- Identify the constant ratio \(k\) for each table, ensuring that \(\frac{a}{b}\) is identical for every column.
- Verify equivalence by checking that each column is a multiple of the base ratio \((a:b)\).
- Construct extension tables for problems 5 and 6 by applying a consistent multiplier to both parts of the ratio.
- The equivalent ratios represent the same proportional relationship throughout the tables.
Explanation
- Analysis of Ratio Tables
A ratio table displays equivalent ratios, where each column pair \((a, b)\) maintains a constant relationship, meaning \(\frac{a}{b} = k\) for some constant \(k\). To find missing values or verify equivalence, we identify the scaling factor between rows or columns. In the given tables, each entry is generated by multiplying both values in the base ratio by an integer \(n\). For problems 5 and 6, we will construct logical tables based on the descriptions provided.
- Evaluating Tables 1-4
- Table 1 (Cars:Trucks = 3:5):
- Ratios are \(\frac{3}{5}\) and \(\frac{6}{10}\).
- Since \(3 \times 2 = 6\) and \(5 \times 2 = 10\), these are equivalent.
- Equivalent ratios: \(3:5 = 6:10\).
- Table 2 (TVs:Computers = 2:7):
- Ratios are \(\frac{2}{7}\), \(\frac{4}{14}\), and \(\frac{6}{21}\).
- Each column is scaled by \(1, 2, 3\): \(2 \times 2 = 4\), \(7 \times 2 = 14\); \(2 \times 3 = 6\), \(7 \times 3 = 21\).
- Equivalent ratios: \(2:7 = 4:14 = 6:21\).
- Table 3 (Dogs:Paws = 4:16):
- Simplify the ratio: \(4:16 = 1:4\).
- Given pairs are \((4, 16)\), \((8, 32)\), \((12, 48)\).
- Each column is scaled by \(1, 2, 3\): \(4 \times 2 = 8, 16 \times 2 = 32\); \(4 \times 3 = 12, 16 \times 3 = 48\).
- Equivalent ratios: \(4:16 = 8:32 = 12:48\).
- Table 4 (Apples:Oranges = 5:3):
- Ratios are \(\frac{5}{3}, \frac{10}{6}, \frac{15}{9}, \frac{20}{12}\).
- Each column is scaled by \(1, 2, 3, 4\): \(5 \times 4 = 20, 3 \times 4 = 12\).
- Equivalent ratios: \(5:3 = 10:6 = 15:9 = 20:12\).
- Completing Problems 5 and 6
- For every 2 cars you wash, let's assume you earn \$10 (a common simple ratio).
| Cars | 2 | 4 | 6 |
| :--- | :--- | :--- | :--- |
| Dollars | 10 | 20 | 30 |
- Equivalent ratios: \(2:10 = 1:5 = 4:20 = 6:30\).
- Your closet has 5 shirts for every 2 pairs of pants.
| Shirts | 5 | 10 | 15 |
| :--- | :--- | :--- | :--- |
| Pants | 2 | 4 | 6 |
- Equivalent ratios: \(5:2 = 10:4 = 15:6\).
Examples
Ratio tables are used every day to maintain proportions. For instance, if you are following a pancake recipe that requires 2 cups of flour for every 3 eggs, you can create a ratio table to determine the ingredients needed for larger batches. If you want to make a batch for a party, you can scale the table by multiplying both numbers by a constant, such as 2 or 3, to ensure the flavor remains consistent while cooking for more people. This simple mathematical tool helps in cooking, budgeting, and scaling projects efficiently.