(Below is a challenging math problem, who can correctly solve it by showing all work and explaining their thinking? Please share, discuss, and read all students' comments)
In football, the player who kicks the ball is referred to as the punter. During a recent football game, the punter, Khan Bootit, kicked the ball five times. His longest kick was 44 yards and he averaged 35 yards per kick. Each of his 5 kicks was a different whole number distance.
Determine the minimum possible distance of Khan’s shortest kick.
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ReplyDeleteI already answered the question but then it got deleted. So i found that the lowest kick was 5. It seems unrealistic but it is technically the lowest shot he could have gotten.
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ReplyDeleteI think the answer is 26 yards. I got the answer 26 by figuring out what is the total yardage during the game, by multiply the average value, which is 35, by the total numbers of kicks, which is 5times, and I got 35 x 5 = 175yrds.
ReplyDeletewhat i did next is subtract 44 from 175, and i got 131 (total of the remaining 4 kicks.) the average of 131 is about between 20-40; therefore I used guess-and-check and I determined that remaining kicks are: 26, 32, 34, 39. and I verified my answers by adding them up.
that's my two cents. peace.
John
hi John,
Deletethank you for starting this conversation.
I got the same answer; however, I used a different method to figure it out.
I figure that the difference between the highest (44) and the average (35) is 9 yards; therefore If I take 9 from 35, i got 26 as well, which should be the shortest.
back at ya.
Jatin
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DeleteHey, I think I have the answer.
DeleteHis lowest kick was 5 yards. Since his average was 35, I multiplied 35 by 5 and got 175, his total yards. I subtracted 44, 43, 42, and 41, the four highest whole numbers, with 44 being the highest. This left 5, which was his lowest kick. I verified by adding 5, 44, 43, 42, and 41, with the total being 175, and dividing by 5, leaving 35, his supposed average.
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ReplyDeleteHello fellow classmates
ReplyDeletei agree with John's method but this way, the answer could be anything. Well, almost anything. it all has to be lower than 44. The numbers could be ridiculous such as 2, 43, 42 and 44. the average it still 35 but the lowest is 2 (which is almost impossible for a football player). All these numbers add up to 131 which is the total for 4 kicks.
How i checked was i added up all 5 kicks and divided it by three which is obviously 35.
So my answer is there is no exact answer. It just has to be under 44. If there is an answer, we'll just have to find out next week.
Sophia :)
I have a question. Why did you divide all five kicks by three?
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ReplyDeleteThe shortest punt possible was 5 yards. His average was 35 yards. He kicked 5 times so 35 times 5 equals 175. His longest punt was 44 yards. 175-44=131. All of his punts were different whole numbers so he could have kicked 43 yards. 131-43=88. 88-42=46. 46-41=5. His shortest possible punt was 5 yards
ReplyDeleteThanks Richard for your answer. I like how you assumed that 'All of his punts were different whole numbers so he could have kicked...'.
DeleteHi there.
ReplyDeleteI think the minimal possible distance of Khan's shortest kick is 5 yards. I came up with this answer by first multiplying the AVERAGE (35) by how many times he kicked the ball (5). I am using the average because I know by calculating average, you need to add all the numbers and divide the sum by how many numbers are in the set. With this in mind, if I work backwards and use the average to find the sum instead of using the sum to find the average, I can easily identify possible kick distances.
The product of 35 x 5 is 175. With this answer, I subtracted 175 by the LONGEST KICK (44) because 44 was already one kick recorded out of the other 4 kicks. The difference of 175 - 44 is 131.
As said before, with using the average to find the sum, I can easily find possible kick distances. So, from here, I started to eliminate possible answers. My idea was to put and find 4 different numbers (all of which were lower than 44) and add them together to equal 131. So, I started with the least possible number (1) and worked my way up (2,3,4..) while trying to find a sum of 131. When I tried numbers 1 to 4 (along with other numbers), it did equal 131, but, the other numbers were higher than 44. Therefore, 1 to 4 as the shortest possible distance are incorrect answers.
However, when I tried 5 yards as the shortest possible distance with numbers 43,42 and 41, it did work. The sum was 131 and all numbers were below 44. Since 5 is the lowest answer, 5 yards is the minimum possible distance of Khan's kick.
The whole process used in this question was a mixture of Elimination/Trail and Error.
I know this is correct because...:
35 x 5 = 175
175 - 44 = 131
43 + 42 + 41 = 126
126 + 5 = 131
The whole addition sentence looks like this:
44+43+42+41+5 = 175
Average:
175 / 5 = 35
wow nice explanation of your work Jonathan
DeleteGood job and thank you for your co-operation!!
DeleteI like how you explained your process thoroughly.
DeleteNice job putting the answer plus show a lot of detail to support your work.
DeleteThe answer that I got was 5 yards. The first thing I did was reverse the operation of finding a mean, since they already stated the average. So, first I multiplied 35 (average) by 5 (total amount of kicks) and got 175, which is the total yardage for all 5 kicks.
ReplyDeleteThe strategy I used to find Khan's shortest kick was to make the other 4 kicks as high as they could possibly go (below 44, as stated in the question). I determined that the first four kicks would be 44,43,42,41, which in total was 170. So, the remaining kick's amount would have to be 5 yards to complete the the total amount, 175.
So, Khan's 5 kicks are 44,43,42,41, and 5. Therefore, 5 yards is Khan's shortest kick.
Rachel
I liked how you explained your process of identifying the different possible kicks. Thank you for your contribution.
DeleteI liked how you explained it!!!
DeleteGood WORK!!!
Just like Johnathon, Thank you for your contribution.
Hehe, I got some help and with some talking, we come to this conclusion:
ReplyDeleteFigure out the total of the unknown kicks: (a,b,c,d - d is minimum kick)
35 = (44+a+b+c+d)/5
35 x 5 = (44+a+b+c+d)
175 = (44+a+b+c+d)
175 - 44 = (a+b+c+d)
131 = (a+b+c+d)
Now that we know that, we also know that a,b, and c are greater than d they are whole numbers and not the same. So, we figure out the maximum possible kicks to figure out the sole minimum kick - the numbers right under 44.
Suppose:
a = 43
b = 42
c = 41
And then:
131 = (41+42+43+d)
131 - (41+42+43) = d
131 - 126 = d
5 = d
Therefore, the minimum possible shortest kick would be 5 yards (though I think it's unlikely considering he's a pro football player ^^;)
I like how you used a different method than everyone else(algebraic terms)
DeleteI like how you used algebraic terms to solve the problem
DeleteI think that his shortest kick was 5 yards our of 5 shots. To find out this answer, I first got the average of all 5 numbers (35) and multiplied it by 5 which equals 175. Since we know his longest kick (44), I subtracted 44 from 175 which is 131. Once I subtracted it, I then did 131 divided by 4 (since we subtracted the longest kick which leaves 4 kicks left) which is 32.75. Just to make sure I didn't make any mistakes, I did 32.75 x 4 to see if it equals 131 (which it does). So now that I have the average of the shots per kick (32.75) I divided this number into 4 sections (32.75 / 32.75 / 32.75 / 32.75) because there are 4 shots that he shot. Using what we know, (44 is his largest shot he made) I started subtracting numbers from one of the 32.75's by 27.75 which then equals (5 / 32.75 / 32.75 / 32.75). Since I had a remainder of 27.75, I started adding this number to each and every one of the other numbers (32.75's). I added 8.25 to the second 32.75 (41), 9.25 to the third 32.75 (42) and 10.25 to the forth 32.75 (43). To check if I wasn't off, I added 8.25, 9.25 and 10.25 together to see if it equals 27.75 (the remaining numbers). (8.25 + 9.25 + 10.25 = 27.75).
ReplyDeleteNow the final step, to find if my answer is correct:
5 / 41 / 42 / 43 (My four Numbers)
5 + 41 + 42 + 43 = 131 (This is correct because we subtracted 44 to make 131)
131 + 44 = 175
175 divided by 5 = 35 (Divide 5 because we added 44 to get 175)
35 is the total average so this answer is valid and since the numbers cannot be the same, 5 yards is the minimum possible distance he shot.
I have a question. Why did you divide 131 by 4? I understand how you used it to find the average per kick, but the average was already provided.
DeleteAlso, where did you get 27.75 from to subtract it from 32.75?
Perfect!
DeleteBut, you should make it a little less confusing, by not adding so many decimals!
But, good job explaning
Wow. Some fantastic answers! I love seeing different problem-solving strategies, like using algebraic terms or using guess and check. Below is the correct solution:
ReplyDeleteSolution
To determine an average, determine the sum of the numbers in the set and divide by the number of numbers in the set. It follows that the sum of the numbers in a set is the average times the number of numbers in the set.
Since Khan’s average was 35 yards per kick and he kicked the ball five times, the total length of all his kicks was 5 × 35 = 175 yards. His longest kick was 44 yards, so his remaining kicks covered a total of 175 − 44 = 131 yards.
To find the minimum length of his shortest kick make the other three kicks as long as possible. Each of these kicks must be a different integer length less than 44 yards long. So for the three kicks to be as long as possible, they must be 43 yards, 42 yards, and 41 yards, respectively.
The minimum length of the shortest kick is therefore 131 − 43 − 42 − 41 = 5 yards.
If he had been able to have different kicks with the same integer length, then the minimum length of his shortest kick would be 175 − 4 × 44 = −1 yard. (It would have travelled backwards.) But the kicks all had to be positive integer lengths so his minimum shortest kick would have been 1 yard with his other kicks being such that the total yards travelled by all of his other four kicks would be 174 yards. There would be a few different possible kick lengths that would make this possible. For example, his five kicks could be 1, 44, 44, 44, and 42 yards each, respectively or 1, 44, 44, 43, and 43 yards each, respectively.
hello
ReplyDeleteI think the minimum possible distance of Khan's shortest kick is 5. Since I know that the highest kick is 44 yards i know that the rest have to be a lower number. So I did the next 3 kicks as one lesser than each other and the last few was 5. the set of number i used was 44,43,42,41,5. It decreased by one from 44,43,42,41 then i added it all and the number left over was 5. At first i started to work backwards and I multiplied 35 times 5 and i got 175, so i know that the sum of the set of numbers need to be 175. It asks for the least possible kick and each number has to be different. So i just subtracted one from next 3 numbers, that way i could get to the least possible number at the end.
good job explaining,but shouldn't 35 be one of these numbers.
DeleteYour #'s were 44,43,42,41,5 but shouldn't one of these numbers be 35???
Hi Kajana. Thanks for your contribution! Could you explain why you did the next 3 kicks as one lesser than each other? Why did you come up with this assumption?
DeleteThank you all who participated in this week's POTW! We are hoping for a bigger turn out next week! You all did a great job explaining your thinking and putting yourself out there!
ReplyDeletePlease remember that it's okay if you come up with the same answer with another person; we are most interested in how you reach the conclusions. cheers!
the answer is 26 yards multiply the average value, which is 35, by the total numbers of kicks, which is 5 x 35
ReplyDeleteHey Artin. Thanks for your answer. I couldn't really understand your answer. Why did you multiply the average value by 5 to get 26?
DeleteCould you explain more. For example, why!?
Deletei like these calculations
ReplyDeleteThis is how i did it.
ReplyDeleteI used a simple way (cause of time)
well i just subtracted 44-35=9
so, i just subtracted 9 each time.
so, 44-9=35-9=26-9=17-9=8
i stopped once i got to the 5th number.
so, my final answer is 44,35,26,17 and 8.
i kept it simple and easy
not like the rest of you who divided and mulitiplied.
Thanks Krupa for your answer! Can you explain why you used these numbers in your equation? What is your reasoning? How do these numbers help you in finding the answer?
Delete5 was the answer I arrived at: I continuously subtracted 44, then 43, then 42, and so on until I reached 5. Although this question, which I have determined is not a "definite answer" question.
ReplyDeleteSorry for the late post.
Hello,
ReplyDeleteThis is Hiya. As per my assumptions and methods, I think the minimum possible distance of Khan's shortest kick was 5 yards because the method I utilized was trial and error or you can choose any number but it has to fit under the characteristics of the number being less than 44 yards.
1. I started out by finding the total amount of yards Khan took. To identify that I used multiplication as my method. I found the product of the average yard of kicks (35) by the amount of kicks taken. I did this because the average is showing the mean of the set of data and therefore to find the shortest kick I have to find the total.
35*5= 175--> Khan took 175 yards in total.
2. With the 175 I subtracted the longest kick of 44 yards therefore I can eliminate 1/5 kick leaving 4/5 kicks to find.
175-44=131 ---> This is the leftover amount of the data which I have to use to find the rest of the 4 numbers.
3. Now this step was purely based on trial and error because you can choose any number but in the range of 44 yards and not more than that. Basically I started to subtract random numbers like 42, 43, 37 and etc.
175-44= 131
131-42=89
89-44= 45
45-40=5
My set of data of Khan's scores were
44, 42, 44, 40, 5
TO check I found the mean (average of it)
44+42+44+5=175
175/5= 35