Dividing One by a Fraction
Dividing 1 by a fraction always gives the reciprocal of that fraction. So 1 divided by 2/3 equals 3/2, because you flip the fraction. This shortcut works because division by a fraction is the same as multiplying by its reciprocal. The concept appears in Chapter 3 of Saxon Math Course 2 and builds the foundation for 7th grade fraction division, where students extend this idea to divide any number or fraction by another fraction.
Key Concepts
Property When you divide the number 1 by any fraction, the answer is always the reciprocal of that fraction. So, for any fraction $\frac{a}{b}$, the result of $1 \div \frac{a}{b}$ is simply $\frac{b}{a}$.
Examples How many $\frac{2}{3}$s are in 1? $1 \div \frac{2}{3} = \frac{3}{2}$ How many $\frac{9}{10}$s are in 1? $1 \div \frac{9}{10} = \frac{10}{9}$.
Explanation Imagine asking, “How many $\frac{1}{4}$s are in 1?” The answer is 4, which is the reciprocal of $\frac{1}{4}$. This trick works every time! Dividing 1 by a fraction is as simple as flipping that fraction upside down.
Common Questions
What happens when you divide 1 by a fraction?
You get the reciprocal of that fraction. For example, 1 divided by 2/3 equals 3/2. Simply flip the numerator and denominator of the fraction.
Why does dividing 1 by a fraction give the reciprocal?
Because dividing by a fraction means multiplying by its reciprocal. Since 1 times anything is itself, 1 x (b/a) = b/a, which is just the flipped fraction.
How many 2/3s are in 1?
There are 3/2, or 1 and 1/2, groups of 2/3 in 1. You can verify: (2/3) x (3/2) = 6/6 = 1.
How does dividing 1 by a fraction relate to dividing other numbers by fractions?
Once you know that 1 divided by a/b equals b/a, you can extend this to any dividend. To divide n by a/b, multiply n by b/a. The principle is the same.
What are common mistakes when dividing 1 by a fraction?
Students sometimes flip the wrong fraction or forget to flip at all and instead multiply 1 by the original fraction. Always remember: dividing means flipping and multiplying.
Is this skill covered in 7th grade math?
Yes. Saxon Math Course 2, Chapter 3, teaches dividing 1 by a fraction as the entry point to understanding all fraction division problems.