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Ordering fractions

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Fractions

Ordering fractions

You cannot compare fifths with eighths directly. Make them the same first โ€” either same denominator, or both as decimals.

Add: 1/9 + 3/9 = 4/9Subtract: 5/6 โˆ’ 2/6 = 3/6add / subtract the top numbers โ€” keep the bottom the same

Two ways to compare

To put fractions in order of size you must first make them comparable. There are two ways: 1. Write them with the SAME DENOMINATOR. 2. Write them all as DECIMALS. Either works โ€” pick whichever is easier for the numbers you have.

Using a common denominator

Compare 3/4 and 5/8. The denominators are 4 and 8. Since 8 is a multiple of 4, use 8: 3/4 = 6/8 Now compare 6/8 and 5/8. Same denominator, so just compare the numerators: 6 > 5. So 3/4 is bigger than 5/8.

Using decimals

Divide the numerator by the denominator. 3/4 = 0.75 and 5/8 = 0.625, so 3/4 is bigger. Some fractions give decimals that go on forever: 1/7 = 0.1428โ€ฆ The three dots (an ellipsis) show it continues. Some repeat a pattern: 2/3 = 0.6666โ€ฆ These are RECURRING decimals. You can write 0.6666โ€ฆ as 0.6ฬ‡ โ€” the dot above shows which digit repeats. For 0.567567567โ€ฆ you mark the first and last of the repeating block.

Mixed numbers

With mixed numbers, compare the WHOLE-NUMBER parts first. Only if those are equal do you need to compare the fractional parts. 3 1/5 is bigger than 2 9/10, because 3 > 2 โ€” you never need to look at the fractions at all.

Worked examples

Compare 3/4 and 5/8 with a common denominator

  1. 8 is a multiple of 4, so use eighths.
  2. 3/4 = 6/8.
  3. 6 > 5.

Answer: 3/4 is bigger

Compare 3/4 and 5/8 using decimals

  1. 3 รท 4 = 0.75.
  2. 5 รท 8 = 0.625.

Answer: 3/4 is bigger โ€” same answer, other method

Write 2/3 as a decimal

  1. 2 รท 3 = 0.6666โ€ฆ
  2. The digits repeat forever.

Answer: 0.6666โ€ฆ, a recurring decimal

Compare 3 1/5 and 2 9/10

  1. Compare the whole-number parts first: 3 and 2.
  2. 3 > 2.

Answer: 3 1/5 is bigger โ€” the fractions never matter

Compare 2/5 and 1/2

  1. Use tenths: 2/5 = 4/10 and 1/2 = 5/10.
  2. 5 > 4.

Answer: 1/2 is bigger

What does 0.6ฬ‡ mean?

  1. The dot sits above the digit that repeats.

Answer: 0.6666โ€ฆ going on forever

Where you see this in real life

  • Two recipes ask for 3/4 cup and 5/8 cup. Converting both to eighths shows the first needs more.
  • 1/3 of a pizza sounds less than 3/8, but 1/3 = 8/24 and 3/8 = 9/24 โ€” so 3/8 really is the bigger slice.

Key words

Numerator
the top number of a fraction
Denominator
the bottom number of a fraction
Common denominator
the same bottom number for two or more fractions
Recurring decimal
a decimal with a digit or block that repeats forever
Mixed number
a whole number and a fraction together
Improper fraction
a fraction whose numerator is bigger than its denominator
Equivalent fraction
a different way of writing the same fraction
Simplest form
a fraction written with the smallest possible numbers
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