+0

# The town of Hamlet has $3$ people for each horse, $4$ sheep for each cow, and $3$ ducks for each person. Which of the following could not po

+2
238
5

The town of Hamlet has  3 people for each horse, 4 sheep for each cow, and  3 ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?

A. 41

B. 47

C. 59

D. 61

E. 66

Nov 24, 2018
edited by Guest  Nov 24, 2018

#1
+6097
+2

you don't give any choices...

total = p + h + s + c + d = 3h + h + 4c + c + 3(3h) = 13h + 5c

here's the first 100 possible totals

18, 23, 28, 31, 33, 36, 38, 41, 43, 44, 46, 48, 49, 51, 53, 54, 56,
57, 58, 59, 61, 62, 63, 64, 66, 67, 69, 70, 71, 72, 74, 75, 76, 77,
79, 80, 82, 83, 84, 85, 87, 88, 89, 90, 92, 93, 95, 96, 97, 98, 100,
101, 102, 103, 105, 106, 108, 109, 110, 111, 113, 114, 115, 116, 118,
119, 121, 122, 123, 124, 126, 127, 128, 129, 131, 132, 134, 135, 136,
137, 139, 140, 141, 142, 144, 145, 147, 149, 150, 152, 154, 155, 157,
160, 162, 165, 167, 170, 175, 180

Nov 24, 2018
#2
+1

How do you find the first 100 possible totals from the equation?

Nov 24, 2018
#4
+6097
+2

you just plug in h=1,10, c=1,10

and sort them

I guess what I'd do is solve 13h+5c = (41,47,59,61,66)

and remember that h and c have to be non-negative integers

$$13h+5c = 41\\ c(h) = \dfrac{41-13h}{5}\\ c(2) = 3\\ \text{so 41 is a legit choice}$$

$$13h+5c=47\\ c(h)=\dfrac{47-13h}{5}\\ \text{there are no integer solutions to this equation}\\ \text{so this must the the non legit total}$$

checking for thoroughness

$$13\cdot 3 + 5\cdot 4 = 59$$

$$13\cdot 2 + 5 \cdot 7 = 61$$

$$13 \cdot 2 + 5 \cdot 8 = 66$$

Rom  Nov 24, 2018
edited by Rom  Nov 24, 2018
edited by Rom  Nov 24, 2018
#3
+1

For example, if I were taking a test, what would be an easier way to get my answer choice, which is B, instead of writing out all the possible numbers?

Nov 24, 2018
#5
+107480
+2

The town of Hamlet has  3 people for each horse, 4 sheep for each cow, and  3 ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?

A. 41

B. 47

C. 59

D. 61

E. 66

1Horse, 3People, 9ducks = some multiple of 13

1Cow, 4 sheep = some multiple of 5

So just like Rom said the answer will be such some multiple of     13H+5C

Now the multiples of 13 are   0, 13, 26, 39, 42, 55, 68 and 68 is bigger than any choice so I do not need to go further

5C will end in 0 or 5 so I can add  0 or 5 to the last digit of the multiples of 13 above

41=26+15

47= ???  to get an answer of 47 the multiple of 13 would have to end in 7 or 2 and none do so it is not a possible answer.

Melody  Nov 24, 2018
edited by Melody  Nov 24, 2018