All Answers 358267 Answers

Obviously, \(k=0 \) is an answer. Let's try to find some more (or prove that k=0 is the only answer). Let the base of the triangle be the segment from (1, 0) to (0, 1). The length of this segment will then be \(\sqrt{2}\). We can see that the height of the triangle be \(\frac{\sqrt{2}}{2}\). Since the third point is (k, k), the point must lie on the line x=y which means that k can be 0 or 1.

 The three boys in Josh’s family own a total of 28 games.     

 The youngest, Kyle, has 8 games. Josh has four times as many games as his older brother, Marcus.     

 How many games does Josh own?      

 J + M = 20     

 J = 4M     

                            K = 8    (given)     

                      J + M = 20     

                   4M + M = 20     

                          5M = 20     

                            M = 4     

                      J + M = 20     

                      J + 4  = 20     

                            J  = 16     

_.     

 The proper divisors of 12 are 1, 2, 3, 4 and 6.     

 A proper divisor of an integer N is a positive divisor of N that is less than N.     

 What is the sum of the proper divisors of the sum of the proper divisors of 280?      

 The proper divisors of 280 are     

 1, 2, 4, 5, 7, 8, 10, 14, 20, 28, 35, 40, 56, 70, & 140.  The sum of these is 440.     

 The proper divisors of 440 are     

 1, 2, 4, 5, 8, 10, 11, 20, 22, 40, 44, 55, 88, 110, & 220.  The sum of these is 640.     

_.     

 The store is selling lemons at $0.53 each.     

 Each lemon yields about 2 tablespoons of juice.   

 How much will it cost to buy enough lemons to make two 9-inch lemon pies, each requiring half a cup of lemon juice?      

 One pie requires 1/2 cup of juice, so 2 pies require 1 cup.     

 The internet says that one cup cosists of 16 tablespoons.     

 One lemon yields 2 tablespoons, so it will take 8 lemons.     

                               (8 lemons) • (0.53 $ / lemon)  =  $ 4.24     

_.     

Adult tickets for the school musical sold for 3.50 and student tickets sold for 2.50.     

Three hundred twenty-one tickets were sold all together for 937.50.     

How many of each kind of ticket were sold?     

Call the number of Adult tickets A     

Call the number of Student tickets S     

(3.50) • A  +  (2.50) • S  =  937.50                   —>   (3.50) • A  +  (2.50) • S  =  937.50     (eq 1)     

A + S  =  321       multiply both sides by 2.50  —>   (2.50) • A  +  (2.50) • S  =  802.50     (eq 2)     

                            subtract (eq 2) from (eq 1)           (1.00) • A                        =  135.00     

                                                                                                                    A  =  135 Adult tickets        

substitute 135 for A in A + S = 321                                                   135 + S  =  321     

                                                                                                                    S  =  186 Student tickets       

check answer     

Does  (3.50) • (135)  +  (2.50) • (186)  =  937.50   ?     

                  472.50     +     465.00         =  937.50     

                                                937.50  =  937.50   True     

_.     

What is the probability that at least one of them picks pineapple, and at least one of them picks pumpkin pie?

The probability that at least one of them picks pineapple is \(\color{blue}\frac{1}{4}\cdot 3 =\frac{3}{4}\ ,\)
the probability that at least one of them picks pumpkin pie is \(\color{blue}\frac{1}{3}\cdot 3=1\ .\)

  !

Two real numbers are chosen at random between 0 and 2. What is the probability that the sum of their squares is no more than 1? Express your answer as a common fraction in terms of pi.     

Assuming the numbers are integers,     

the following are the possible choices.     

       0 0    0 1    0 2     

       1 0    1 1    1 2     

       2 0    2 1    2 2     

And the following are the sums of their squares.     

          0       1       4     

          1       2       5     

          4       5       8     

The highlighted numbers are the sums which are no more than 1.     

There are three out of nine, which makes the probability 33.33%.    

π has nothing to do with this problem.     

_.     

Elita bought a jacket that was on sale for 45% off the original price. If she paid $110 for the jacket when she checked out, what was the original price of the jacket?     

Let P stand for the original price                  P – (0.45 • P)  =  110     

                                                                              0.55 • P  =  110     

                                                                                         P  =  110 / 0.55     

                                                                                         P  =  200     

_.     

 Nora and Renata had an equal number of polished stones at first.     

 After Nora received 65 more stones and Renata lost 23 stones, Nora had three times as many stones as Renata.     

 How many stones did they have altogether at first?      

 N = Nora's original number of stones     

 R = Renata's original number of stones     

 N = R     

 (N + 65)  =  3 • (R – 23)     

 N = R so substitute it          (N + 65)  =  3 • (N – 23)     

                                               N + 65  =  3N – 69     

                                                   –2N  =  –134     

                                                       N  =  67    

 since N = R                                    R  =  67     

 check answer     

 does (67 + 65)  =  3 • (67 – 23) ?     

                  132  =  201 – 69     

                  132  =  132   True      

_.     

What is the solution set for (2x - 7)^2 = 25?     

                                            (2x – 7)²  =  25     

sqrt both sides                    +(2x – 7)  =  +5     

four possibilities                  +(2x – 7)  =  +5     —>     x  =  +6     

                                            +(2x – 7)  =  –5     —>     x  =  +1     

                                            –(2x – 7)  =  +5     —>     x  =  –6     

                                            –(2x – 7)  =  –5     —>     x  =  –1     

_.     

     A list of 10 positive integers, each less than 15, has an average of 6.     

     What is the maximum number of ones (the number 1, not the digit) that could be in the list?      

     The 10 integers average 6, so the total of the 10 integers is 60.     

     If you have 9 1's, can you make up the other 51 with 1 number?     No     

     If you have 8 1's, can you make up the other 52 with 2 numbers?   No     

     If you have 7 1's, can you make up the other 53 with 3 numbers?   No     

     If you have 6 1's, can you make up the other 54 with 4 numbers?   Yes - three 14's & one 12     

_.     

Two real numbers are chosen at random between 0 and 2. What is the probability that their sum is no more than 1?    

Assuming the numbers are integers, the following are all the possible choices.     

       0 0    0 1    0 2     

       1 0    1 1    1 2     

       2 0    2 1    2 2     

The highlighted numbers are the pairs whose sum is no more than 1.     

Sum no more than 1       3     

All possible choices        9          ——>>    probability is 3 / 9  or  1 / 3     

                                                                   or, as a percentage,  33.33%     

_.     

Jamie's office recycled a total of 10 kilograms of paper over 2 weeks. After 3 weeks, how many kilograms of paper will Jamie's office have recycled? Solve using unit rates.     

      10 kg     

     ––––––  =  5 kg/wk          Therefore   —>>   (5 kg/wk) • (3 wk)  =  15 kg     

        2 wk     

_.     

Buns are sold in a packs of 8 and hot dogs are sold in packs of 12. What is the least number of hot dogs you can make if you don’t want any leftovers? How many packs of each should you buy? If you are holding a party and you expect between 80 and 100 people, how many should you buy?     

The worst case (or best, depending on your viewpoint) is that you have to feed 100 people.     

You can buy 12 pks of buns and 8 pks of dogs, and feed 96 people, if there are 4 no-shows.     

But, to ensure you can feed all 100 people, you have to get 15 pks buns and 10 pks dogs to     

make 120 sandwiches.  Then 80 people get one sandwich and 20 people get two sandwiches.     

_.     

The lines -2x + y = k and 0.5x + y = 14 intersect when x = -8. What is the value of k?     

y  =  k + 2x          —> when x = –8  then                           —>   y = k – 16     

y  =  14 – 0.5x     —> when x = –8  then  y = 14 – (–4)     —>   y = 18     

                                                                                                   k – 16  =  18     

                                                                                                           k  =  18 + 16     

                                                                                                           k  =  34     

_.     

Each row of a seating arrangement seats 7 or 8 people. $47$ people are to be seated. How many rows seat exactly 8 people if every seat is occupied?     

In the 8-person rows, you can put 40 people into 5 rows     

and then, the remaining 7 people sit in one 7-person row.     

_.     

 The angles of a quadrilateral are xº, 2xº, 4xº, 8xº.  Find x     

The sum of the internal angles of a quadrilateral is 360^o.     

Therefore, (x + 2x + 4x + 8x)  =  360     

                                        15x  =  360     

                                            x  =  24     

.     

Find the largest value of n such that 5x^2+nx+40 can be factored as the product of two linear factors with integer coefficients.     

I think the factors would be     (5x + 1)(x + 40)     

                        multiplies to      5x² + 201x + 40     

                                        so      n = 201     

_.     

    The perimeter of a rectangle is 24 inches. What is the number of inches in the maximum diagonal for this rectangle?     

    It will have the longest diagonal when you scrunch it down to a long, narrow figure.

    Assuming integer lengths, the dimensions are 11 x 1 of this long, narrow rectangle.

    Diagonal  =  sqrt(11² + 1²)     

                    =  sqrt(122)     

                    =  11.045     

_.     

1)   I do not know..... I have never paid anything but I have only answered uestions ( thousands) 

2)  you get a point or two for posting an answer ( give one to yourself !) ...other users can give you a 'thumbs up' point too...and moderator CPhil will often give you a point and a recognition as 'Best  Answer' 

3) 'Top users'  only shows the top users who have logged on recently..... after a while of no activity the user will not show up unitl login on again

4)  Nope ....not that I am aware of..... didn't use to be that way.....things have kind of gone downhill on this website since AI was introduced...but I think they did it to cut down on abuse. bullying and nuisance posts.

5) No idea.... the number of moderators has not changed since I became a memeber a LONG time ago

6) 'onlie user' shows users who are actually logged on at the time

7) All sorts of topics are asked, but it is best if you put non-math topic in the headline when posting

This site used to be a very active AND interactive community with literally hundreds of questions posted everyday.  Everone was free to post what they thought to be the answer and you were able to edit your answers if you made a mistake .   You could also 'talk ' to people via the posts....

 all of this no longer exists and there are VERY few posts of questions these days.....yours is the first I have seen in over a week.

Maybe I'll see ya around ! 

   ~ EP 

That's a very good question! The area of the polygon is 14. I actually did not understand this problem at first, but then I found a way.

Solution: We are looking for positive integers (x, y) such that x^2 + y^2 = 5. We find the solutions (x,y) =

(1,2), (1, -2), (-1, 2), (-1, -2), (2, 1), (2, -1), (-2, 1), (-2, -1), which forms a convex octagon.

To calculate the area of the octagon, you can subtract the area of the right triangles from the area of the square with vertices of (+-2, +-2), where +-a represents a and -a. The area of the square is 16, while the distance between an octagon vertice and a vertice of the square is 1, so the area of each of the triangles is 1/2, so 4 of the triangles has area 2.

The answer is 16 - 2 = 14.

WARNING: Do not assume the octagon was regular! It was equiangular, with 4 sides of length sqrt2 and 4 alternating sides of length 2.

I hope you enjoy the solution! Thank you for reading it!

There are 6 and 2 over 3 pounds of walnuts in a container, which will be divided equally into containers that hold 1 and 1 over 2 pounds. This would fill ____ and 4 over 9 containers.    

6 ²/₃  divided by  1 ¹/₂      —>>    ²⁰/₃  divided by  ³/₂     

invert divisor and multiply            ²⁰/₃  times  ²/₃   =   ⁴⁰/₉     

                       reduces to                                      =   4 ⁴/₉     

_.     

Noah was asked to compute the product of two numbers.     

He misread the problem and entered the sum of the two numbers instead.     

Luckily he still got the right answer. One of the two numbers is $8$. What is the other number?      

                                                         8 + x  =  8 • x     

divide both sides by x to obtain     8/x + 1  =  8     

                                                             8/x  =  7     

                                                              7x  =  8     

                                                                x  =  8/7     

check answer     

8 + 8/7  =  64/7     

8 •  8/7  =  64/7     

1.  Nine points are arranged in a 3 x 3 grid, as shown below.     

     In how many different ways can you choose three points, so that they form a triangle?     

     76     

     Let me tell you how I arrived at that number.     

     I postulated that any combination of three dots in that 3x3 grid will form a triangle, except     

     the eight combinations that lie in a straight line; namely, 3 vertical, 3 horizontal, 2 diagonal.     

     I calculated that ₉C₃ = 84 and then subtracted the 8 straight lines that would be formed.     

     I posted this before, but made a careless mistake.  I said  84 – 8 = 72   Sheesh.     

2.  A right isosceles triangle has a hypotenuse of 1.  What is the area of the triangle?     

     As I always say, sketch it.  Call the legs a and b and call the hypotenuse c.     

     By Pythagoras' Theorem, a² + b² = c²     

     a² + b² = c²     since a=b and c=1 then     2a² = 1     

                                                                         a² = 1/2     

                                                                         a¹ = sqrt(1/2)  and b also = sqrt(1/2)     

     Area  =  (1/2) • base • height     

               =  (1/2) • sqrt(1/2) • sqrt(1/2)     

               =  (1/2) • (1/2)     ——>>                   Area = 1/4     

_.     

what do we do? what do u mean explore? what are we asking for?

Sarah bought a binder for $7.25 and six folders that totaled $9.05.     

How much did each folder cost?     

                                                B + 6F  =  9.05     

                                                B           =  7.25     

subtract to obtain                           6F  =  1.80     

                                                         F  =  0.30     

_.     

  In triangle ABC, we have angle A = 90 degrees and sin B = 4/7. Find sin C.     

  Sketch it.   AC = 4     

                    BC = 7     

                    AB = sqrt(7² – 4²) = sqrt(33)     

  sin(C) = AB / BC     

  sin(C) = sqrt(33) / 7     

_.     

Write down the quadratic equation whose roots are x = -7 and $x = 11$ and the coefficient of x^2 is 1.     
Enter your answer in the form "x^2 + bx + c = 0".     

(x + 7)(x – 11)  =  x² – 4x – 77     

_.     

 A deep-sea exploring ship is 250 feet below sea level.     

 It rises at a rate of 5 meters per minute.     

 How far below sea level is the ship after 2 minutes?      

 1 meter = 3.28084 feet     —>>     5 meters = 16.4042 feet     

 250 ft – (16.4042 ft / min • 2 min)  =     

 250 ft –   32.8084 ft                        =  217.1916 ft     

_.     

The answer is \(14\)

Given a right triangle CAB with the right angle at B and angle bisector AD, let AD = AC.

Solution:

The question is nonsensical.

The length of the angle bisector of an acute angle in a right triangle is not equal to the length of the hypotenuse.

 !

Among all rectangles that have a perimeter of 84, find the dimensions of the one whose area is largest. Write your answers as fractions reduced to lowest terms.     

The rectangle that has the highest ratio of area to perimeter is a square.     

Therefore, when its perimeter is 84, the sides of that square are 21x21.     

_.     

Wat number can 6 and 9 go into  equally?     

Not counting zero, the first one is 18 . . . but there are infinitely more of them.     

_.     

  Nine points are arranged in a 3 x 3 grid, as shown below.     

  In how many different ways can you choose three points, so that they form a triangle?     

  72     

  Let me tell you how I got that number.     

  I think that any combination of three dots in that 3x3 grid will form a triangle, except     

  the eight triples that lie in a straight line; namely, 3 vertical, 3 horizontal, 2 diagonal.     

  So I went ₉C₃ = 84 and then subtracted the eight straight lines that would be formed.     

  I posted this answer a while ago, but made a typo.  I said 9x9 grid.  Should been 3x3.     

_.     

  Nine points are arranged in a 3 x 3 grid, as shown below.     

  In how many different ways can you choose three points, so that they form a triangle?    

  72     

  Let me tell you how I got that number. 

  I think that any combination of three dots in that 9x9 grid will form a triangle, except     

  the eight triples that lie in a straight line, namely, 3 vertical, 3 horizontal, 2 diagonal.     

  So I went ₉C₃ = 84 and then subtracted the eight straight lines that would be formed.     

_.