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Their two different solutions here and I'm not sure which one seems correct. Can someone else pls break the tie?

Yes, it is! Your solution is really helpful!

You are wwelcome.  Do you know if my answer is correct?

Thank you so much Melody!

Blue is '3' out of  1 + 2 +3 +4  =10    or  3/10ths     3/10 * 180 = 54 blue marbles 

The coefficient of y^4 is 126.

[ 10  +  8  -  1] C 10 ==17 C 10 ==19,448 ways

Sum=F / [1 - r]

Sum=42 / [1 - 1/7]

Sum=42 / (6/7) =42 x 7/6=49 - your solution.

Sum =(7/8) / [1 - 7/8]

Sum =(7/8) / [1/8]==7/8 x 8/1==56 /8 ==7 - solution above yours.

(a) Let x be the amount of red potion that must be added to 500 mL of blue potion. The total volume of the mixture will be x+500 mL. The total amount of magical syrup in the mixture will be 0.6x+0.3(500)=0.6x+150 mL. We want the mixture to be 40% magical syrup, so we have the equation:

0.6x+150 = 0.4(x+500)

1.2x = 200

x = 166.67 mL

Therefore, 166.67 mL of red potion must be added to 500 mL of blue potion in order to produce a potion that is 40% magical syrup by volume.

(b) Let x be the amount of red potion and y be the amount of blue potion that are combined to produce 100 mL of a potion that is 54% magical syrup by volume. The total volume of the mixture will be x+y mL. The total amount of magical syrup in the mixture will be 0.6x+0.3y mL. We want the mixture to be 54% magical syrup, so we have the equation:

0.6x+0.3y = 0.54(x+y)

0.06x = 0.18y

x = 3y

Since we want the total volume of the mixture to be 100 mL, we have the equation:

x+y = 100

Substituting x=3y into the second equation, we get:

3y+y = 100

4y = 100

y = 25

Since x=3y, we have x=3(25)=75​.

(c) I'll leave you to think about this one.

The given equation is a geometric series with first term b and common ratio 1/7​. The sum of an infinite geometric series is equal to a/(1-r)​, where a is the first term and r is the common ratio. In this case, the sum is equal to 7, so we have:

b/7 + b^2/7^2 + b^3/7^3 + ... = 7

\frac{b}{1-\frac{1}{7}} = 7

\frac{b}{6} = 7

b = \(\boxed{42}\)

Without uploading the "diagram", you will be stuck forever, lazy !

sumfor(b, 1, 1000, (6.125^b / (7^b))==7

b ==6 + 1/8 ==6.125

These are the first 10 terms of GS:

(7 / 8, 49 / 64, 343 / 512, 2401 / 4096, 16807 / 32768, 117649 / 262144, 823543 / 2097152, 5764801 / 16777216, 40353607 / 134217728, 282475249 / 1073741824)

The given equation is a geometric series with first term b and common ratio 71​. The sum of a geometric series is given by the formula S=a​(1−rn)/(1-r)​, where a1​ is the first term, r is the common ratio, and n is the number of terms. In this case, we have S=7, a_1​=b, and r=1/7​.

Solving for n, we get n=−log(1−r)/log(r)​=log(1/7)/log(7)​=−log(7)/log(7)​=−1. Substituting this value of n into the formula for the sum of a geometric series, we get S=a(1-r^n)/(1-r)​=b(1-(1/7)^(-1))/(1-1/7)​=b(1−7)/(-6)​=7. Solving for b, we get b=49/6​​.

ab^4 = 48     then a = 48/b^4

sub into the second equation

48/b^4  * b = 72

48/72 = b^3 

b = .87358      then    a = 82.4194

x/y = 2/3.

There are 12 divisors of 72 as follows:

(1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72)

There are: 2^12 ==4,096 subsets including the empty set.

There are: 2^11 + 2^10 ==3,072 subsets that have either a"2" or a "3" or both in them.

Therefore: There are: 4,096  -  3,072 ==1,024 subsets that contain only composite numbers.

There are 36 - 20 = 16 people who brought yoga mats but no water bottles. There are 26 - 20 = 6 people who brought water bottles but no yoga mats. Therefore, 16 + 6 = 22 people had to suffer through class having neither a yoga mat nor a water bottle. So the answer is 22

The area of rectangle ABCD is 8×4=32 square centimeters. The area of triangle AMC is 1/2​×8×4=16 square centimeters. The area of triangle MNC is 1/2​×4×4=8 square centimeters. Therefore, the area of region AMCN is 16+8=24 square centimeters.

It's actually a nice question. 

But there are typo's, the last line should be

Prove that

\((x+h)^{2}+(y+h)^{2}=(a+b)^{2}.\)

x and y are AH and BH., h is CH.

For the first question, by stars and bars, there are C(11,3) = 165 ways.

For the second question, if each school must receive 2 teachers, there are 4! = 24 possible divisions.

The reason for the difference is that in the first case, the teachers can be divided among the schools in any way. In the second case, the teachers must be divided in such a way that each school receives 2 teachers. This limits the number of possible divisions.

Y=Yellow,   G==Green

G ==Y + 28........................(1)

G - 3/7G ==4/7G - green balls remain
Y - 1/3Y ==2/3Y - yellow balls remain.

4/7G ==2/3Y......................(2), solve for G, Y

G ==196 - original number of green balls
Y ==168 - original number of yellow balls

Total =196 + 168 ==364 green and yellow balls at start

35.00  x  0.90 ==$31.50 after 10% coupon discount.

$31.50  x  1.08 ==$34.02 - her final price

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