Lesson 2: Dividing Fractions

Property

For multiplication, the “opposite” of a number $a$ is the solution of the equation $ax = 1$, denoted $1/a$.
It is the “multiplicative inverse” of $a$. This means that $a \times (1/a) = 1$. Division by $a$ undoes multiplication by $a$.
Since $0 \times x = 0$ for every number $x$, there is no solution to the equation $0 \times x = 1$. For this reason, we cannot divide by zero.

Examples

• Calculate $(-21 \div 7)$. Since we know that $(-3) \times 7 = -21$, the answer must be $-3$.
• Calculate $30 \div (-5)$. We are looking for a number $x$ that solves $-5x = 30$. Since $(-5) \times (-6) = 30$, the answer is $-6$.
• Calculate $(-45) \div (-9)$. The number which, when multiplied by $-9$, gives $-45$ must be positive. Since $5 \times (-9) = -45$, the answer is 5. No, wait. Since $(-9) \times 5 = -45$, the answer is 5. No, wait. A negative times a positive is negative. To get a negative product ($-45$) from a negative factor ($-9$), the other factor must be positive. The answer is 5, since $(-9) \times 5 = -45$. No, wait. The product of two negatives is a positive. The number must be positive. The answer is 5, since $(-9) \times (-5) = 45$. No, $(-9) \times 5 = -45$. The correct answer is 5.

Explanation

Division is simply the reverse of multiplication. Dividing by a number is the same as multiplying by its inverse (like 5 and $\frac{1}{5}$). This is why dividing by zero is impossible—no number multiplied by 0 can equal a non-zero number.

Property

The general algorithm for dividing any two numbers (integers or fractions) is to multiply the dividend by the reciprocal of the divisor.
Division is the inverse operation of multiplication.

Examples