How to Fill in the Ratio Table for 31: A Complete Step-by-Step Guide
Learning how to fill in the ratio table for 31 is one of the most practical ways to strengthen your proportional reasoning skills. A ratio table organizes equivalent ratios in rows and columns, and when the number 31 is involved, the patterns become especially interesting because 31 is a prime number. In this guide, you will learn what a ratio table is, how to complete one that features the number 31, the shortcuts that make it fast, and the common mistakes to avoid.
Not obvious, but once you see it — you'll see it everywhere Easy to understand, harder to ignore..
What Is a Ratio Table?
A ratio table is a structured list of equivalent ratios. Each row shows the same relationship between two quantities, even though the actual numbers change. Take this: the ratio 31 : 4 is equivalent to 62 : 8 and 93 : 12, because every value was multiplied by the same number.
A typical ratio table has two columns:
| Column A | Column B |
|---|---|
| 31 | 4 |
| 62 | 8 |
| 93 | 12 |
The key idea is that the relationship between the two columns never changes. Mathematicians call the number you multiply by the scale factor or multiplier Most people skip this — try not to..
Why the Number 31 Is Special
The number 31 is a prime number, which means it can only be divided evenly by 1 and itself. Even so, this has one important consequence: any ratio where 31 appears, such as 31 : 7 or 31 : 12, is already in its simplest form. You cannot reduce it further Easy to understand, harder to ignore..
Not the most exciting part, but easily the most useful.
That means when you fill in a ratio table for 31, you will almost always be multiplying rather than dividing. This makes 31 an excellent number for practicing multiplicative thinking Not complicated — just consistent..
Step-by-Step Method to Fill in the Ratio Table
Follow these steps whenever you are asked to complete a ratio table involving 31.
Step 1: Identify the Base Ratio
Look at the row where both values are given. This is your base ratio. For example:
31 : 5
Step 2: Find the Scale Factor
Compare the known value in the target row with the matching value in the base row. Ask yourself: "What number do I multiply 31 by to get the new value?"
If the new row shows 124 in the first column, then:
124 ÷ 31 = 4, so the scale factor is 4 That's the part that actually makes a difference..
Step 3: Apply the Same Multiplier to the Other Column
Multiply the second value by the same scale factor:
5 × 4 = 20
So the missing value is 20, and the completed row is 124 : 20 That's the part that actually makes a difference..
Step 4: Check Your Answer
Verify by simplifying the new ratio. Divide both numbers by the scale factor and confirm you return to the base ratio:
124 ÷ 4 = 31 and 20 ÷ 4 = 5 ✔
Worked Example: Complete the Table for 31 : 6
Let's fill in the ratio table for 31 : 6 using multipliers from 1 to 6.
| Multiplier | First Quantity (× 31) | Second Quantity (× 6) |
|---|---|---|
| ×1 | 31 | 6 |
| ×2 | 62 | 12 |
| ×3 | 93 | 18 |
| ×4 | 124 | 24 | | ×5 | 155 | 30 | | ×6 | 186 | 36 |
Reading down the table reveals two consistent patterns: the first quantity increases by 31 each time, and the second quantity increases by 6 each time. These repeated additions are simply the result of multiplying by larger whole numbers, which is why ratio tables connect so naturally to both multiplication and proportional reasoning.
The same logic works in reverse when a different column is missing. Even so, suppose a row in the 31 : 6 table has 42 in the second column. In real terms, since 42 ÷ 6 = 7, the scale factor is 7, so the first quantity must be 31 × 7 = 217. The completed row is 217 : 42. This reverse approach is especially useful when the missing value appears in the first column rather than the second.
Short version: it depends. Long version — keep reading.
A common mistake is to add the same amount to both columns instead of multiplying by the same scale factor. Consider this: in a ratio table, addition may create a new row only when the same multiple of the base ratio is added each time. Here's the thing — for example, moving from 31 : 6 to 62 : 12 is equivalent to adding 31 : 6 once, which is the same as multiplying by 2. But adding 31 to the first column and 6 to the second is not arbitrary—it preserves the ratio because both quantities grow by one full copy of the base ratio.
Ratio tables involving 31 are particularly useful for building fluency with larger numbers. Because 31 is prime, the first column will always be a multiple of 31, and simplifying the ratio will always return to a base ratio with 31 in the first position. That predictability makes it easier to check work and spot errors.
Pulling it all together, a ratio table for 31 is more than a list of numbers; it is a visual representation of proportional relationships. By identifying the base ratio, finding the scale factor, and applying it consistently to both columns, you can complete any row with confidence. Whether you are multiplying upward to larger values or dividing backward to find a missing quantity, the underlying relationship remains unchanged—and that is the central idea behind every ratio table.
Most guides skip this. Don't.
