If a ∶ b = c ∶ d = e ∶ f = 5 ∶ 7, then what is the ratio (3a + 5c + 11e) ∶ (3b + 5d + 11f)?

Question

Accepted Answer

Given that a : b = c : d = e : f = 5 : 7, it means that each pair (a,b), (c,d), and (e,f) are in the ratio 5 : 7.

Therefore, we can write:
- a = 5k, b = 7k
- c = 5m, d = 7m
- e = 5n, f = 7n

Now, consider the ratio (3a + 5c + 11e) : (3b + 5d + 11f).
Substitute the values:
= (3*5k + 5*5m + 11*5n) : (3*7k + 5*7m + 11*7n)
= (15k + 25m + 55n) : (21k + 35m + 77n)

Factor out 5 from numerator and 7 from denominator:
= 5(3k + 5m + 11n) : 7(3k + 5m + 11n)

Since (3k + 5m + 11n) is common and non-zero, it cancels out, leaving:
= 5 : 7

Hence, the required ratio is 5 : 7.

Answer

Answer

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