Understanding the Mathematical Subject Reciprocal Fraction

The reciprocal of a fraction is found by inverting the fraction.  This is done by interchanging the numerator and denominator.  For example, the fraction ¾ has a numerator of 3 and a denominator of 4.  The reciprocal of ¾ has the opposite: a numerator of 4 and denominator of 3, which is 4/3. 

More examples are (5/10)^-1=10/5, (6/3)^-1=3/6, (9/11)^-1=11/9.  The negative one in the expressions means to the power of negative one, which is the reciprocal.  The first two expressions can be reduced to lower terms, but this does not change the value of the fraction: 5/10=1/2, (1/2)^-1=2=10/5 and 6/3=2, 2^-1=1/2=3/6.        

If the fraction is negative, the reciprocal is also negative.

The reciprocal of a fraction with the value one in the numerator is always an integer. This is because inverting the numerator and denominator will always place a one in the denominator and any fraction with a denominator of one is an integer. Examples are (1/5)^-1=5/1=5, (1/100)^-1=100/1=100, and (1/75)^-1=75/1=75.  

Mixed fractions can also be inverted by changing them to an improper fraction, then inverting it as usual. For example, 4 7/8 can be changed to an improper fraction. First, since 4×8=32, the improper fraction is found using 32/8+7/8=39/8. Switching the numerator and denominator makes it 8/39, which is the reciprocal of 4 7/8.

The next example is 7 9/10. The denominator is what is used to figure out what to do with 7. Since 7=70/10, the setup is 70/10+9/10=79/10. Now switching the numerator and denominator makes it 10/79, which is the reciprocal of 7 9/10.

Any fraction times its reciprocal equals one. This is because the reciprocal has the values of the numerator and denominator reversed, causing the same value to be in the denominator as in the numerator in the product. And any integer divided by itself equals one.  For example, the inverse of 4/5 is 5/4. The product of 4/5×5/4=4×5/5×4=20/20=1.

Reciprocals are used to divide fractions. When dividing two fractions, invert the fraction in the denominator, then multiply. This is done by finding the reciprocal of the fraction in the denominator and then multiplying that reciprocal by the fraction in the numerator. For example, dividing 7/3 by 6/5 is done as follows: 

(7/3)/(6/5)=(7/3)x(6/5)^-1=(7/3)x(5/6). Next, the numerators and denominators are multiplied together to find the respective numerator and denominator of the inverted fraction: (7×3)/(5×6)=35/18.

One important thing to remember is to reduce the fraction to its lowest terms before inverting a fraction to find its reciprocal. The first thing to do is find the prime factorization, then cancel out any primes that are both in the numerator and denominator. For example, the fraction 35/15 has the prime factorization 35=7×5 and 15=5×3, so the prime number 5 is canceled. This makes the fraction 35/15 equal to 7/3 in its lowest terms. The reciprocal is 3/7.