Question

If $*$ is a binary operation defined by $a*b=\frac{a}{b}+\frac{b}{a}+\frac{1}{ab}$ for positive integers $a$ and $b$, then $2*5$ is equal to

• A. 4
• B. 3
• C. 2
• D. 1
• E. 5

Hint:

Calculation Tip: Always find the lowest common denominator (LCD) before adding fractions to ensure accurate and easily simplified results.

Answer 1

The Correct Option is B

Concept:
A binary operation $*$ combines two elements according to a specific given rule. Here, the rule is: $$a*b=\frac{a}{b}+\frac{b}{a}+\frac{1}{ab}$$

Step 1: Identify the values for a and b.
We need to evaluate the expression for $2*5$. Comparing this to $a*b$, we assign: $$a=2$$ $$b=5$$

Step 2: Substitute values into the operation.
Substitute $a=2$ and $b=5$ into the defined formula: $$2*5=\frac{2}{5}+\frac{5}{2}+\frac{1}{(2)(5)}$$

Step 3: Simplify the expression.
Multiply the terms in the denominator of the third fraction: $$2*5=\frac{2}{5}+\frac{5}{2}+\frac{1}{10}$$

Step 4: Find a common denominator.
To add the fractions, find the least common multiple of the denominators 5, 2, and 10, which is 10. Convert each fraction: $$\frac{2}{5}=\frac{4}{10}$$ $$\frac{5}{2}=\frac{25}{10}$$ The third fraction is already $\frac{1}{10}$.

Step 5: Add the fractions and select the answer.
Combine the numerators over the common denominator: $$2*5=\frac{4+25+1}{10}$$ $$2*5=\frac{30}{10}$$ $$2*5=3$$ Hence the correct answer is (B) 3.