Three simple ratios are given: 5 : 6, 7 : 10 and 6 : 5. What is the equivalent compound ratio formed by combining these three ratios in order?

Introduction / Context:
This question is about compound ratios. When several simple ratios are given, we can combine them to form an equivalent single ratio by multiplying corresponding terms. This concept is useful in advanced ratio and proportion problems where multiple relations need to be summarised into one collective ratio.

Given Data / Assumptions:

• First ratio: 5 : 6.
• Second ratio: 7 : 10.
• Third ratio: 6 : 5.
• We are asked for the equivalent compound ratio of these three ratios.
• The compound ratio is formed by multiplying all antecedents together and all consequents together.

Concept / Approach:
For ratios a : b, c : d and e : f, the compound ratio is (a * c * e) : (b * d * f). This simply extends the idea that ratios compare multiplicative factors. After multiplying, we may simplify the resulting ratio by dividing both sides by their greatest common divisor to obtain a reduced form.

Step-by-Step Solution:
First ratio = 5 : 6. Second ratio = 7 : 10. Third ratio = 6 : 5. Multiply all antecedents: 5 * 7 * 6 = 210. Multiply all consequents: 6 * 10 * 5 = 300. So the compound ratio is 210 : 300. Find a common factor of 210 and 300, which is 30. Divide: 210 / 30 = 7 and 300 / 30 = 10. Therefore, the simplified compound ratio is 7 : 10.

Verification / Alternative check:
We can check the simplification by observing that 210 and 300 both end in zero, so dividing by 10 yields 21 : 30. Then noticing that both are multiples of 3, dividing by 3 gives 7 : 10 directly. This is the same result obtained by dividing by 30 in one step, confirming that 7 : 10 is fully simplified.

Why Other Options Are Wrong:
Ratios such as 7 : 9 and 9 : 7 involve incorrect products or incomplete simplification. The option 7 : 4 reverses and miscalculates the effect of multiplying the given ratios. Option 10 : 7 is simply the inverse of the correct ratio and does not represent the compound ratio in the given order.

Common Pitfalls:
One common mistake is to add the ratios term by term rather than multiplying them, which is not how compound ratios work. Others may incorrectly match antecedent with consequent when multiplying. Remember that we always multiply first terms together and second terms together to retain the correct structure of the ratio.

Final Answer:
The equivalent compound ratio of 5 : 6, 7 : 10 and 6 : 5 is 7 : 10.