Question
If the pattern in the table is extended to represent more equivalent ratios for 2:6, which pair of numbers would be in the columns?
A multiplication table. In the column labeled 2, the numbers 2, 4, 6, 8, 10, 12, 14, 16, and 18 are highlighted. In the column labeled 6, the numbers 6, 12, 18, 24, 30, 36, 42, 48, and 54 are highlighted.
20 would be in the column for 2, and 60 would be in the column for 6.
20 would be in the column for 6, and 60 would be in the column for 2.
20 would be in the column for 2, and 56 would be in the column for 6.
20 would be in the column for 6, and 56 would be in the column for 2.
A multiplication table. In the column labeled 2, the numbers 2, 4, 6, 8, 10, 12, 14, 16, and 18 are highlighted. In the column labeled 6, the numbers 6, 12, 18, 24, 30, 36, 42, 48, and 54 are highlighted.
20 would be in the column for 2, and 60 would be in the column for 6.
20 would be in the column for 6, and 60 would be in the column for 2.
20 would be in the column for 2, and 56 would be in the column for 6.
20 would be in the column for 6, and 56 would be in the column for 2.
Answers
Answered by
GPT-4o mini
To find pairs of numbers that represent equivalent ratios to 2:6, we can note that the ratio simplifies to 1:3. This means that for every 1 unit of the first number, the second number should have 3 units.
Using the pattern from the table, if we identify that 2 corresponds to 6, then when we multiply both numbers in the ratio 2:6 by the same multiplier \( k \), we can generate equivalent ratios.
The pairs will therefore be of the form:
\[
(2k, 6k)
\]
For example:
- If \( k = 0 \): \( (2 \times 0, 6 \times 0) = (0, 0) \)
- If \( k = 1 \): \( (2 \times 1, 6 \times 1) = (2, 6) \)
- If \( k = 2 \): \( (2 \times 2, 6 \times 2) = (4, 12) \)
- If \( k = 3 \): \( (2 \times 3, 6 \times 3) = (6, 18) \)
- If \( k = 4 \): \( (2 \times 4, 6 \times 4) = (8, 24) \)
- If \( k = 5 \): \( (2 \times 5, 6 \times 5) = (10, 30) \)
- If \( k = 10 \): \( (2 \times 10, 6 \times 10) = (20, 60) \)
Now analyzing the options:
1. **20 would be in the column for 2, and 60 would be in the column for 6.**
This is true since \( 20 = 2 \times 10 \) and \( 60 = 6 \times 10 \), which maintains the ratio \( 20:60 = 2:6\).
2. **20 would be in the column for 6, and 60 would be in the column for 2.**
This is incorrect.
3. **20 would be in the column for 2, and 56 would be in the column for 6.**
This is incorrect since \( 56 \) does not maintain the \( 6k \) for any integer \( k\).
4. **20 would be in the column for 6, and 56 would be in the column for 2.**
This is also incorrect for the same reason as option 3.
Therefore, the correct statement is:
**20 would be in the column for 2, and 60 would be in the column for 6.**
Using the pattern from the table, if we identify that 2 corresponds to 6, then when we multiply both numbers in the ratio 2:6 by the same multiplier \( k \), we can generate equivalent ratios.
The pairs will therefore be of the form:
\[
(2k, 6k)
\]
For example:
- If \( k = 0 \): \( (2 \times 0, 6 \times 0) = (0, 0) \)
- If \( k = 1 \): \( (2 \times 1, 6 \times 1) = (2, 6) \)
- If \( k = 2 \): \( (2 \times 2, 6 \times 2) = (4, 12) \)
- If \( k = 3 \): \( (2 \times 3, 6 \times 3) = (6, 18) \)
- If \( k = 4 \): \( (2 \times 4, 6 \times 4) = (8, 24) \)
- If \( k = 5 \): \( (2 \times 5, 6 \times 5) = (10, 30) \)
- If \( k = 10 \): \( (2 \times 10, 6 \times 10) = (20, 60) \)
Now analyzing the options:
1. **20 would be in the column for 2, and 60 would be in the column for 6.**
This is true since \( 20 = 2 \times 10 \) and \( 60 = 6 \times 10 \), which maintains the ratio \( 20:60 = 2:6\).
2. **20 would be in the column for 6, and 60 would be in the column for 2.**
This is incorrect.
3. **20 would be in the column for 2, and 56 would be in the column for 6.**
This is incorrect since \( 56 \) does not maintain the \( 6k \) for any integer \( k\).
4. **20 would be in the column for 6, and 56 would be in the column for 2.**
This is also incorrect for the same reason as option 3.
Therefore, the correct statement is:
**20 would be in the column for 2, and 60 would be in the column for 6.**