Ordering Fractions, Decimals and Percentages: Quick Comparison Methods

The challenge

Exam questions often ask you to put a mixed set of values in order:

Arrange 3/8, 0.4, 35%, and 2/5 from smallest to largest.

The values are in three different formats. To compare them, you need a common format.

Method 1: Convert everything to decimals

This is usually the fastest approach.

1. Convert each value to a decimal.
2. Compare the decimals.
3. Write the original values in the correct order.

Worked example

Order: 3/8, 0.4, 35%, 2/5.

• 3/8 = 3 ÷ 8 = 0.375
• 0.4 = 0.4 (already a decimal)
• 35% = 35 ÷ 100 = 0.35
• 2/5 = 2 ÷ 5 = 0.4

Now order the decimals: 0.35, 0.375, 0.4, 0.4.

Notice 0.4 and 2/5 are equal. So the final order is:

35% < 3/8 < 0.4 = 2/5.

Method 2: Convert everything to fractions with a common denominator

This avoids decimals entirely and gives exact comparisons.

Worked example

Order: 2/3, 5/8, 3/4.

Find the LCM of 3, 8, and 4. LCM = 24.

• 2/3 = 16/24
• 5/8 = 15/24
• 3/4 = 18/24

Compare numerators: 15, 16, 18.

Order: 5/8 < 2/3 < 3/4.

Method 3: Convert everything to percentages

Some students prefer working in percentages.

Worked example

Order: 1/3, 0.3, 32%.

• 1/3 = 33.33%
• 0.3 = 30%
• 32% = 32%

Order: 0.3 (30%) < 32% < 1/3 (33.33%).

Which method should you use?

• Decimals work best when the fractions convert cleanly and you are comfortable with decimal comparison.
• Common denominators work best when you want exact comparisons and the denominators are small.
• Percentages work well when some values are already given as percentages.

In practice, most students find the decimal method fastest.

Comparing just two fractions

If you only need to compare two fractions, the cross-multiplication shortcut is efficient:

Compare a/b and c/d by computing a × d and c × b.

• If a × d > c × b, then a/b > c/d.
• If a × d < c × b, then a/b < c/d.
• If equal, the fractions are equal.

Example: Compare 3/7 and 5/12.

3 × 12 = 36, 5 × 7 = 35. Since 36 > 35, 3/7 > 5/12.

Handling negative values

Negative fractions, decimals, and percentages follow the same ordering rules. Remember that -0.5 is less than -0.3 (further from zero means smaller for negatives).

Common mistakes when ordering

Comparing numerators without common denominators. You cannot say 3/8 > 2/5 just because 3 > 2. The denominators matter.

Forgetting to convert percentages. 35% is 0.35, not 35. Comparing 0.4 to 35 without converting is a common exam error.

Decimal place confusion. 0.4 and 0.40 are the same, but 0.4 and 0.04 are very different. Be careful when adding trailing zeros for comparison.

Mixing up ascending and descending. Read the question carefully. "Smallest to largest" is ascending. "Largest to smallest" is descending.

Exam-style practice questions

Question 1

Put in ascending order: 7/20, 0.36, 1/3.

Working:

• 7/20 = 0.35
• 0.36 = 0.36
• 1/3 = 0.333...

Order: 1/3, 7/20, 0.36.

Question 2

Put in descending order: 60%, 5/8, 0.59, 3/5.

Working:

• 60% = 0.6
• 5/8 = 0.625
• 0.59 = 0.59
• 3/5 = 0.6

Order (descending): 5/8, 60% = 3/5, 0.59.

Question 3

Which is larger: 4/9 or 45%?

4/9 = 0.444... = 44.4%.

45% > 44.4%, so 45% is larger.

Using simplification to help

Sometimes simplifying a fraction first reveals a familiar value. 8/20 simplifies to 2/5 = 0.4, which is much easier to compare. Use the Reducing Fractions Calculator when you are not sure of the GCF.

Placing values on a number line

Drawing a rough number line from 0 to 1 (or wider if needed) and marking each value is a visual way to order them. This works especially well for students who find numerical comparison confusing.

0------|------|------|------|------|------|------|------|------|------|------1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Mark each converted decimal on this line and read them off left to right for ascending order.

FAQ

Can I always convert to decimals? Yes. Every fraction and percentage has a decimal equivalent. Some decimals repeat, but for ordering purposes you only need enough decimal places to distinguish the values.

How many decimal places do I need? Usually 2 or 3 is enough. If two values are very close (like 0.333... and 0.335), use more places until the difference is clear.

What if two values are equal? State that they are equal. In an ordering question, place them next to each other. For example: 0.4 = 2/5.

Does this work for negative numbers? Yes. Convert to decimals and use the standard rules for ordering negatives. For the full conversion toolkit, see the Fractions to Decimals and Percent guide.