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Comparing Fractions

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description

  • Comparing like fractions with same denominators
  • Comparing unlike fractions with same numerators
  • Comparing unlike fractions with different numerators

notes

Comparing Fractions:

Consider `1/2 and 1/3`

The portion of the whole corresponding to `1/2` is clearly larger than the portion of the same whole corresponding to `1/3`. But often it is not easy to say which one out of a pair of fractions is larger. We should, therefore, like to have a systematic procedure to compare fractions. It is particularly easy to compare like fractions.

1. Comparing like fractions with the 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.

Between `4/5 and 3/5, 4/5` is greater.

Between `11/20 and 13/20, 13/20` is greater.

2. Comparing unlike fractions 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`.

  • In the same way, we can say `2/3 > 2/5`. In this case, the situation is the same as in the case above, except that the common numerator is 2, not 1. The whole is divided into a large number of equal parts for `2/5` than for `2/3`. Therefore, each equal part of the whole in the case of `2/3` is larger than that in the case of `2/5`. Therefore, the portion of the whole showing `2/3` is larger than the portion showing `2/5` and hence, `2/3 > 2/5`.

    We can see from the above example that if the numerator is the same in two fractions, the fraction with the smaller denominator is greater of the two.

    Thus, `1/8 > 1/10, 3/5 > 3/7, 4/9 > 4/11` and so on.

3. Comparing unlike fractions 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.

Example

Find answers to the following. Write and indicate how you solved them.

Is `5/9 "equal to" 4/5` ?

`5/9, 4/5`

Converting these into like fractions,

`5/9 = 5/9 xx 5/5 = 25/45`.

`4/5 = 4/5 xx 9/9 = 36/45`.

As `36/45 ≠ 25/45`

Therefore, `5/9  "is not equal to"  4`.

Example

Find answers to the following. Write and indicate how you solved them.

Is `9/16 "equal to" 5/9` ?

`9/16, 5/9`.

Converting these into like fractions,

`9/16 = 9/16 xx 9/9 = 81/144`.

`5/9 = 5/9 xx 16/16 = 80/144`.

As `81/144 ≠ 80/144`,

Therefore, `9/16  "is not equal to"  5/9`.

Example

Find answers to the following. Write and indicate how you solved them.

Is `4/5 "equal to" 16/20`?

`4/5, 16/20`

`16/20 = (4 xx 4)/(5 xx 4) = 4/5`.

Therefore, `4/5 = 16/20`.

Example

Ila read 25 pages of a book containing 100 pages. Lalita read `2/5` of the same book. Who reads less?

Numbers of pages read by Lalita = `2/5 xx 100` = 40

Number of pages read by Ila = 25

Hence, Ila has read less number of pages.

Example

Asha and Samuel have bookshelves of the same size partly filled with books. Asha’s shelf is `5/6`th full and Samuel’s shelf is `2/5`th full. Whose bookshelf is more full? By what fraction?

Fraction of Asha’s shelf = `5/6`

Fraction of Samuel’s shelf = `2/5`

Converting these into like fractions,

`5/6 = 5/6 xx 5/5 = 25/30`.

`2/5 = 2/5 xx 6/6 = 12/30`

`25/30 > 12/30`.

Clearly, Asha’s bookshelf is more full.

Difference = `5/6 - 2/5 = 25/30 - 12/30 = 13/30`.

Example

Jaidev takes `2 1/5` minutes to walk across the school ground. Rahul takes `7/4` minutes to do the same. Who takes less time and by what fraction?

Time taken by Jaidev = `2 1/5 "minutes" = 11/5` min

Time taken by Rahul = `7/4` min

Converting these into like fractions,

`11/5 = 11/5 xx 4/4 = 44/20`

`7/4 = 7/4 xx 5/5 = 35/20`

As 44 > 35,

`11/5 > 7/4`

Hence, Rahul takes lesser time.

Difference = `11/5 - 7/4`

= `44/20 - 35/20 = 9/20` min.

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