Question

How do you order the fractions from least to greatest: `11/12, 7/8, 15/16`?

Answer 1

`7/8 < 11/12 < 15/16`

Explanation:

To determine which between two fractional numbers `A` and `B` is larger, perform cross-multiplication.

1. Multiply the numerator of `A` with the denominator of `B`. The resulting product would be `A`'s weighted value relative to `B`.

`A = 11/12`

`B = 7/8`

`W_(A_B) = 11* 8 = 88`

1. Similarly, multiply the numerator of `B` with the denominator of `A`. The resulting product would be `B`'s weighted value relative to `A`

`W_(B_A) = 7 * 12 = 84`

1. Compare the weighted value of `A` and `B`. Whichever has the higher weighted value, is larger.

`W_(A_B) = 88 ? W_(B_A) = 84`

`W_(A_B) > W_(B_A)`

`=> A > B`

Note the use of the word relative. This is because the computed weighted value is only valid between the two numbers. If a third number is involved, the above steps need to be performed for each possible pairing

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Do the same for `A` and `C`

`A = 11/12`

`C = 15/16`

`W_(A_C) = 11 * 16 = 176`

`W_(C_A) = 12 * 15 = 180`

`W_(A_C) = 176 ? W_(C_A) = 180`

`=> W_(A_C) < W_(C_A)`

`=> A < C`

`=> C > A`

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Since A > B, and C > A, it should follow that C > B.
Hence, we will no longer check which is bigger between B and C.

`7/8 < 11/12 < 15/16`

Answer 2

`7/8 < 11/12 < 15/16`

Explanation:

You cannot compare fractions which have different denominators.
Convert all the fractions to a common denominator.

In this case the `LCM = 48`

`11/12 xx4/4" " 7/8 xx6/6" " 15/16 xx3/3`

`=44/48" "=42/48" "=45/48`

It is now possible to compare and arrange them in order:

`42/48 < 44/48 < 45/48`

Using the original fractions:

`7/8 < 11/12 < 15/16`

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Another method works in this case if you realise that each fraction is only 1 part short of being a whole number.

`11/12` is `1/12` less than 1,

`7/8` is `1/8` less than 1

`15/16` is `1/16` less than 1.

Arranging in size `1/8 > 1/12 > 1/16`

Therefore `7/8`is furthest from 1 and must be the smallest.

`15/16` is closest to 1 and is therefore the biggest.