If m : n = 11 : 6, then find (11m + 6n) : (11n + 6m)

Sure, let's solve this step by step:

Step 1: We are given that m : n = 11 : 6. This means that the ratio of m to n is the same as the ratio of 11 to 6.

Step 2: We are asked to find the ratio (11m + 6n) : (11n + 6m).

Step 3: We can rewrite this ratio by factoring out the common terms. This gives us:

(11m/11n + 6n/6m) : 1

Step 4: Simplifying this gives us:

(m/n + n/m) : 1

Step 5: But we know from the given that m/n = 11/6. So we can substitute this into the equation:

(11/6 + 6/11) : 1

Step 6: Simplifying this gives us:

(121/66 + 36/66) : 1

Step 7: Adding the fractions gives us:

157/66 : 1

Step 8: Simplifying this gives us the final answer:

157 : 66

So, (11m + 6n) : (11n + 6m) = 157 : 66 when m : n = 11 : 6.