What happens to the quotient when you divide a fraction by a fraction and why?

Answer 1

To divide by a fraction, it is inverted and multiplied.

You can see this by using the example 2/3 ÷ 3/7:

1. Write the division as a fraction with the first fraction as the numerator and the second fraction as the denominator;
2. Convert this new fraction into an equivalent fraction by multiplying both numerator and denominator by the divisor turned upside down (with the denominator as the numerator and the numerator as the denominator)
3. Do the multiplication in the denominator as normal
4. Simplify the denominator and the overall fraction

1. 2/3 ÷ 3/7 = (2/3)/(3/7)
2. (2/3)/(3/7) × (7/3)/(7/3) = (2/3 × 7/3)/(3/7 × 7/3)
3. (2/3 × 7/3)/(3/7 × 7/3) = (2/3 × 7/3)/((3×7)/(7×3) = (2/3 × 7/3)/(21/21)
4. (2/3 × 7/3)/(21/21) = (2/3 × 7/3)/1 = (2/3 × 7/3) ÷ 1 = 2/3 × 7/3

Thus 2/3 ÷ 3/7 = 2/3 × 7/3 which can be done by normal multiplication.

When a fraction is so inverted it is called the reciprocal of the fraction*.

Through this inversion of the divisor, and conversion to a multiplication you can now see what happens to the dividend to create the quotient:

Ignoring the sign:

• if divisor < 1, its reciprocal > 1 → quotient > dividend
• if divisor = 1, its reciprocal = 1 → quotient = dividend
• if divisor > 1, its reciprocal < 1 → quotient < dividend

----------------------

* strictly speaking the reciprocal of a fraction (or any number) is 1 divided by the fraction. Following the above, eg:

1 ÷ 3/7 = 1/(3/7) = (1×(7/3))/((3/7)×(7/3) = (7/3)/(21/21) = (7/3)/1 = 7/3 ÷ 1 = 7/3

This also follows from whole numbers being the same as a fraction with a denominator of 1, eg:

1 ÷ 24 = 1 ÷ 24/1 = 1 × 1/24 = 1/24 → reciprocal of 24 is 1/24

1 ÷ (1/24) = 1 × 24/1 = 24/1 = 24 → reciprocal of 1/24 is 24.

Answer 2

Supose the divisor is q. If

• q < -1 : the quotient has a smaller magnitude but changes sign.
• q = -1 : the quotient changes sign.
• -1 < q < 0 : the quotient has a larger magnitude but changes sign.
• q = 0 : the quotient is not defined.
• 0 < q < 1 : the quotient has a larger magnitude and has the same sign.
• q = 1 : the quotient remains unchanged.
• q > 1 : the quotient has a smaller magnitude but the same sign.

All of these follow from the basic rules of division.