Equivalent Ratios and Ratio Tables: Fifth Grade Mathematics

Equivalent ratios maintain an identical proportional relationship, organized into ratio tables to solve real-world rate problems.

What Are Equivalent Ratios?

Two ratios are equivalent if one can be obtained from the other by multiplying or dividing both quantities by the same non-zero number. Equivalent ratios behave identically to equivalent fractions.

+--------------------------------------------------------------------------+
|                  EQUIVALENT RATIOS GENERATION                            |
|                                                                          |
|            2 : 3  =  (2 * 4) : (3 * 4)  =  8 : 12                        |
|                                                                          |
|       Both terms must be multiplied or divided by the IDENTICAL factor!   |
+--------------------------------------------------------------------------+

Structure of a Ratio Table

A ratio table displays equivalent ratios in rows or columns, allowing you to scale quantities up or down to find missing values.

+--------------------------------------------------------------------------+
|                  LEMONADE RECIPE RATIO TABLE                             |
+---------------------+-----+-----+-----+-----+----------------------------+
| Cups of Lemon Juice |  2  |  4  |  6  |  8  | Multiplier: * 2            |
+---------------------+-----+-----+-----+-----+----------------------------+
| Cups of Cold Water  |  5  | 10  | 15  | 20  | Multiplier: * 5            |
+---------------------+-----+-----+-----+-----+----------------------------+
| Proportional Ratio  | 2:5 | 4:10| 6:15| 8:20| All ratios equal 2/5!      |
+---------------------+-----+-----+-----+-----+----------------------------+

Scaling Up and Scaling Down

  • Scaling Up: Multiply both terms by a whole number to find quantities for larger batches.
  • Scaling Down: Divide both terms by a common factor to find unit rates or simpler ratios.
Solving with Intermediate Scaling (Build-Up Strategy):
  Problem: 6 shirts cost $48. How much do 10 shirts cost?
  Step 1: Scale down to 1 shirt: $48 / 6 = $8 per shirt.
  Step 2: Scale up to 10 shirts: 10 * $8 = $80.

Exercises and Solutions

Exercise 1: Completing a Ratio Table. Complete the ratio table for a car that travels 110 miles for every 2 gallons of fuel: Gallons: 2, 4, 6, 8. Miles: 110, ?, ?, ?.

Solution 1: Find the unit rate (miles per 1 gallon): 110 / 2 = 55 miles per gallon. Now multiply each gallon quantity by 55: For 4 gallons: 4 * 55 = 220 miles. For 6 gallons: 6 * 55 = 330 miles. For 8 gallons: 8 * 55 = 440 miles. The completed table pairs are (2, 110), (4, 220), (6, 330), and (8, 440).

Exercise 2: Scaling Down to Scale Up. A baker uses 12 cups of flour to make 18 loaves of bread. How many cups of flour are needed to make 30 loaves of bread?

Solution 2: The ratio of flour to loaves is 12:18. Scale down by dividing both terms by 6 to find the simplest equivalent ratio: 12 / 6 = 2 cups of flour for every 18 / 6 = 3 loaves of bread (ratio 2:3). Next, scale up to 30 loaves: multiply 3 loaves by 10 to reach 30 loaves. Multiply the flour by 10: 2 * 10 = 20 cups of flour. The baker needs 20 cups of flour to make 30 loaves.