All About Fractions: Division and Multiplication

A division of a number by a fraction can be represented in several ways. The division of fractions is a very important topic of mathematics. It is denoted just by placing a horizontal bar between them. However, when dividing fractions, it is often helpful to use this bar. The bit being divided should have a horizontal bar placed over it, and the dividend or numerator should be underlined:

place | x | under

Division involving an even number of integer values can also be shown using a vinculum (a horizontal line joining the topmost element with the bottommost)—divisor/dividend.

Methods of Multiplying Fractions

There are several methods using which one can multiply fractions. For example, Fractions A and B can be multiplied using cross multiplication or by multiplying the numerators and denominators separately.

  • The simplest method of multiplying two fractions is to multiply their numerators and denominators and simplify if possible.
  • If this is not possible (the results would yield a fraction with more than three terms), then another approach must be taken. One such method is to find equivalent fractions that have a common factor, then reduce the fractions to the lowest terms, and finally use cross multiplication.
  • Another method of multiplying fractions is known as “cancelling,” where like factors in the numerator and denominator are cancelled out.
  • The last method of multiplying fractions is known as “column multiplication,” where the numerators and denominators are multiplied separately, then reduced to lowest terms.

Division of Fractions

One can also express the division of fractions numerically by writing the dividend over the divisor with a horizontal line. For example, A mistake often made in algebra is trying to divide by zero. An expression containing zero is undefined, so the phrase has no value (the answer is “irrational”). The number 0/0 (called “indeterminate”) represents an ill-defined or meaningless quantity. Therefore, any attempt to use this result will yield an incorrect result. It would be helpful to remember that dividing by zero cannot be done; however, there are some rules for how it should be written:

  • If one tries to divide by zero, it changes into one divided by (which equals 0) or 0 divided by 0, which equals 0/0. It then becomes a question of what you would like to do with that value.
  • In most cases, one would prefer to have the answer as a fractional value (that is 1/1 or .5), but in some other cases it might be easier to use another form (for example, 1 has infinite decimal digits, whereas 0.000… = 0). This can lead to differences when working with fractions.
  • The second case seems less intuitive because canceling out factors does not seem obvious. Furthermore, the first form may look confusing because the denominator disappears. However, this is how division works; one can take five away from 10 and end up with 2, 3/5 can be written as, and so on.
  • It is also possible to create a mixed fraction whenever the numerator and denominator are not integers. For example: If this is done, then another rule must be kept in mind. If one cannot cancel out like factors (as was shown earlier), then one would need to reduce the value under the bar to the lowest terms:


  • In general, when multiplying fractions, the answer is always a fraction in the lowest terms.
  • To figure out when dividing fractions, it’s helpful to know when the denominator becomes 0/0.
  • Dividing fractions involves knowing what to do with mixed numbers and how to simplify them.

A real-life example:

When it comes to dividing fractions, some people might prefer using cancelling factors. For example, if there were two pizzas and five friends, one could divide the first pizza so that everyone gets an equal slice of pizza (see picture below). If the slices were not equal, either someone would get more or less than 1/5 of a pizza, which is unfair.

By dividing the first pizza into five equal slices, everyone gets their slice. By dividing by five, all of the slices are cut to be equal in length. If, for some reason, there were only four people eating pizza, then the equation would need to be changed to 5/4 so that one person still receives an equal slice of pizza.

Cuemath is the one website that provides accurate and efficient information on all mathematics topics. They have expert teachers that help the students understand mathematics easily.

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