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Comparison of Rational Numbers

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Comparison of Rational Numbers:

1. Comparing Two Positive Rational Number:

a. Comparing like a rational number with same denominators:

Let us compare two like rational numbers: `3/8 and 5/8`.

In both the fraction, the whole is divided into 8 equal parts. For `3/8 and 5/8`,

We take 3 and 5 parts respectively out of the 8 equal parts. Clearly, out of 8 equal parts, the portion corresponding to 5 parts is larger than the portion corresponding to 3 parts.
Hence, `5/8 > 3/8`.

Note the number of the parts taken is given by the numerator. It is, therefore, clear that for two fractions with the same denominator, the fraction with the greater numerator is greater.

b. Comparing unlike rational number with the same numerators:

Which is greater `1/3 or 1/5`?

In `1/3`, we divide the whole into 3 equal parts and take one. In `1/5`, we divide the whole into 5 equal parts and take one. Note that in `1/3`, the whole is divided into a smaller number of parts than in `1/5`. The equal part that we get in `1/3` is, therefore, larger than the equal part we get in `1/5`. Since in both cases we take the same number of parts (i.e. one), the portion of the whole showing `1/3` is larger than the portion showing `1/5`, and therefore `1/3 > 1/5`.

c. Comparing unlike rational number with different numerators:

Compare `5/6 and 13/15`.

Solution:

The fractions are unlike. We should first get their equivalent fractions with a denominator which is a common multiple of 6 and 15.

Now, `(5 × 5)/(6 × 5) = 25/30, (13 × 2)/(15 × 2) = (26)/(30)`

Since, `(26/30) > (25/30) "we have" (13/15) > (5/6)`

Why LCM?

The product of 6 and 15 is 90; obviously 90 is also a common multiple of 6 and 15. We may use 90 instead of 30; it will not be wrong. But we know that it is easier and more convenient to work with smaller numbers. So, the common multiple that we take is as small as possible. This is why the LCM of the denominators of the fractions is preferred as the common denominator.

2. Comparing Two Negative Rational Number:

To compare two negative rational numbers, we compare them ignoring their negative signs and then reverse the order.

For example, to compare `-7/5 and -5/3`, we first compare `7/5 and 5/3`.

We get `7/5 < 5/3` and conclude that `(-7)/5 > (-5)/3`.

  • To compare rational numbers `(-3)/(-5) and (-2)/(-7)` reduce them to their standard forms and then compare them.

    `(-3)/(-5) = ((-3) xx (-1))/((-5) xx (-1)) = 3/5`.

    `(-2)/(-7) = ((-2) xx (-1))/((-7) xx (-1)) = 2/7`.

    `3/5 > 2/7`.

3. Comparing Positive Rational Number and Negative Rational Number:

  • Comparison of a negative and a positive rational number is obvious. A negative rational number is to the left of zero whereas a positive rational number is to the right of zero on a number line. So, a negative rational number will always be less than a positive rational number.

    Thus, `- 2/7 < 1/2`.

Example

Compare the numbers `5/4 and 2/3`. Write using the proper symbol of <, =, >.

`5/4 = (5 xx 3)/(4 xx 3) = 15/12.`

`2/3 = (2 xx 4)/(3 xx 4) = 8/12.`

`15/12 > 8/12`

∴ `5/4 > 2/3`.

Example

Compare the rational numbers `(-7)/9 and 4/5`.

A negative number is always less than a positive number.

Therefore, `(-7)/9 < 4/5`.

Example

Compare the numbers `(-7)/3 and (-5)/2`.

Let us first compare `7/3 and 5/2`.

`7/3 = (7 xx 2)/(3 xx 2) = 14/6`,

`5/2 = (5 xx 3)/(2 xx 3) = 15/6`

`14/6 < 15/6`

∴ `7/3 < 5/2`

∴ `(-7)/3 < (-5)/2`.

Example

`3/5 and 6/10` are rational numbers. Compare them.

`3/5 = (3 xx 2)/(5 xx 2) = 6/10`

∴ `3/5 = 6/10`

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