How can I use a ratio table to find equivalent pairs?

Make one column for each quantity and put the starting pair together in the first row. Keep the order consistent: red counters first, blue counters second. Then choose a scale factor and apply it to both numbers. Equal ratios have the same quotient; multiplying both parts of a ratio by the same factor keeps that relationship unchanged. [S1]

For example, multiply both 2 and 3 by 2 to get 4 and 6. The table organizes those paired values; the important part is keeping the same multiplier for both entries.

Here is a small table using whole-number scale factors:

Scale factor | Red counters | Blue counters
1            | 2            | 3
2            | 4            | 6
3            | 6            | 9
4            | 8            | 12

Each row comes from the original pair, 2:3. For instance, the row with factor 3 comes from 2 × 3 = 6 red counters and 3 × 3 = 9 blue counters, so it gives 6:9. The pattern is that each step here adds 2 red counters and 3 blue counters. That pattern is visible in this table; the reliable check is that both entries in each row use the same scale factor as the starting pair. [S1]

Compare the known value with the corresponding value in the starting pair, not with the other column. If the red-counter value changes from 2 to 10, the factor is 10 ÷ 2 = 5. Apply that factor to the blue-counter value too: 3 × 5 = 15. So the matching pair is 10 red counters to 15 blue counters.

A common slip is to add the same amount to both entries instead. Adding 2 to 2:3 gives 4:5, which does not use one shared multiplier. Check your result by comparing the ratios: 10:15 has the same quotient as 2:3. [S1]

Use this routine:

1. Write the starting pair in a consistent order, such as apples then oranges.
2. Choose a scale factor, or work out the factor from the known value and its matching starting value.
3. Multiply or divide both starting values by that same factor.
4. Put the resulting values in the same row and label the quantities.
5. Check that the pair still represents the same ratio.

For example, if a new starting pair is 4 apples to 5 oranges and you choose factor 3, the row is 12 apples to 15 oranges. Keep the labels and order clear so you do not accidentally match a value from one column with the wrong starting value.