The difference of $1\frac{3}{16}$ and its reciprocal is equal to

Concept & Formula The reciprocal of a fraction $$\frac{a}{b}$$ is $$\frac{b}{a}$$. To find the exact difference between a mixed fraction and its reciprocal, you must first convert the mixed fraction into an improper fraction before finding a common denominator to subtract them. Step-by-Step Solution * First, convert the mixed fraction into an improper fraction: $$1\frac{3}{16} = \frac{19}{16}$$ * Find the reciprocal of this improper fraction: $$\text{Reciprocal} = \frac{16}{19}$$ * Calculate the difference by subtracting the smaller value from the larger value: $$\text{Difference} = \frac{19}{16} - \frac{16}{19}$$ * Find the common denominator ($$16 \times 19 = 304$$) and cross-multiply to subtract the numerators: $$\frac{19^2 - 16^2}{304}$$ * Simplify using the difference of squares algebraic identity ($$a^2 - b^2 = (a-b)(a+b)$$): $$19^2 - 16^2 = (19 - 16)(19 + 16) = 3 \times 35 = 105$$ * The exact difference is $$\frac{105}{304}$$. * Since this calculated fraction is not present among any of the given specific options, the correct choice is "None of these". Exam Strategy & Shortcut Use the algebraic identity $$a^2 - b^2 = (a-b)(a+b)$$ to evaluate the numerator quickly. Calculating $$(19-16) \times (19+16)$$ mentally is vastly faster than manually squaring 19 and 16 and then subtracting them ($$361 - 256$$). Common Pitfall A prevalent mistake is finding the reciprocal incorrectly by only inverting the fractional part of the mixed number (e.g., treating the reciprocal of $$1\frac{3}{16}$$ as $$1\frac{16}{3}$$). Always convert fully to an improper fraction before flipping the numerator and denominator. Final Answer Therefore, the correct answer is **None of these**.