Multiply by the reciprocal
Multiplying by the reciprocal is a Grade 7 algebra strategy from Saxon Math, Course 2, Chapter 7, used to solve equations where a fractional coefficient is multiplied by a variable. Instead of dividing by the fraction, you multiply both sides by its reciprocal (the flipped version), which cancels the coefficient and isolates the variable. For example, to solve (3/4)x = 9/10, multiply both sides by 4/3 to get x = 6/5. This technique makes solving fraction equations clean and efficient.
Key Concepts
Property Instead of dividing by a fractional coefficient, you can multiply by the reciprocal of the coefficient. The goal is to make the coefficient become $1$.
Examples To solve $\frac{3}{4}x = \frac{9}{10}$, multiply both sides by the reciprocal, $\frac{4}{3}$, to get $x = \frac{6}{5}$. For $\frac{2}{5}y = 8$, multiply both sides by $\frac{5}{2}$: $\frac{5}{2} \cdot \frac{2}{5}y = \frac{5}{2} \cdot 8$, which simplifies to $y = 20$. In $\frac{7}{2}z = \frac{3}{4}$, multiply by $\frac{2}{7}$: $\frac{2}{7} \cdot \frac{7}{2}z = \frac{2}{7} \cdot \frac{3}{4}$, so $z = \frac{3}{14}$.
Explanation When an equation has a fraction for a coefficient, dividing can feel clunky. Here’s a pro move: multiply both sides by the reciprocal (the flipped version) of that fraction! This magical step instantly cancels out the coefficient, turning it into $1$ and making it super simple to solve for the variable without getting tangled in complex division.
Common Questions
What does it mean to multiply by the reciprocal?
Multiplying by the reciprocal means multiplying by the flipped version of a fraction. For example, the reciprocal of 3/4 is 4/3. Multiplying a fraction by its reciprocal always gives 1, which cancels the coefficient in an equation.
How do you use the reciprocal to solve an equation?
Identify the fractional coefficient, find its reciprocal, then multiply both sides of the equation by that reciprocal. This isolates the variable. For example, (2/5)y = 8 → multiply by 5/2 → y = 20.
Why do we multiply by the reciprocal instead of dividing?
Multiplying by the reciprocal achieves the same result as dividing but is often simpler when dealing with fractions, avoiding complex fraction-divided-by-fraction steps.
What is the reciprocal of a whole number?
The reciprocal of a whole number n is 1/n. For example, the reciprocal of 5 is 1/5 because 5 × (1/5) = 1.
Where is multiplying by the reciprocal taught in Saxon Math Course 2?
This strategy appears in Chapter 7 of Saxon Math, Course 2, as part of Grade 7 equation-solving techniques.
What is the difference between a fraction and its reciprocal?
A fraction and its reciprocal have their numerator and denominator swapped. For example, 2/3 and 3/2 are reciprocals. Their product is always 1.
What mistakes do students make when multiplying by the reciprocal?
Common mistakes include forgetting to apply the multiplication to both sides of the equation, flipping the wrong fraction, or failing to simplify the resulting fraction.