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There are 1200/3=400 multiples of 3 and 1200/4=300 multiples of 4 makings 7. But we also have to account for the numbers that are in both sets, which are all the multiples of 12. 1200/12 is 100. So the number of multiples of 3 and 4 between 1 and 1200 should be 300+400-100 or 600. There are 1200 numbers through 1-1200, so 1200-600= 

600 numbers remaining.

Yep I do!! Thank you so much!

Oh, dw about it. 

Do you understand the logic behind the explanation?

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Since the problem only asks for the difference in the \(y \)-coordinates, we can ignore the \(x\)-coordinates.
They originally agreed to meet at the midpoint of \((5,-11)\) and \((-7,13)\), so the -coordinate of the planned location is \( \frac{-11+13}{2}=1\).
The correct meeting location should be at the midpoint of \((5,-11)\) and \((-5,5)\), so the -coordinate should be at \(\frac{-11+5}{2}=-3\). The positive difference is \(1-(-3)=\boxed{4}\).
Alternatively, notice that an 8-unit change in the y-coordinate of Barbara's location resulted in a 4-unit change in the midpoint since the 8 gets divided by 2. \(\frac{-11+5}{2}=\frac{-11+13}{2}-\frac{8}{2}=1-\boxed{4}\).

Wait im sorry it was supposed to be 40 not 49!! Sorry about that mistake!

Using the midpoint formula, you can see that the midpoint of these two locations are (-1.5,2.5), then the difference of the x coordinates is simply -10-(-1.5), which is -11.5, and absolute value is 11.5.

Hello Guest!

I'm a bit confused with part of your question, "If she sells them at 50 dollars a bouquet, she makes a 10 dollar profit for each sale." Was 10 a mistypo of 1? I'm going to assume that it was a typo since I can't figure out what if would mean otherwise. 

Variables

First, let's set x as the number of dollars she raises above $50. 

That would mean that she sells each bouquet at 50 + x, meaning that her profit would be 50 + x - 49 = 1 + x. 

It would also mean that the number of bouquets she's able to sell is 500 - 10x. 

Then, the profit in total is (500 - 10x)(1 + x)

Equation

The goal is to find the value of x that creates the largest answer in this equation, (500 - 10x)(1 + x). 

You could simply graph it out, or you can start simplifying. 

(10)(50 - x)(1 + x)

We want to make it so that (50 - x) is as close to (1 + x) as possible. 

50 - x = 1 + x

51 = 2x

25.5 = x

Answer

Since we're looking for her selling price (50 + x). 

We just need to plug x in. 

50 + 25.5 = 75.5

I hope this helped. :)))))

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The only possible values of w^n + 1/w^n are 1 and -1.

M - m = 2*sqrt(5) - 4.

Hint: use the quadratic formula

I found the solution:

\[\frac{1}{x} + \frac{1}{y} = \frac{1}{2/3} + \frac{1}{3/4} = \frac32 + \frac43 = \frac{9}{6} + \frac{8}{6} = \frac{17}{6}.\] 

Writing $\frac{17}{6}$ as a mixed number gives $\boxed{2\frac{5}{6}}$.

Hello guest!

We can use the chart to help!

Since triangle BCD is isosceles with BC = DC ,  ∠DBC = ∠BDC = 70°

Since ∠BDC and ∠CDA form a straight line ^^

∠BDC + ∠CDA  =  180°

                                               Subtract  ∠BDC  from both sides of the equation

∠CDA  =  180° - ∠BDC

                                              ∠BDC = 70°

∠CDA  =  180° - 70°

∠CDA  =  110°

Since there are 180° in every triangle (as shown)

∠CDA + ∠DAC + ∠ACD  =  180°

                                                         ∠CDA = 110°

110° + ∠DAC + ∠ACD  =  180°

                                                         Since triangle ACD is isosceles with CD = AD,  ∠ACD = ∠DAC.

110° + ∠DAC + ∠DAC  =  180°

110° +       2∠DAC        =  180°

                 2∠DAC        =   70°

                   ∠DAC        =   35°  =  ∠BAC

If you need a more in debt answer just ask!

-wolfie

These angles will sum to  180°

that means

x  + 4x  = 180

5x  = 180

x = 36° measure of smaller interior angle Should be close

If you need a more in debt answer just ask

-wolfie

yeah sorry 

Angles in a quadrilateral add up to 360. In this case, since it's an isosceles trapezoid, the 2 biggest angles are going to be congruent and so are the 2 smallest angles.

Therefore, we can set up an equation where x is equal to the smallest angle:

x+x+5x+5x=360

12x=360

x=30

Therefore, the angle of the smallest interior angle in the trapezoid is 30 degrees.

Hi Guest!

Sadly, I wasn't able to answer your problem, however I was able to run a quick code in Python that gave me the answer. 

https://repl.it/join/tghttcsn-160782

The total number of possible combinations are 6891. 

6891/117649 is our answer. 

However, I'm not the most sure. 

Sorry I wasn't able to help more.

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Hi Wolfie,

I think you may have answered the wrong problem. In 2019 State Sprint Round, there is a problem like this "Eliza creates a custom 6-sided die by randomly choosing six distinct integers from 1 to 7, inclusive, to paint onto the sides of a blank cube. She tells Philip that the faces of her die have a sum of 24. Philip rolls the die. What is the probability that Philip’s die shows a prime number on the top face? Express your answer as a common fraction."

However, this seems to be a different version of the problem since the numbers don't have to be distinct. 

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I hope everyone has a fantastic 2021. :))))

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Hello Melody, 

Here's the answer explained by AoPS. 

https://artofproblemsolving.com/wiki/index.php/2014_AMC_12A_Problems/Problem_25

It seems like the answer is 40. 

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