Solving with Reciprocals
Solving with reciprocals identifies equations of the form (a/b)n = 1, where the unknown n must be the reciprocal of a/b. In Grade 6 Saxon Math Course 1, students learn that flipping a fraction's numerator and denominator gives its reciprocal, and any number times its reciprocal equals 1. For (3/8)n = 1, the answer is n = 8/3. This principle extends to negative fractions: −(5/12)k = 1 gives k = −12/5. Reciprocals are the foundation of fraction division.
Key Concepts
Property When two numbers are multiplied and the product is 1, the two factors must be reciprocals. To solve an equation like $\frac{a}{b}n = 1$, find the reciprocal of the known factor. The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$.
Examples Example: In the equation $\frac{3}{8}n = 1$, the unknown $n$ must be the reciprocal of $\frac{3}{8}$, so $n = \frac{8}{3}$. Example: To solve $\frac{7}{4}x = 1$, you simply flip the fraction $\frac{7}{4}$ to find the answer. Thus, $x = \frac{4}{7}$. Example: For $5y=1$, think of 5 as $\frac{5}{1}$. The reciprocal is $\frac{1}{5}$, so $y = \frac{1}{5}$.
Explanation When a fraction multiplied by an unknown number equals 1, you've found a shortcut! The unknown number must be the reciprocal, or the 'flipped' version, of the known fraction. This happens because multiplying a number by its reciprocal always cancels everything out perfectly, resulting in the number 1.
Common Questions
What is the reciprocal of 3/8?
Flip numerator and denominator: 8/3. Check: (3/8) × (8/3) = 24/24 = 1. ✓
What is the reciprocal of the whole number 5?
Write 5 as 5/1; reciprocal is 1/5. Check: 5 × 1/5 = 1.
Solve: (3/8)n = 1
n must be the reciprocal of 3/8. Answer: n = 8/3.
Solve: −(5/12)k = 1
k = reciprocal of −5/12 = −12/5.
Why does multiplying by a reciprocal equal 1?
(a/b) × (b/a) = ab/ab = 1. A fraction times its flip always equals 1, the multiplicative identity.