Explanation:
mean value : the sum of all data points divided by the number of data points.
median value : the data point, where half of the other data points are smaller, and the other half larger than that. for an even number of data points it is the mean value of the 2 "middle values".
we have 6 data points.
the mean value is
(178+142+112+150+206+130)/6 = 918/6 = 153
for the median value let's sort the data points :
112 130 142 150 178 206
so, the 2 "middle values" are 142 and 150. the mean value between these 2 (and therefore the median value) is
(142+150)/2 = 146
now, we add 2 wrestlers. that means we have then 8 data points. and the mean value stays the same.
(178+142+112+150+206+130+w1+w2)/8 = 153
(178+142+112+150+206+130+w1+w2) = 153×8 = 1224
918 + w1 + w2 = 1224
w1 + w2 = 306
the median value increases by 3 pounds to 149.
let's see, how the sorted lineup of the data points could look like :
112 130 142 150 w1 w2 178 206
that cannot be right, because the new median would be between 150 and w1, and that would be larger than 149.
the same problem for all other cases, where w1 and w2 are both on the right side of 150.
if they are both on the left side of 150, they cannot reach together 306 pounds.
112 130 142 w1 150 w2 178 206
that is possible : the new median would be between w1 and 150. to get 149 as result, w1 must be 148 (mean value between 148 and 150 is 149).
w1 = 148
w1 + w2 = 306
w2 = 306 - 148 = 158
if w1 would be further left of 150, then the new median would still be the mean value between 142 and 150 (146).
and if w2 would be further to the right of 150, then w1 + w2 would be higher than 306.
so, yes,
w1 = 148
w2 = 158
is the only correct solution.