Equivalent Ratios & Ratio Tables - Sixth Grade Mathematics

Equivalent ratios maintain a constant multiplicative relationship between quantities, organized within structured ratio tables to model proportional growth and solve missing value problems.

What Are Equivalent Ratios?

Two ratios are equivalent if they express the exact same relationship between quantities. Equivalent ratios are formed by multiplying or dividing both terms of a ratio by the same non-zero number.

Equivalent Ratio Family for 2 : 3:
- Multiply both by 2:  (2 * 2) : (3 * 2)  =  4 : 6
- Multiply both by 3:  (2 * 3) : (3 * 3)  =  6 : 9
- Multiply both by 5:  (2 * 5) : (3 * 5)  = 10 : 15
Family: { 2:3, 4:6, 6:9, 10:15, ... }

The Multiplicative Constant

In any family of equivalent ratios a:b, the quotient a/b (or b/a) is strictly constant. For 2:3, 4:6, and 10:15: 3/2 = 6/4 = 15/10 = 1.5. This constant multiplier is the unit rate.

Ratio Tables: Structure and Navigation

A ratio table is a structured grid displaying columns and rows of equivalent ratios. Ratio tables can be navigated through horizontal scaling (across columns) and vertical scaling (between rows).

+-------------------------------------------------------------------------+
|                        SAMPLE RATIO TABLE: BAKING BREAD                 |
+---------------------+-----+-----+-----+-----+-----+-----+---------------+
| Cups of Flour (x)   |  3  |  6  |  9  | 12  | 15  | 30  |      x        |
+---------------------+-----+-----+-----+-----+-----+-----+---------------+
| Loaves of Bread (y) |  1  |  2  |  3  |  4  |  5  | 10  |    x / 3      |
+---------------------+-----+-----+-----+-----+-----+-----+---------------+

Horizontal Strategies: Scaling Across Columns

In horizontal scaling, observe what factor multiplies or divides a term in one column to produce the term in another column. Apply that exact same factor to the other row. If 3 cups make 1 loaf, how many cups make 10 loaves? From 1 loaf to 10 loaves, multiply by 10. Apply the same factor to flour: 3 * 10 = 30 cups.

Vertical Strategies: Unit Rate Across Rows

In vertical scaling, observe the multiplicative relationship between the top and bottom rows. In the table above, dividing the top row by 3 produces the bottom row (or multiplying the bottom row by 3 produces the top row). This vertical multiplier holds true across every column in the table.

The Additive Combination Property of Ratio Tables

Because equivalent ratios represent proportional quantities, you can add or subtract corresponding columns in a ratio table to generate new equivalent ratios without multiplying from scratch.

Adding Columns:
If 3 cups flour = 1 loaf (Column 1)
and 6 cups flour = 2 loaves (Column 2)
Then: (3 + 6) cups = (1 + 2) loaves  -->  9 cups = 3 loaves! (Column 3)

Scaling Down to Scale Up

Sometimes an intermediate unit rate is a fraction that is awkward to work with. In such cases, scale DOWN to a friendly intermediate common factor, and then scale UP to the target value.

Problem: 12 tickets cost $48. How much do 15 tickets cost?
Step 1: Scale DOWN to find the cost of 1 ticket:
        Divide by 12: 12 / 12 = 1 ticket; $48 / 12 = $4 per ticket.
Step 2: Scale UP to 15 tickets:
        Multiply by 15: 1 * 15 = 15 tickets; $4 * 15 = $60.
Conclusion: 15 tickets cost $60.

Exercises

Exercise 1. Complete the ratio table to find the missing values: Cups of Concentrate: 2 | 4 | 6 | ? | 20 Cups of Water: 5 | ? | 15| 25| ?

Exercise 2. A machine stamps 42 bottle caps in 6 seconds. Construct a ratio table to determine how many bottle caps the machine stamps in 45 seconds.

Exercise 3. Verify whether the ratios 16:24 and 20:30 are equivalent using two different methods (simplifying to lowest terms and cross-multiplying).

Solutions

Solution to Exercise 1. Find the base unit ratio: 2 cups concentrate to 5 cups water. For Column 2: concentrate is 4 (multiplied by 2); water = 5 * 2 = 10. For Column 3: concentrate is 6 (multiplied by 3); water = 5 * 3 = 15 (matches). For Column 4: water is 25 (multiplied by 5); concentrate = 2 * 5 = 10. For Column 5: concentrate is 20 (multiplied by 10); water = 5 * 10 = 50. Missing values in order: water = 10; concentrate = 10; water = 50.

Solution to Exercise 2. Find the unit rate: 42 caps in 6 seconds means 42 / 6 = 7 caps per second. To find caps in 45 seconds: multiply unit rate by 45: 7 * 45 = 315 bottle caps. In 45 seconds, the machine stamps 315 caps.

Solution to Exercise 3. Method 1 (Simplifying): For 16:24, divide by GCF 8: 16/8 = 2, 24/8 = 3, giving 2:3. For 20:30, divide by GCF 10: 20/10 = 2, 30/10 = 3, giving 2:3. Because both simplify to 2:3, they are equivalent. Method 2 (Cross-Multiplication): Write as fractions 16/24 and 20/30. Compute cross-products: 16 * 30 = 480; 24 * 20 = 480. Because both cross-products equal 480, the ratios are confirmed to be equivalent.