BrainDen.com - Brain Teasers # Donald Cartmill

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## Everything posted by Donald Cartmill

1. The % of the pay out x money in the pot is correct. In the 2nd method you are taking the same % as before of a larger number composed of real \$ and imaginary money. The larger % will always produce the larger quotient and therefore when the missing \$ in each case is subtracted the larger quotient will appear to have earned more. This appearance can actually be reversed depending on the amount each fails to include. Example:A) puts in \$25; B) puts in \$15 ; C) puts in \$0 but agrees to add \$19 later ; D) is such a pitiful golfer He only owes \$1 (to be paid later ) All agree the payout will be 0.6 % and 0.4% : They play and C) wins the 0.6 ; and D) wins the 0.4. OK there is \$40 actual so C) gets 0.6 x 40 = \$24; D) gets 0.4 x 40 =\$16. Now if they were to calculate using the imaginary amount it would be C) 0.6 x 60 = \$36 ; D) 0.4 x 60 =\$24. Now you take \$19 away from C) \$36 wins \$17 ; D) \$24 - \$1 wins \$ 23 A bit wordy and over done Bottom line is there are 2 different %'s to establish payout. The larger the disparity of "imaginary \$ "between the two winners determines what the false winning amount is in each instance
2. The hat should contain \$20 ( but only \$10 ) By proportionality then \$12 becomes \$6 and \$8 becomes \$4. The 2nd method does NOT work as the following ex: will show Assuming The winner is to receive \$16 and the trash to get \$4. \$16 - \$5 =\$11 ( only \$10 in the hat ) The trash \$ 4- \$5 Means he would owe \$1 Where in contrast by proportionality \$16 represents 8/10 of \$20 and would receive \$8 of the \$10 ,and 4/20 = 2/`10 = \$2
3. Problem does NOT state how the first \$ 10 was split , but if split according to initial premise dad should get \$6 out of every \$10 The 1st \$10 is split Dad = 6 ; Larry =4 Now they split the 2nd \$10 \$6 \$4 Therefore at this point Dad has \$12 Larry has \$8 Subtracting out the \$5 in each case Dad wins \$7 Larry wins \$3 Nowhere does it say Larry wins \$4. The mention of \$4 out of the 2nd \$ 10 is misdirection
4. Started with 760 buttons; Problem is solved by working backward. X/2 -10 = 5 X - 20 = 10 , X = 30. X/5 -10 =30 ; X - 50 =150 X = 200 X/4 -10 =200 X -40 = 800 X = 760
5. I want to share with you the solution to any odd numbered grid,where by the columns , rows ,and diagonals each has the same total. we have all seen the very simple 3 x 3 grid filled with # 1 thru 9 ; Much more difficult would be a 5 x 5 grid with numbers 1 thru 25 ; A 7 x 7 grid would be virtually impossible ,but with the few simple rules below any odd numbered grid can be filled as fast as you can write and all cols ,rows and diagonals total the same 1a) Always start in the square to right of center square !) you are always traveling diagonally down to the right; when you exit the right side,you move down one row then jump left to farthermost open square 2) moving diagonally down to the right.you will exit the bottom..move one column right and jump up to the highest open square 3)When you are trying to move down and right and you encounter a numbered square ...you go in the upper left corner and out the upper right corner ending on the same row you started 3 a) If the above move puts you out side the main grid then you jump back to the left as far as poss on that row 3 b) If you are still inside the big grid,then you move down to the right as before 3 c) when you exit on the main diagonal Move to the left as far as possible in the bottom row 4) The above are all repetitive moves and allow you fill in any odd numbered grid such that all columns ,all rows and the diagonals add up to the same sum
6. A monk started out at 7:00 am at the bottom of the mountain and climbed to the top arriving at 7:00 pm He prayed all thru the night and at 7:00 am started down the mountain arriving at the bottom at 7:00 pm. Is there a point on the mountain trail where the monk was at the same time going up and coming down Proof please
7. Fold rope in 1/2 then again in 1/4 allow 1/4 to burn =15 minutes
8. 60-40= 20 car lengths; therefor man walked only 10 car lengths. ; mans rate is 3 mph; t = d / speed ; We have to look at the 1st car in each direction .as a unit of length Now assume 10 cl to be 1 mile 1m / 3 mph =.333 hrs; traveling in same direction train traveled = 40 + 10 = 50 lts =5 miles ; s = d / t 5 m/ .33 hrs = 15 mph train traveling opposite direction =60 - 10 = 50 lts = 5m; 5m / .33 hrs = 15 mph So let us assume a different value for 10cl = 0.5 miles ; 0.5 / 3mph = 016666667 hrs ; 50 cl = 2.5 m; 2.5 m / 0.16666666667 = 15 mph Regardless of what value in miles is assumed for 10 car lengths the answer is 15 mph If you cannot assume both are traveling at the same speed,then it is unsolvable. or actually a solutions for every increment of time he walks i.e. he walks 15 seconds and in that time 40 passes one way and 60 passes the other way...Obviously they would really be flying and one going approximately 50% faster
9. 60-40= 20 car lengths; therefor man walked only 10 car lengths. ; mans rate is 3 mph; t = d / speed ; Now assume 10 cl to be 1 mile . / 3 mph =.333 hrs; traveling in same direction train traveled = 40 + 10 = 50 lts =5 miles ; s = d / t 5 m/ .33 hrs = 15 mph train traveling opposite direction =60 - 10 = 50 lts = 5m; 5m / .33 hrs = 15 mph So let us assume 10cl = 0.5 miles ; 0.5 / 3mph = 016666667 hrs ; 50 cl = 2.5 m; 2.5 m / 0.16666666667 = 15 mph Regardless of what value in miles is assumed for 10 car lengths the answer is 15 mph 60-40= 20 car lengths; therefor man walked only 10 car lengths. ; mans rate is 3 mph; t = d / speed ; Now assume 10 cl to be 1 mile . / 3 mph =.333 hrs; traveling in same direction train traveled = 40 + 10 = 50 lts =5 miles ; s = d / t 5 m/ .33 hrs = 15 mph train traveling opposite direction =60 - 10 = 50 lts = 5m; 5m / .33 hrs = 15 mph So let us assume 10cl = 0.5 miles ; 0.5 / 3mph = 016666667 hrs ; 50 cl = 2.5 m; 2.5 m / 0.16666666667 = 15 mph Regardless of what value in miles is assumed for 10 car lengths the answer is 15 m time = distance / speed; "t " will be in hours. speed = distance / time Regardless of what value in miles is assumed for 10 car lengths the answer is 15 mph
10.     90 -70= 20 deg using side of square as "a" , we have tan20 = o/a and then O= atan 20 or 0.3639a; The right leg of the  triangle and  the top portion of the right side               20 +45 =65 ; 90-65  =     25 degrees  the other angle at top left and therefore the 3rd angle is 65;                                 tan 25 = o/a  or  o = a tan25;        o = 0.4663a.   since a  is the length of a side then the bottom leg of the lower right triangle = 1 - 0.4663 = 0.5336a ;          the vertical leg of the lower triangle is 1 -   0.3639a  =     0.6360a.  we will designate the left angle of the bottom triangle as Y ,  So we can say Tan Y = o/a or 0.6360a / 0.5336a;      tan Y = 1.1917;  Y = 50    Adding the bottom angles we have 65 + 50 = 115 :   180 - 115 =65 degrees              X = 65 degrees

11. Machine A worked 40 hours. 45x = 40(x + 5 ); 45x = 40x +200.; 5x = 200 ;x = 40
12. Comfortable
13. The area would be that of the original triangle,however that area would vary as determined by the length of the 3rd side
14. operates on the 1st ; Turns the gloves inside out for # 2; operates bare handed on the 3rd
15. If the question is : What is the ratio of the arc length to the perpendicular length from the center of one side of the square to the arc ??? Then assuming diameter of 1" ; the diagonal of the largest square would also be 1 ; this divides the square into 2 isosceles triangles ,whose angles are 45,45,90 In a 45 degree triangle the sides are = Hsin angle or 1 sin 45 = 0.707; therefore i/2 of the lt of a side is 0.353 . The radius is 0.5. 0.5 - 0.353 = the rise 0.147 the distance from arc to square Now the circumference is pie D or 3.14 D, 3.14 (1) = 3.14; acr length 3.14 divided by 4 =0.7853 ratio of arc length to rise ....0.7853/0.147 = 5.342 :1 Previously I assumed the wrong question ,However assuming a diameter of 1" R = 0.5" . I calculated the rise ( r ),which is the distance from the one side of the square at the mid point of that side to the arc or circumference . That value r =0.147 therefore the ratio of r/R = 0.147/0.5 0.147 /0.5 = 1/X; 0.147X = 1x 0.5; X = 1 /0.147; Ratio r / R is 1 /6.802 Quote
16. total of \$ 0.75 ; Arrange 4 pennies in a circle around a penny in the center (A ) such that none of the outside pennies are on the same diameter passing thru (A) Let's assume that as we look at our screen (B) is directly below (A) at a distance of the radius ; Moving CW up to the left more than 90 deg is (C) ; continuing CW approx 70 degrees which stops us short of being diametrically opposite (B) is location of (D) : continue CW about 90 degrees for (E). You have 4 triangles which do NOT over lap ABC; ACD ; ADE ; AEB . You then have 6 triangles created by lines from(B) to (D) and from (C) to (E) These triangles being BCD ; BDE; BDA; CDE ; CEA ; CEB. So we have 4 @ 0.10 = \$040 and 6 @ .05 = 0.30 for a total of \$ 0.70 plus the 5 pennies = 75 cents
17. Total of \$ 0.70 ; Arrange 4 pennies in a circle around a penny in the center (A ) such that none of the outside pennies are on the same diameter passing thru (A) Let's assume that as we look at our screen (B) is directly below (A) at a distance of the radius ; Moving CW up to the left more than 90 deg is (C) ; continuing CW approx 70 degrees which stops us short of being diametrically opposite (B) is location of (D) : continue CW about 90 degrees for (E). You have 4 triangles which do NOT over lap ABC; ACD ; ADE ; AEB . You then have 6 triangles created by lines from(B) to (D) and from (C) to (E) These triangles being BCD ; BDE; BDA; CDE ; CEA ; CEB. So we have 4 @ 0.10 = \$040 and 6 @ .05 = 0.30 for a total of \$ 0.70
18. If the question is : What is the ratio of the arc length to the perpendicular length from the center of one side of the square to the arc ??? Then assuming diameter of 1" ; the diagonal of the largest square would also be 1 ; this divides the square into 2 isosceles triangles ,whose angles are 45,45,90 In a 45 degree triangle the sides are = Hsin angle or 1 sin 45 = 0.707; therefore i/2 of the lt of a side is 0.353 . The radius is 0.5. 0.5 - 0.353 = the rise 0.147 the distance from arc to square Now the circumference is pie D or 3.14 D, 3.14 (1) = 3.14; acr length 3.14 divided by 4 =0.7853 ratio of arc length to rise ....0.7853/0.147 = 5.342 :1
19. Suppose i have a circle. I cut off its arcs such that it became the biggest possible square i could make from that circle. This is clear Several clarifications are required: What's the ratio of the edge of the circle .....(1) edge of the circle a) the circumference or b) one of the arcs clarification required: to the middle of the edge of the square ? (assume minimum length) to the radius of the circle. . Is the question : What is the ratio of the arc length to the perpendicular length from the center of one side of the square to the arc ??? Suppose i have a circle. I cut off its arcs such that it became the biggest possible square i could make from that circle. What's the ratio of the edge of the circle to the middle of the edge of the square (assume minimum length) to the radius of the circle.
20. OK I disagree with our most sage and top contributor rocdocmac on 5, 8 , 12 is ok if you add 30 seconds ,17 ,20 . In addition I will be making corrections to #1 . What is the value of x If 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9, 9/10 then x/1000? x = 500,000 CORRECTION 2 = 999 2. If 4 and 2= 26, 8 and 1 = 79 and 6 and 5 = 111. Then, what is 7and3?...410 3. What 3 positive numbers give the same result when multiplied and added together? 2 mis interpreted the Q 1,2,3 is correct 4. What number do you get when you multiply all of the numbers on a telephone's number pad? zero 5. Here is a light switch. Note the order of the positions. If the light is now at medium and it is switched 3922 times what will be the position of the switch? medium I assumed in the absence of additional info there were 3 positions . i.e low , med , high. any even number of flips will put the sw back on med 6. Can you arrange four nines to make it equal to 100. 99 9/9 8. If 1/2 of 5 is 3, then what is 1/3 of 10? 3.67 This is NOT a math calculation I envisioned 1/2 of 5 = 2 1/2 + 1/2 = 3; therefore 1/3 of 10 = 3.33 + 1/3 =3.367 9. 100 students entered college. 55 of them chose music. 44 of them chose sports. 20 of them chose both. How many of them chose neither music nor sports? 21 Of the 55 ,20 choose both leaving 35 ; of the 44 ,20 choose both leaving 24; 35 music only ;24 sports only ; 20 M&S total=79 100 - 79 = 21 who do neither 10. Four friends are going to a concert. When they arrive, there are only five seats together left in the theater. The manager will let all four friends in for free if one of them can tell her how many different seating arrangements are possible for four people with five empty seats. 120 5 seats taken 4 at a time 5 ! All four are let in free. Could you have given the correct answer? 120 5 seats taken 4 at a time 5 ! 12. At six o’clock the wall clock struck 6 times. Checking with my watch, I noticed that the time between the first and last strokes was 30 seconds. How long will the clock take to strike 12 at midnight? 5 hrs 59 1/2 minutes NOTE: the question is not how long it chimes at 12 :00 ,but how long from the 30 seconds of chiming at 6:00 3. There are several books on a bookshelf. If one book is the 4th from the left and 6th from the right, how many books are on the shelf? 9 14. John has been hired to paint the numbers 1 through 100 on 100 apartments. my bad 20 How many times he has to paint 8? 20 15. You have two books. One of them is upside-down and the other is rotated so the top of the book is facing you. What is the sum of the first page in each book? 1 +1 Up side down ,laying on its side ,propped at an angle the first page is 1 16. There is a certain club which is for men only. There are 600 men who belong to this club and 5% of these men wear one earring. Of the other 95% membership, half wear two earrings and the other half wear none. How many earrings are being worn in this club? 600 17. Does a pound of gold or a pound of feathers weight more? feathers A pound of feathers weighs 16 ounces avoirdupois; Gold is weighed on the troy system whereby a troy pound is 12 oz ; 1 troy oz = 1.0971 oz avoirdupois therefore 1 pound of gold weighs 13.1652 oz avoirdupois 18. There is a chain nailed to the wall. The chain is 10 feet long and the center of the chain dips down 5 feet from where each side of the chain is nailed to the wall. How far are the 2 ends of chain from each other? on the same nail 19.Little Johnny is walking home. He has \$300 he has to bring home to his mom. While he is walking a man stops him and gives him a chance to double his money. The man says "I'll give you \$600 if you can roll 1 die and get a 4 or above, you can roll 2 dice and get a 5 or 6 on at least one of them, or you can roll 3 dice and get a 6 on at least on die. If you don't I get your \$300." What does Johnny do to have the best chance of getting home with the money? keep on walking 20. If you have 6 women and 2 friends, how many women do each of your friends get if the friends are two of the 6 women;i.e. women themselves , then each is entitled to herself, or one each/ If the two friends are male then no women. Quote
21. . What is the value of x If 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9, 9/10 then x/1000? x = 500,000 CORRECTION x =900 2. If 4 and 2= 26, 8 and 1 = 79 and 6 and 5 = 111. Then, what is 7and3?...410 3. What 3 positive numbers give the same result when multiplied and added together? 2 mis interpreted the Q 1,2,3 is correct 4. What number do you get when you multiply all of the numbers on a telephone's number pad? zero 5. Here is a light switch. Note the order of the positions. If the light is now at medium and it is switched 3922 times what will be the position of the switch? medium I assumed in the absence of additional info there were 3 positions . i.e low , med , high. any even number of flips will put the sw back on med 6. Can you arrange four nines to make it equal to 100. 99 9/9 8. If 1/2 of 5 is 3, then what is 1/3 of 10? 3.67 I envisioned 1/2 of 5 = 2 1/2 + 1/2 = 3; therefore 1/3 of 10 = 3.33 + 1/3 =3.367 9. 100 students entered college. 55 of them chose music. 44 of them chose sports. 20 of them chose both. How many of them chose neither music nor sports? 21 Of the 55 ,20 choose both leaving 35 ; of the 44 ,20 choose both leaving 24; 35 music only ;24 sports only ; 20 M&S total=79 100 - 79 = 21 who do neither 10. Four friends are going to a concert. When they arrive, there are only five seats together left in the theater. The manager will let all four friends in for free if one of them can tell her how many different seating arrangements are possible for four people with five empty seats. 120 5 seats taken 4 at a time 5 ! All four are let in free. Could you have given the correct answer? 120 5 seats taken 4 at a time 5 ! 12. At six o’clock the wall clock struck 6 times. Checking with my watch, I noticed that the time between the first and last strokes was 30 seconds. How long will the clock take to strike 12 at midnight? 5 hrs 59 1/2 minutes NOTE: the question is not how long it chimes at 12 :00 ,but how long from the 30 seconds of chiming at 6:00 3. There are several books on a bookshelf. If one book is the 4th from the left and 6th from the right, how many books are on the shelf? 9 14. John has been hired to paint the numbers 1 through 100 on 100 apartments. my bad 20 How many times he has to paint 8? 20 15. You have two books. One of them is upside-down and the other is rotated so the top of the book is facing you. What is the sum of the first page in each book? 1 +1 Up side down ,laying on its side ,propped at an angle the first page is 1 16. There is a certain club which is for men only. There are 600 men who belong to this club and 5% of these men wear one earring. Of the other 95% membership, half wear two earrings and the other half wear none. How many earrings are being worn in this club? 600 17. Does a pound of gold or a pound of feathers weight more? feathers A pound of feathers weighs 16 ounces avoirdupois; Gold is weighed on the troy system whereby a troy pound is 12 oz ; 1 troy oz = 1.0971 oz avoirdupois therefore 1 pound of gold weighs 13.1652 oz avoirdupois 18. There is a chain nailed to the wall. The chain is 10 feet long and the center of the chain dips down 5 feet from where each side of the chain is nailed to the wall. How far are the 2 ends of chain from each other? on the same nail 19.Little Johnny is walking home. He has \$300 he has to bring home to his mom. While he is walking a man stops him and gives him a chance to double his money. The man says "I'll give you \$600 if you can roll 1 die and get a 4 or above, you can roll 2 dice and get a 5 or 6 on at least one of them, or you can roll 3 dice and get a 6 on at least on die. If you don't I get your \$300." What does Johnny do to have the best chance of getting home with the money? keep on walking 20. If you have 6 women and 2 friends, how many women do each of your friends get if the friends are two of the 6 women;i.e. women themselves , then each is entitled to herself, or one each/ If the two friends are male then no women.
22. . What is the value of x If 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9, 9/10 then x/1000? x = 500,000 2. If 4 and 2= 26, 8 and 1 = 79 and 6 and 5 = 111. Then, what is 7and3?...410 3. What 3 positive numbers give the same result when multiplied and added together? 2 mis interpreted the Q 1,2,3 is correct 4. What number do you get when you multiply all of the numbers on a telephone's number pad? zero 5. Here is a light switch. Note the order of the positions. If the light is now at medium and it is switched 3922 times what will be the position of the switch? medium I assumed in the absence of additional info there were 3 positions . i.e low , med , high. any even number of flips will put the sw back on med 6. Can you arrange four nines to make it equal to 100. 99 9/9 8. If 1/2 of 5 is 3, then what is 1/3 of 10? 3.67 I envisioned 1/2 of 5 = 2 1/2 + 1/2 = 3; therefore 1/3 of 10 = 3.33 + 1/3 =3.367 9. 100 students entered college. 55 of them chose music. 44 of them chose sports. 20 of them chose both. How many of them chose neither music nor sports? 21 10. Four friends are going to a concert. When they arrive, there are only five seats together left in the theater. The manager will let all four friends in for free if one of them can tell her how many different seating arrangements are possible for four people with five empty seats. 120 5 seats taken 4 at a time 5 ! All four are let in free. Could you have given the correct answer? 120 5 seats taken 4 at a time 5 ! 12. At six o’clock the wall clock struck 6 times. Checking with my watch, I noticed that the time between the first and last strokes was 30 seconds. How long will the clock take to strike 12 at midnight? 5 hrs 59 1/2 minutes 13. There are several books on a bookshelf. If one book is the 4th from the left and 6th from the right, how many books are on the shelf? 9 14. John has been hired to paint the numbers 1 through 100 on 100 apartments. my bad 20 How many times he has to paint 8? 20 15. You have two books. One of them is upside-down and the other is rotated so the top of the book is facing you. What is the sum of the first page in each book? 1 +1 16. There is a certain club which is for men only. There are 600 men who belong to this club and 5% of these men wear one earring. Of the other 95% membership, half wear two earrings and the other half wear none. How many earrings are being worn in this club? 600 17. Does a pound of gold or a pound of feathers weight more? feathers 18. There is a chain nailed to the wall. The chain is 10 feet long and the center of the chain dips down 5 feet from where each side of the chain is nailed to the wall. How far are the 2 ends of chain from each other? on the same nail 19.Little Johnny is walking home. He has \$300 he has to bring home to his mom. While he is walking a man stops him and gives him a chance to double his money. The man says "I'll give you \$600 if you can roll 1 die and get a 4 or above, you can roll 2 dice and get a 5 or 6 on at least one of them, or you can roll 3 dice and get a 6 on at least on die. If you don't I get your \$300." What does Johnny do to have the best chance of getting home with the money? keep on walking 20. If you have 6 women and 2 friends, how many women do each of your friends get if the friends are two of the 6 women;i.e. women themselves , then each is entitled to herself, or one each/ If the two friends are male then no women.
23. . What is the value of x If 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9, 9/10 then x/1000? x = 500,000 2. If 4 and 2= 26, 8 and 1 = 79 and 6 and 5 = 111. Then, what is 7and3?...410 3. What 3 positive numbers give the same result when multiplied and added together? 2 4. What number do you get when you multiply all of the numbers on a telephone's number pad? zero 5. Here is a light switch. Note the order of the positions. If the light is now at medium and it is switched 3922 times what will be the position of the switch? medium 6. Can you arrange four nines to make it equal to 100. 99 9/9 8. If 1/2 of 5 is 3, then what is 1/3 of 10? 3.67 9. 100 students entered college. 55 of them chose music. 44 of them chose sports. 20 of them chose both. How many of them chose neither music nor sports? 21 10. Four friends are going to a concert. When they arrive, there are only five seats together left in the theater. The manager will let all four friends in for free if one of them can tell her how many different seating arrangements are possible for four people with five empty seats. 240 All four are let in free. Could you have given the correct answer? 240 12. At six o’clock the wall clock struck 6 times. Checking with my watch, I noticed that the time between the first and last strokes was 30 seconds. How long will the clock take to strike 12 at midnight? 5 hrs 59 1/2 minutes 13. There are several books on a bookshelf. If one book is the 4th from the left and 6th from the right, how many books are on the shelf? 9 14. John has been hired to paint the numbers 1 through 100 on 100 apartments. 11 How many times he has to paint 8? 11 15. You have two books. One of them is upside-down and the other is rotated so the top of the book is facing you. What is the sum of the first page in each book? 1 +1 16. There is a certain club which is for men only. There are 600 men who belong to this club and 5% of these men wear one earring. Of the other 95% membership, half wear two earrings and the other half wear none. How many earrings are being worn in this club? 600 17. Does a pound of gold or a pound of feathers weight more? feathers 18. There is a chain nailed to the wall. The chain is 10 feet long and the center of the chain dips down 5 feet from where each side of the chain is nailed to the wall. How far are the 2 ends of chain from each other? on the same nail 19.Little Johnny is walking home. He has \$300 he has to bring home to his mom. While he is walking a man stops him and gives him a chance to double his money. The man says "I'll give you \$600 if you can roll 1 die and get a 4 or above, you can roll 2 dice and get a 5 or 6 on at least one of them, or you can roll 3 dice and get a 6 on at least on die. If you don't I get your \$300." What does Johnny do to have the best chance of getting home with the money? keep on walking 20. If you have 6 women and 2 friends, how many women do each of your friends get if the friends are two of the 6 women;i.e. women themselves , then each is entitled to herself, or one each/ If the two friends are male then no women.
24. 90 students however there are 95 involvements i.e. ( 30 fb; 40 vb; 25 bb ). That there are 23 who do not participate is NOT crucial to the solution. A ) 15 fb + vb ; B) 10 fb + bb ; C) 8 bb + vb ;; 15 fb+ 10 fb =25 fb combined plus 5 ; 15 vb + 8 vb = 23 vb plus 17 ; 10 bb + 8 bb = 18 bb plus 7 The plus 5 ; plus 17; plus 7 represents respectively those who play only 1 or all 3 sports . = 29 Now 25 + 23 + 18 = 66 involvements; plus 29 involvements = 95 involvements Therefore the plus 5 ; plus 17; plus 7 represents respectively those who play only 1 sport and there were none who played all 3 sports I yield to Rubaidi's Venn diagram above great solution
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