Sporties are pieces of chocolate in the shape of golf balls, tennis balls and footballs. All golf ball sporties are 5 g, all tennis ball sporties are 6 g and all football sporties are 8 g. Bags of sporties contain exactly 125 g of chocolate and always include at least 2 of each type. What is the minimum number of sporties that a 125 g bag can contain?
Answers
Answer:
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Given : Sporties are pieces of chocolate in the shape of golf balls, tennis balls and footballs.
All golf ball sporties are 5 g,
all tennis ball sporties are 6 g and
all football sporties are 8 g.
Bags of sporties contain exactly 125 g of chocolate and always include at least 2 of each type.
To Find : the minimum number of sporties that a 125 g bag can contain?
Solution:
golf ball sporties = G
tennis ball sporties = T
football sporties = F
Total weight = 5G + 6T + 8F = 125
G , T , F ≥ 2 and are integers
=> 5(2) + 6(2) + 8(2) = 38 gm
x = G - 2 , y = T - 2 , z = F - 2
Total number of sporties = 2 + 2 + 2 + x + y + z = 6 + x + y + z
5x + 6y + 8z = 125 - 38
=> 5x + 6y + 8z = 87
x , y , z ≥ 0 and are integers
To Get minimum value z should be maximum and for a given z ( y should be maximum)
z = 10
=> 5x + 6y + 80 = 87 =>5x + 6y = 7 no possible solution
z = 9
=> 5x + 6y + 72 = 87 => 5x + 6y = 15 => x = 3 , y = 0
z = 9 and x = 3 , y = 0 is one of the possible solution
=> x + y + z = 12
z = 8
=> 5x + 6y + 64 = 87 => 5x + 6y = 23 => x = 1 , y = 3
=> z = 8 , x = 1 , y = 3 is another solution
=> x + y + z = 12
z = 7
=> 5x + 6y + 56 = 87 => 5x + 6y = 31 => x =5 , y = 1
=> x + y + z = 13
z = 6
=> 5x + 6y + 48 = 87 => 5x + 6y = 39 => x = 3 , y = 4
=> x + y + z = 13
z = 5
=> 5x + 6y + 40 = 87 => 5x + 6y = 47 => x = 1 , y =7
=> x + y + z = 13
z = 4
=> 5x + 6y + 32 = 87 => 5x + 6y = 55 => x = 5 , y =5
=> x + y + z = 14
and further keep increasing
Least number is 6 + x + y + z = 6 + 12 = 18
Possible cases are :
G T F Total
5 2 11 18
3 5 10 18
minimum number of sporties that a 125 g bag can contain = 18
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