Final answer:
The sum of a third of (y = (a+b)) and a fifth of (y = (a-b)) simplifies to (2y/3), which does not match any of the provided multiple-choice options. There may be an error in the provided answers or question interpretation.
Step-by-step explanation:
The question asks to find the sum of a third of (y = (a+b)) and a fifth of (y = (a-b)). To solve this, we will first take each expression and find the respective fractions:
- One third of (a+b) is (1/3)(a+b) or (a+b)/3.
- One fifth of (a-b) is (1/5)(a-b) or (a-b)/5.
Next, we combine these fractions:
(a+b)/3 + (a-b)/5
To add these fractions, they need a common denominator, which is 15. We get:
(5(a+b) + 3(a-b)) / 15
Expanding and simplifying the numerator gives us (5a+5b+3a-3b) / 15 which further simplifies to:
(8a + 2b) / 15
Since y equals both (a+b) and (a-b) in either case, we replace a and b with y. Doing a simple rearrangement, we get:
(8y/15) + (2y/15) = (10y/15) or (2y/3).
However, none of the given options matches this result. There may have been a mistake in the given answers or in the interpretation of the question's instructions.