Comparing Fractions Using Half (1/2) as a Benchmark - Third Grade Mathematics

When fractions have different numerators AND different denominators, comparing them directly can seem tricky at first. Fortunately, mathematicians have a trusty reference point called a benchmark fraction. The most useful benchmark in the world is one-half (1/2). By comparing each fraction to 1/2, you can quickly decide which fraction is larger without drawing complicated pictures or calculating common denominators.

What Is a Benchmark?

A benchmark is a familiar, well-known standard that you use to judge and compare other things. Think of a benchmark like a height mark on a wall:
- If Child A is taller than the line, and Child B is shorter than the line, you know immediately that Child A is taller than Child B!
In fractions, 1/2 acts as that magic dividing line.

0                         1/2                         1
+--------------------------+--------------------------+
<-- Less than 1/2 -------->|<------- Greater than 1/2 ->

How to Determine If a Fraction Is Less Than, Equal To, or Greater Than 1/2

To test any fraction against 1/2, look at its denominator:
1. Find what half of the denominator would be.

2. Compare the numerator to that half number:
- If the numerator is LESS than half the denominator -> Fraction < 1/2. - If the numerator is EXACTLY half the denominator -> Fraction = 1/2. - If the numerator is GREATER than half the denominator -> Fraction > 1/2.

Let us test several fractions:

Fraction: 3/8
- Denominator is 8. Half of 8 is 4.
- Numerator is 3. Since 3 < 4, 3/8 is LESS than 1/2.

Fraction: 4/8
- Numerator is 4. Since 4 = 4, 4/8 is EQUAL to 1/2.

Fraction: 5/8
- Numerator is 5. Since 5 > 4, 5/8 is GREATER than 1/2.

Fraction: 2/6
- Denominator is 6. Half of 6 is 3.
- Numerator is 2. Since 2 < 3, 2/6 is LESS than 1/2.

Fraction: 4/6
- Numerator is 4. Since 4 > 3, 4/6 is GREATER than 1/2.

Comparing Two Fractions Using the Benchmark 1/2

Now watch the benchmark strategy solve a comparison problem with different numerators and denominators!

Problem: Compare 2/6 and 5/8.
- Step 1: Compare 2/6 to 1/2:
Half of 6 is 3. Since 2 < 3, 2/6 is less than 1/2.
- Step 2: Compare 5/8 to 1/2:
Half of 8 is 4. Since 5 > 4, 5/8 is greater than 1/2.
- Step 3: Draw your conclusion:
Since 2/6 is less than 1/2, and 5/8 is greater than 1/2: 2/6 < 5/8!

0               2/6        1/2        5/8             1
+----------------+----------+----------+--------------+
                ^                      ^
          (Below 1/2)            (Above 1/2)

You solved the comparison without doing any complicated arithmetic!

What About 0 and 1 as Benchmarks?

You can also use 0 and 1 as benchmarks:
- Close to 0: unit fractions with large denominators, like 1/8 or 1/10.

- Close to 1: fractions where the numerator is just 1 less than the denominator, like 7/8 or 5/6.


If you compare 1/8 and 5/6:
- 1/8 is very close to 0.

- 5/6 is very close to 1.
Clearly, 5/6 > 1/8!

Chapter Practice Exercises

  1. Sort each fraction into the correct column: "Less than 1/2", "Equal to 1/2", or "Greater than 1/2": 3/6, 2/8, 5/8, 1/4, 4/8, 5/6, 2/5, 7/10
  2. Compare using the benchmark 1/2 (write >, <, or =): a. 1/4 ___ 5/8 b. 3/6 ___ 2/4 c. 4/6 ___ 3/8 d. 7/8 ___ 2/6
  3. Maya read 3/8 of her book. Noah read 4/6 of his book. Who read more than half of their book? Who read less than half?
  4. Explain how you can tell without drawing a picture that 5/8 is greater than 2/5. (Hint: compare both to 1/2).
  5. Name a fraction that has a denominator of 8 and is greater than 1/2. Name a fraction that has a denominator of 6 and is less than 1/2.

Solutions and Step-by-Step Answers

  1. Sorting fractions:
  2. Less than 1/2: 2/8 (half of 8 is 4; 2 < 4); 1/4 (half of 4 is 2; 1 < 2); 2/5 (half of 5 is 2.5; 2 < 2.5).
  3. Equal to 1/2: 3/6 (3 is half of 6); 4/8 (4 is half of 8).
  4. Greater than 1/2: 5/8 (5 > 4); 5/6 (5 > 3); 7/10 (7 > 5).
  5. Comparisons using benchmark: a. 1/4 < 5/8 (1/4 < 1/2, while 5/8 > 1/2). b. 3/6 = 2/4 (both equal 1/2). c. 4/6 > 3/8 (4/6 > 1/2, while 3/8 < 1/2). d. 7/8 > 2/6 (7/8 > 1/2, while 2/6 < 1/2).
  6. Maya read 3/8 (less than half, since 3 < 4). Noah read 4/6 (more than half, since 4 > 3). Noah read more than half, and Maya read less than half.
  7. For 5/8: half of 8 is 4; since 5 > 4, 5/8 > 1/2. For 2/5: half of 5 is 2.5; since 2 < 2.5, 2/5 < 1/2. Since 5/8 is greater than 1/2 and 2/5 is less than 1/2, 5/8 > 2/5.
  8. Greater than 1/2 with denominator 8: 5/8, 6/8, or 7/8. Less than 1/2 with denominator 6: 1/6 or 2/6.