If ` a : b = 3 : 4 " and " x : y = 5 : 7`, then find the value of ` (3ax - by) : (4by - 7ax)`.

Explanation

Given that ` a : b = 3 : 4`
or, ` a/b = 3/4 " or, " a = (3b)/4 ` .....................(1)
Again , `x : y = 5 : 7 " or, " x/y = 5/7 " or, " x = (5y)/7` .....................(2)
Now, `(3ax-by) : (4by - 7ax)`
` = (3ax - by)/(4by - 7ax) = (3 xx(3b)/4 xx (5y)/7 - by)/(4by - 7 xx (3b)/4 xx(5y)/7)`
` = (45/28 by -by)/(4by - (15by)/4) = ((45by-28by)/28)/((16by - 15by)/4) = ((17by)/28)/((by)/4)`
` = (17by)/28 xx 4/(by) = 17/7`
Hence the required value ` = 17/7`.
Aliter : Given that ` a : b = 3 : 4`
`:. " Let " a = 3k_(1) , b = 4k_(1)`
Again, ` x : y = 5 : 7`
` :. " Let " x = 5k_(2), y = 7k_(2) . [k_(1), k_(2) ne 0]`
` :. ` Given quantity ` = (3ax - by) : (4by - 7ax)`
` = (3ax - by )/(4by - 7ax) = (3 xx 3k_(1) xx 5k_(2) - 4k_(1) xx 7k_(2))/(4 xx 4k_(1) xx 7k_(2) - 7 xx 3k_(1) xx 5k_(2))`
` = (45 k_(1) k_(2) - 28 k_(1) k_(2))/(112 k_(1)k_(2)- 105 k_(1)k_(2)) = (17 k_(1)k_(2))/(7 k_(1) k_(2)) = 17/7`.
Hence the required value = `17/7`.