Suppose five pumpkins are weighed two at a time in all possible ways. The weights in pounds are recorded as 16, 18, 19, 20, 21, 22, 23, 24, 26, and 27. How much does each individual pumpkin weigh?

\boxed{a=16-b} \\ \\ b+c=18=>\boxed{c=18-b} \\ \\c+d=19 \\ \\ a+c=21 \\ \\ d+e=20 \\ \\ a+d=22 \\ \\ a+e=23 \\ \\ b+d=24 \\ \\ b+e=26 \\ \\ \\ a+c=21 \\ 16-b+18-b=21 \\ 32-2b=21 \\ -2b=21-34 \\ -2b=-13 \\ \\ \boxed{b=\frac{-13}{-2}=6.5} \\ \\ a=16-6.5 \\ \\ \boxed{a=9.5} \\ \\ c=18-6.5 \\ \\ \boxed{c=11.5} \\ \\ d=19-11.5 \\ \\ \boxed{d=7.5} \\ \\ e=20-7.5 \\ \\ \boxed{e=12.5}"> a + b = 16 => a = 16 − b ​ b + c = 18 => c = 18 − b ​ c + d = 19 a + c = 21 d + e = 20 a + d = 22 a + e = 23 b + d = 24 b + e = 26 a + c = 21 16 − b + 18 − b = 21 32 − 2 b = 21 − 2 b = 21 − 34 − 2 b = − 13 b = − 2 − 13 ​ = 6.5 ​ a = 16 − 6.5 a = 9.5 ​ c = 18 − 6.5 c = 11.5 ​ d = 19 − 11.5 d = 7.5 ​ e = 20 − 7.5 e = 12.5 ​

Answered by mariamikayla | 2024-06-10

The individual weights of the five pumpkins are: 6.5 pounds, 9.5 pounds, 11.5 pounds, 12.5 pounds, and 14.5 pounds. This was determined by setting up equations for each pair of pumpkin weights based on the provided sums and solving them systematically. Each step of substitution led us to find the weight of each individual pumpkin accurately.
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Answered by mariamikayla | 2024-09-06