How do you order these fractions smallest to largest: 22/25, 8/9, 7/8?

2 Answers
May 15, 2015

The answer is : 7/8 < 22/25 < 8/9.

In order to see which one is larger than another, you need to put them all to the same denominators :

22/25 = (22*9*8)/(25*9*8) = 1584/1800

8/9=(8*25*8)/(9*25*8) = 1600/1800

7/8=(7*25*9)/(8*25*9)=1575/1800

Therefore, 7/8 = 1575/1800 < 22/25 = 1584/1800< 8/9=1600/1800.

May 16, 2015

Same idea, but less actual arithmetic. (Is it easier? Probably not, but the arithmetic is easier, because we don't finish much of it.)

22/25, 8/9, 7/8

We can order them, pairwise (two at a time).

First the easy pair 8/9, 7/8

The least common denominator is 9xx8, I don't care what the number really is. It's the numerators I need to compare.

8/9 = (8xx8)/(9xx8) = 64/(9xx8)

7/8 = (7xx9)/(9xx8) = 63/(9xx8)

So 7/8 < 8/9
(At the end, we won't need this as a separate step, but it's not difficult to do.)

Second Pair
(Note: it is even quicker to observe that 7/8 is 1/8 less than 1, while 8/9 is 1/9 less than 1, so 7/8 <8/9)

22/25 , 8/9

The least common denominator is 9xx25, Again, I don't care what the number really is. It's the numerators I need to compare.

22/25 = (22xx9)/(25xx9)

8/9 = (25xx8)/(25xx9)

The numerators are:

22xx9 color(white)"ssssssssssssssssssss"and 25xx8, which we can rewrite as:

22xx(8+1)=22xx8 + 22 and (22+3)8 = 22xx8 +24

Whatever 22xx8 is, adding 24 will give a bigger total than adding 22. The second numerator is greater. So

22/25 < 8/9

Third pair
22/25, 7/8 Denominator 8xx25,

Numerators:

22xx8 color(white)"ssssssssssssssssssss"and 25xx7

22xx8 = 22(7+1)=22(7)+22 and 25xx7=(22+3)7 = 22(7)+21

Adding 22 will give a greater total than adding 21, so the first number is greater:

7/8 < 22/25

Final Answer

7/8 < 22/25 < 8/9