Compare & Order Decimals
Complete Notes & Formulae for 5th Grade Math
Equivalent Decimals
Definition: Equivalent decimals have the same value, even if their digits look different. You can add zeros to the right of the decimal without changing its value.
Rule: $a = a.0 = a.00 = a.000$.
Example: $0.3 = 0.30 = 0.300 = \frac{3}{10}$
Example: $2.4 = 2.40 = 2.400$
Rule: $a = a.0 = a.00 = a.000$.
Example: $0.3 = 0.30 = 0.300 = \frac{3}{10}$
Example: $2.4 = 2.40 = 2.400$
Rules: How to Compare Decimals
- Align decimal points and compare left to right.
- Compare the whole number part first.
- If equal, compare tenths, then hundredths, then thousandths, and so on.
- If one number has fewer decimal places, fill missing places with zeros.
- The first digit from the left that is different determines which is larger.
Example: Compare $5.612$ and $5.071$:
- Whole part: both 5
- Tenths: 6 vs 0 ($6 > 0$), so $5.612 > 5.071$.
- Whole part: both 5
- Tenths: 6 vs 0 ($6 > 0$), so $5.612 > 5.071$.
Example: Compare $4.13$ and $4.257$:
- Whole number: both 4
- Tenths: $4.13$ has 1, $4.257$ has 2 ($1 < 2$), so $4.13 < 4.257$
- Whole number: both 4
- Tenths: $4.13$ has 1, $4.257$ has 2 ($1 < 2$), so $4.13 < 4.257$
Like Decimals: Same number of decimal places (easier to compare)
Unlike Decimals: Add zeros to equalize decimal places and compare.
Unlike Decimals: Add zeros to equalize decimal places and compare.
Compare Decimals Using Grids & Number Lines
- Grid Model: Shade the fraction of the grid to represent each decimal. More shaded = greater value.
Example: On a 10×10 grid, 0.35 means 35 squares shaded. 0.57 means 57 squares shaded, so $0.57 > 0.35$. - Number Line: Decimals to the right are greater.
Example: To compare $0.48$ and $0.52$, find their position between 0 and 1. $0.48$ is left of $0.52$, so $0.48 < 0.52$.
Order Decimals: Ascending & Descending
Ascending Order: Smallest to largest.
Descending Order: Largest to smallest.
Method: List all numbers, add zeros to equalize decimal places, then compare each place left to right.
Descending Order: Largest to smallest.
Method: List all numbers, add zeros to equalize decimal places, then compare each place left to right.
Example: Order $17.102, 17.243, 17.05$
- All whole parts 17
- Tenths: 1, 2, 0 (so $17.05 < 17.102 < 17.243$)
- All whole parts 17
- Tenths: 1, 2, 0 (so $17.05 < 17.102 < 17.243$)
Comparing & Ordering Decimals: Word Problem Strategies
- Identify each decimal in the problem and write them in aligned form (same decimal places).
- Use place value, grids, or number lines to visually compare or order if needed.
- Read the question carefully to see if it asks for largest, smallest, or order/rounding.
- For rounding, apply regular rounding rules (digit right of rounding place: 5 or more → up, else down).
Example Problem 1: Which is greater: $0.7$ L or $0.65$ L?
Write as $0.70$ and $0.65$, compare tenths ($7 > 6$): $0.7 > 0.65$
Example Problem 2: Order $5.62, 5.627, 5.6$ from least to greatest.
- $5.620, 5.627, 5.600$
- Compare tenths ($6$), hundredths ($2,2,0$), thousandths ($0,7,0$): $5.6 < 5.62 < 5.627$
Write as $0.70$ and $0.65$, compare tenths ($7 > 6$): $0.7 > 0.65$
Example Problem 2: Order $5.62, 5.627, 5.6$ from least to greatest.
- $5.620, 5.627, 5.600$
- Compare tenths ($6$), hundredths ($2,2,0$), thousandths ($0,7,0$): $5.6 < 5.62 < 5.627$
Quick Reference Table: Compare & Order Decimals
| Step | What to Do | Example |
|---|---|---|
| 1 | Align decimal points, pad zeros | 0.67 & 0.7 → 0.67 & 0.70 |
| 2 | Compare digit by digit (left to right) | First tenths: 6 vs 7 (7 is greater) |
| 3 | Use >, <, = symbols | $0.67 < 0.70$ |
| 4 | Draw grid or number line if stuck | Both visual methods confirm $0.70 > 0.67$ |
Important Tips & Reminders
- Zeros to the right of a decimal do NOT change its value: $0.4 = 0.40 = 0.400$
- For ordering, pad zeros, then compare by digits from left after decimal point
- Use number lines or grids for tough comparisons
- The greater decimal is always further right on the number line
- Always read the problem for context—money, length, weight
- Check your answer—does it make sense visually?