Improper Fractions

In arithmetic, there are three types of fractions. Proper fraction, improper fraction, and mixed fraction. In this article you fill find all the topics related to improper fractions.

Improper Fraction Definition

It is defined as a fraction whose numerator is greater than the denominator and so the fraction is always greater than one.

Examples of Improper Fractions:

\(\frac{17}{5}\) \(\frac{9}{4}\) \(\frac{16}{3}\) \(\frac{5}{2}\)

Converting Improper Fractions to Mixed Fractions

To convert an improper fraction to a mixed fraction divide the denominator by the numerator. The quotient becomes the whole number, the reminder will be the numerator, and the divisor becomes the denominator.

Let us consider an example. Convert \(\frac{17}{4}\) to a mixed fraction.

When we divide 17 by 4, quotient is 4 and reminder is 1. So the mixed fraction is \(4\frac{1}{4}\).

Although any improper fraction can be converted to a mixed fraction, there are situations where expressing a fraction as an improper fraction minimizes a lot of confusion especially when the fractions are expressed in calculations.

For example, consider \(3 + 6 \frac{2}{3}\).

Is it \(3 + 6 + \frac{2}{3}\) or \(3 + 6 \times \frac{2}{3}\)?

This confusion can be removed by writing it as \(3 + \frac{20}{3}\)

However, improper fraction cannot be used in everyday life. It makes far more sense to say “The drive takes \(4\frac{1}{4}\) hours.” than to say “The drive takes \(\frac{17}{4}\) hours.”

Adding Improper Fractions

Addition of improper fractions can have two scenarios. If the denominators of all the fractions are equal, we can add all the numerators and keep the same denominator.

For example, \(\frac{17}{4}\) + \(\frac{9}{4}\) + \(\frac{5}{4}\)

Since the denominator of all three fractions are equal, we just add all the numerators:

17+9+5 = 31

Therefore,

\(\frac{17}{4}\) + \(\frac{9}{4}\) + \(\frac{5}{4}\) = \(\frac{31}{4}\)

If the denominators of fractions are not equal, the process is slightly different and involves calculation of least common multiple of the denominators.

You can learn the calculation of least common multiples here.

Consider the example,

\(\frac{15}{3}+\frac{3}{4}+\frac{5}{2}\)

The least common multiple of 3, 4, and 2 is 24.

Now the fraction we get by adding all these fractions will have 24 as the denominator.

Divide the LCM by each of the denominators and multiply the quotient by the numerators.

The numerator of the new fraction will be the sum of all the numbers obtained in the previous step.

Example, in the first fraction, the denominator is 3. LCM 24 divided by 3 is 8. Numerator is 15. 15 x 8 is 120. Similarly for other two fractions the numbers are 18, and 60.

Therefore, \(\frac{15}{3}+\frac{3}{4}+\frac{5}{2}\)

= \(\frac{120+18+60}{24}\)

= \(\frac{198}{24}\)

= \(8\frac{1}{4}\)

Subtraction of improper fractions can be carried out similarly. If the previous example was a subtraction problem, instead of adding the numbers 120, 18, and 60, we subtract.

So 120 -18 – 60 = 42. The answer is \(\frac{42}{24}\).

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Practise This Question

The cube of an even natural number is