Two numbers P and Q are such that P : Q = x : y. If 1 is added to both the numbers, the ratio becomes 7 : 10. If 3 is added to both the numbers, the ratio becomes 3 : 4. What is (x + y) : (x - y) equal to?

Let P = x·t and Q = y·t (in lowest form, P/Q = x/y). From (P+1)/(Q+1) = 7/10: 10P + 10 = 7Q + 7, so 10P - 7Q = -3. From (P+3)/(Q+3) = 3/4: 4P + 12 = 3Q + 9, so 4P - 3Q = -3. Solve: from second, 3Q = 4P + 3. Substitute scaled — multiply second by 7/3: doesn't work cleanly. Use elimination: 10P - 7Q = -3 ... (i); 4P - 3Q = -3 ... (ii). Multiply (ii) by 7/3: not integer. Multiply (i) by 3 and (ii) by 7: 30P - 21Q = -9 and 28P - 21Q = -21. Subtract: 2P = 12, so P = 6. Then 4(6) - 3Q = -3 gives Q = 9. So P:Q = 6:9 = 2:3, hence x = 2, y = 3. Then (x+y):(x-y) = 5:(-1) = -5.