Let x and y be the proportionality constants.
Since, the ratio by weight of X : Y in the first alloy is 6 : 5.
\(\therefore\) Weight of metals X and Y in the first alloy is 6x kg and 5x kg respectively.
Also, the ratio by weight of X : Y in the second alloy is 7 : 13.
\(\therefore\) Weight of metals X and Y in the second alloy is 7y kg and 13y kg respectively.
Now, weight of the first alloy is 11 kg.
\(\therefore\) 6x + 5x = 11
\(\therefore\) 11x = 11
\(\therefore\) x = 1
\(\therefore\) first alloy has 6x = 6 \(\times\) 1 = 6 kg of X metal.
Also, weight of the second alloy is 20 kg.
\(\therefore\) 7y + 13y = 20
\(\therefore\) 20y = 20
\(\therefore\) y = 1
\(\therefore\) second alloy has 7y = 7 \(\times\) 1 = 7 kg of Y metal.
Suppose z kg of metal X is melted so as to produce the new alloy.
\(\therefore\) Total weight of X metal in the new alloy
Now, the new alloy contains 40% of metal Y
i.e., it contains 60% of metal X
\(\therefore\) 14 kg of metal X must be melted to produce a new alloy containing 40% of metal Y.
