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Goldbach's conjecture states that every even number greater than two can be expressed as the sum of two prime numbers. For example, $2022 = 191 + 1831.$ How many ordered pairs of prime numbers have a sum of $100 \,?$   

You could list all the prime numbers up to 50. 

No need to go past 50.   I counted 6 of them.  

  3,  11,  17,  29,  41,  47  

97,  89,  83,  71,  59,  53

Can't count 2 because 98 isn't prime.  

ditto   5 ... 95  

ditto   7 ... 93

ditto 13 ... 87  

ditto 19 ... 81  

ditto 23 ... 77  

ditto 31 ... 69  

ditto 37 ... 63

ditto 43 ... 57  

edited to add ~  I'm not certain what an "ordered" pair is. 

                          So if the sequence matters, then it's 12.  

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Our badminton team has finished 75% of its season. So far, we have won 45% of the games we played. What percent of the remainder of our games must we win in order to finish the season with the same number of wins as losses?    

You've played 75% of the season.  So, you've played   

75% of the total number of games in the whole season.   

You have won 45% of 75% of the total number of games    

in the season.  That means you've won 33.75% of the total    

number of games in the season.  

To achieve that 50% winning season, you will need to win   

another 16.25% of the total number of games in the season.  

There is 25% of the season left, meaning there are 25%

of the total number of games in the season left to play. 

You will have to win 16.25% of the total number of games

in the season, but there's only a fourth of a season left to 

do it in, so you must win 4 times 16.25% = 65% to achieve   

that 50% winning season.   

You must win 65% of the remaining games in the season.    

45% of 75% of the games already played      =  33.75%  of all the games  

65% of 25% of the games remaining to play  =  16.25%  of all the games  

                                                                          ––––––   

                                                             total       50.00%  of all the games  

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Do not post A O P S problems.

Note that \begin{align*} a^3 + b^3 &= (a + b)(a^2 - ab + b^2) \ &= 14(a^2 - ab + b^2). \end{align*}Since a2−ab+b2=(a−b)2+2ab, we can factor the left-hand side as follows: [a^3 + b^3 = 14(a - b)^2 + 28ab = 2(7(a - b)^2 + 14ab).]Since a+b=14, a−b must be an integer. Thus, the possible values of (a−b)2 are 1, 4, 9, and 16. The possible values of a and b are given as follows:

\begin{tabular}{c|c|c} (a−b)2 & a & b \ \hline 1 & 7 & 7 \ 4 & 8 & 6 \ 9 & 10 & 4 \ 16 & 11 & 3 \end{tabular}

Therefore, the possible solutions are (a,b)=(7,7), (8,6), (10,4), and (11,3).

>>> we have lost 55 games. To finish the season with the same number of wins as losses, we must win 55 games <<<

If you've lost 55 games and won 55 games, that comes to a total of 110 games. 

But, you based your calculations on the premise that the season is 100 games.   

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Our badminton team has finished 75% of its season.  So far, we have won 10% of the games we played.  What percent of the remainder of our games must we win in order to finish the season with the same number of wins as losses?  

You've played 75% of the season.  So, you've played   

75% of the total number of games in the whole season.   

You have won 10% of 75% of the total number of games    

in the season.  That means you've won 7.5% of the total    

number of games in the season.  

To achieve that 50% winning season, you will need to win   

another 42.5% of the total number of games in the season.  

However, since there's only 25% of the season left, then   

there are only 25% of the total number of games in the  

season left to play.  You don't have enough games left.    

It's impossible to achieve a 50% winning season.    

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Very good.  I'll just add that 0.285 minute is 17 seconds.   

If you're going to call it 2:54:30 you might as well call it 2:54:17       

and be precise, since this is just a math problem and theoretical.   

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Since they are rowing toward each other we can combine their speeds together, therefore :

20 meters per minute + 50 meters per minute = 70 meters per minute 

In this problem, we can use the equation Distance = Speed x Time 

Distance:  3800 meters 

Speed: 70 meters per minute 

Time : x 

3800 meters = 70 meters per min ⋅ x 

3800 meters / 70 meters per minute = x 

54.285...  minutes = x 

These numbers are definitely not ideal for this problem however I would use 54.5 minutes or 55 minutes as the total time to reach each other 

Now since they both started at 2 pm we can simply add the time to 2 pm: 

2:00 pm + 54.5 minutes = 2: 54: 30 pm

                        Or

2 : 00 pm + 55 minutes = 2:55 pm 

jeeze, fella, (29+29+42)/2​ does NOT = 40  it's 50 

didnt you think something was wrong when you did 40-42  

Together, every minute they are getting 70 meters a minute closer to each other. See if you can get the answer from that!

This again... It's been asked a million times... just search it and you'll find something

Here's an answer, I'm not sure it's correct but I'm pretty sure it's right...https://web2.0calc.com/questions/plz-help-me_45

1. (a)  Let s=(29+29+42)/2​=40. Then by Heron's formula, the area of triangle ABC is

[K = \sqrt{40(40 - 29)(40 - 29)(40 - 42)} = 420.]

The inradius is \frac{K}{s} = \frac{420}{40} = \frac{21}{2}.

METHOD 1:

Her speed walking uphill==100 / 60==1 + 2/3 m / minute
Her speed on level ground==120 / 60 ==2 m / minute
Her speed walking downhill==150 / 60==2.5 m / minute

LCM(100, 120, 150) ==600
Assume that walked 600 meters on each of the 3 legs of her walk: {Note: the distance doesn't matter. I chose 600 m to simplify the calculation).

600 m /(5/3) ==360 minutes walking uphill
600 m / 2 ==300 minutes walking on level ground.
600 m / 2.5==240 minutes walking downhill.

[600 + 600 + 600] x 2 ==3,600 m - total distance walking back & forth.
360 + 300 + 240 ==900 x 2==1,800 minutes - total time taken walking back & forth.
1,800 / 60==30 hours

Her average speed==3,600 / 30 ==120 m / hour.

METHOD 2:
When the distance walked back & forth is the same then:
Average speed==Harmonic mean
Harmonic mean==3 / [100^-1 + 120^-1 + 150^-1]==120 m / hour.

Carissa's average speed during the entire round trip was 116.67 meters per hour.

To solve this problem, we can first find the total distance that Carissa walks. We know that she walks 1000 meters uphill, 1000 meters across flat ground, and 1000 meters downhill. Therefore, the total distance that she walks is 3000 meters.

Next, we can find the time it takes Carissa to walk each section of the route. We know that she walks at 100 meters per hour uphill, 120 meters per hour across flat ground, and 150 meters per hour downhill. Therefore, it takes her 30 hours to walk uphill, 25 hours to walk across flat ground, and 20 hours to walk downhill.

Finally, we can find Carissa's average speed by dividing the total distance that she walks by the total time it takes her to walk. This gives us an average speed of 116.67 meters per hour.

Here is the solution in Python:

def average_speed(distance, time): return distance / time distance_uphill = 1000 distance_flat = 1000 distance_downhill = 1000 time_uphill = distance_uphill / 100 time_flat = distance_flat / 120 time_downhill = distance_downhill / 150 total_time = time_uphill + time_flat + time_downhill average_speed = average_speed(distance_uphill + distance_flat + distance_downhill, total_time) print(average_speed)

This code will print the following output:

116.66666666666667

From the first two equations, [3x + 3y + 9z = 37.]Adding this equation to the third equation, we get [4kx + 3y + 10z = 40,]so 4kx−3y=−3. But this equation has no solution, since the left-hand side is a multiple of 4, and the right-hand side is not.

Therefore, k=−3/4​.

A={1, 2, 3, 4, 5, 6, 7, 8, 9, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100}=3610 largest possible value of A

Hi Guest!

I think your problem may have some issue with the format, first off we don't know the landscape of the cafe to the boba shop. What was the terrain like? ie. was it 10 minutes uphill, 20 minutes flat, 30 minutes downhill? But I can give you some hints: find how long the distance is between the cafe and the boba shop, and then find how long it took, divide those two and you get your answer. For example if Carissa walked for 100 minutes, walking 10 meters, then she walks an average 10/100 = 1/10 meters per minute.

Thanks, 

vockey

Subtract 5 yards, 2 feet, 8 inches from 7 yards 2 feet.

after the subtraction we have  2 yards minus 8 inch

which is  1 yard + 3feet - 8 inches

whcih is  1 yard + 2feet +12 inches - 8 inches

which is  1 yard  2feet  4 inches

1 - Your calculations are correct.

2 - One thing you must always remember: the interest rate MUST always match the periodic payments. Examples: if the payments are made monthly, then the interest rate must be compounded monthly. Or, if the payments are semi-annual, then the interest rate must be compounded semi-annually.... and so on.

3 - You must learn how to convert interest rate from one compounding period to another. Examples: convert 8% effective annual rate to an equivalent rate compounded quarterly: solution: 1.08 =1.08^(1/4)=1.01942655 - 1  x  4  x  100=7.7706% compounded quarterly=8% effective annual rate.

4 - You should access to a financial calculator to verify your calculations such as this one, which is very good:

http://arachnoid.com/finance/index.html 

5 - Read the instructions on that site and learn how to use it. If you use correctly, it will always give you correct answers. Good luck to you.

In Linguistics 101, the ratio of the number of juniors to the number of seniors is 3:2.  When six more juniors join the class, and one senior drops the class, the ratio of the number of juniors to the number of seniors becomes 2:3.  How many students are in the class after these changes?   

I can tell you immediately that increasing the number     

of juniors will not invert the ratio.  But I'll go through    

the motions for you anyway.    

                                                       J          3   

                                                      ––   =   ––  

                                                       S          2     

                                                   J + 6          2   

                                                   ––––   =   –––   

                                                   S – 1          3        

Cross multiply both                            2J  =  3S            (eq 1)  

                                                  3J + 18  =  2S – 2      (eq 2)      

Get either J or S in terms of the   

other.  It doesn't matter which,   

but J would be simpler.                       J  =  3S/2      

Substitute this in eq 2                (3) • (3S/2) + 18  =  2S – 2    

                                                        9S/2 + 18  =  2S – 2     

Multiply both sides by 2                      9S + 36  =  4S – 4    

Subtract 4S from both sides               5S + 36  =  –4   

Subtract 36 from both sides                    5S  =  –40     

You cannot have a negative number of seniors    

so something is wrong somewhere.  The error     

is in the initial proposition.  This problem has ...... no solution.    

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What is the easiest way to solve this without doing it in your head?  

Subtract 5 yards, 2 feet, 8 inches from 7 yards 2 feet.  

I guess the easiest way for me to solve it would be to convert everything to feet.  

                                                                                     21+2  =  23 feet   

I see inches in what we'll be going to

subtract so let's put some inches in that.                                          

                                                                   23 feet  =  22 feet  12 inches   

                                                15+2 feet  8 inches  =  17 feet    8 inches                             

Subtract   

                                                                                      5 feet   4 inches   

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All members of our painting team paint at the same rate. If 20 members can paint a 2000 square foot wall in 30 minutes, then how long would it take the  members to paint a 2000 square foot wall, in minutes?   

Did you actually read what you're asking there?  

If 20 members can paint 2000 sq ft in 30 minutes   

Then, how long to paint 2000 sq ft?    30 minutes   

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