25/9

which makes:                                  a  =  -19/9

To find the 23rd term:  a + 22d  =  -19/9 + 22(25/9)  =  ...

You can find all of them by making the discriminant positive: 

               b² - 4·a·c  >=  0      with a = 8 and b = 45

and solving for c.

Or, you can realize that one of the possible correct answers is 0; making the 

product of all of them also 0.

x² - 8x + 1

First:  separate the squared term and the single x-term from the constant:

         (x² - 8x    ) + 1

Second:  complete the square by dividing the coefficient of the x-term by 2 and

                squaring the result:  -8 / 2  =  -4      (-4)² = 16

Third:  add this result into the set of parentheses and subtract this result after the parentheses.

          (x² - 8x + 16) + 1 - 16

Fourth:  factor the expression within the parentheses and simplify the constants:

          (x - 4)² - 15

Can you take it from here?

OP here... I figured out the answer for Part (a), (1 - (1/2 * 1/2 * 1/2))^n = (7/8)^n. But, still need not sure how to do part B.

Thanks! I've just been asked.

You are very welcome.  

Thanks melody! I suspected that the solution would be simpler than I thought. 

or maybe you meant that the factors have to be between 30 and 140 ??    [The meaning of the question is not clear]

1 is a factor for all integers ....  

2210269 = = 31*37*41*47 

the only other factors are 1 and 2210269