Subtracting Different types of Rational Numbers

Absolute value is the distance between 0 to the number on the number line. In other words, it is a number’s magnitude or size that is calculated using a number line. The absolute value (or modulus)  \(\left|a\right|\)  of a real number “a” is its non-negative value, regardless of its sign.

We have to add the opposite of a rational number to subtract a rational number. 

To subtract rational numbers with different signs, subtract the lesser absolute value from the greater absolute value. 

Now, let’s find the subtraction between two rational numbers  \(\frac{1}{3}\)  and  \(-\frac{2}{3}\)  using absolute values. Let’s estimate 0.4 – ( – 0.6 ) = 1

By adding the opposite, rewrite the difference as a sum.

\(\frac{1}{3}\) – (\(~-\frac{2}{3}\)) = \(\frac{1}{3}\) + ( \(\frac{2}{3}\) )

Because the signs are the same, add  \(\left|\frac{1}{3}\right|\)  and  \(\left|~-\frac{2}{3}\right|\).

\(\left|\frac{1}{3}\right|~+~\left|~-\frac{2}{3}\right|~=~\frac{1}{3}~+~\frac{2}{3}\)         (Finding the absolute values)

                                =  \(\frac{1}{3}~+~\frac{2}{3}\)     ( Writing)

                                = \(\frac{3}{3}\)              ( Adding the fractions and simplifying )

                                = 1.

Because  \(-\frac{2}{3}\)  has a negative sign, use a negative sign in the difference.

So, \(\frac{1}{3}~-~(-\frac{2}{3})\)  =  1. 

Check: Estimated value  = solution (   1  =   1 )

Solved Examples:

• Find – 3 \(\frac{3}{4} ~-~ \frac{1}{4}\)  = ?

Solution: Let’s estimate – 3 – 1 = – 4

By adding the opposite, rewrite the difference as a sum.

– 3 \(\frac{3}{4} ~-~ \frac{1}{4}\) = – 3 \(\frac{3}{4} ~+~ ( -\frac{1}{4} )\)

Because the signs are the same, add \(\left|-3~\frac{3}{4}\right|\) and \(\left|-~\frac{1}{4}\right|\).

\(\left|-3~\frac{3}{4}\right|\) + \(\left|-~\frac{1}{4}\right|\) = \(~3~\frac{3}{4}\) + \(\frac{1}{4}\)                    (Finding the absolute values)

                                = 3 + \(\frac{3}{4}\) + \(\frac{1}{4}\)            ( Writing  3 \(\frac{3}{4}\) as 3 + \(\frac{3}{4}\)  )

                                = 3 + \(\frac{4}{4}\)                   ( Adding the fractions and simplifying )

                                = 3 + 1 = 4.

Because – 3 \(\frac{3}{4}\)  and – \(\frac{1}{4}\)  both are negative, use a negative sign in the difference.

So, -3 \(\frac{3}{4}\)  – \(\frac{1}{4}\)  = – 4. 

Check: Estimated value = solution ( – 4 = – 4 )