Part-to-Part vs. Part-to-Whole Ratios - Sixth Grade Mathematics

Understanding the distinction between part-to-part and part-to-whole ratios enables you to analyze compositions of collections, calculate fractions of totals, and scale proportional mixtures.

Two Ways to Compare Collections

When analyzing any collection made of distinct subcategories, ratios can compare subcategories to each other, or subcategories to the entire group.

A fruit basket contains: 5 Apples and 3 Oranges.
Total fruit in the basket = 5 + 3 = 8 pieces of fruit.

Part-to-Part Ratio:     Apples to Oranges = 5 to 3  (5 : 3)
Part-to-Whole Ratios:   Apples to Total Fruit = 5 to 8  (5 : 8, or 5/8)
                        Oranges to Total Fruit = 3 to 8  (3 : 8, or 3/8)

The Structural Matrix: Comparing Ratios

The table below contrasts the properties of part-to-part and part-to-whole relationships.

+-------------------------------------------------------------------------+
|                  PART-TO-PART VERSUS PART-TO-WHOLE                      |
+-------------------+-----------------------+-----------------------------+
| Feature           | Part-to-Part          | Part-to-Whole               |
+-------------------+-----------------------+-----------------------------+
| What is compared? | One subset to another | One subset to the aggregate |
|                   | subset in the group   | total of all subsets        |
+-------------------+-----------------------+-----------------------------+
| Direct Fraction   | CANNOT be directly    | DIRECTLY represents a       |
| Meaning           | interpreted as a      | fraction of the whole; can  |
|                   | fraction of the total | convert to percent! (5/8=62%)|
+-------------------+-----------------------+-----------------------------+
| Summing Terms     | Sum of terms gives the| Denominator IS the total;   |
|                   | total parts: a + b    | Numerator is the subset part|
+-------------------+-----------------------+-----------------------------+

Deriving the Whole from Part-to-Part Terms

If a ratio of red marbles to blue marbles is 4:7, the total collection is partitioned into 4 + 7 = 11 equal parts. From this part-to-part ratio, you can deduce two part-to-whole ratios: Red marbles to total marbles = 4 to 11 (4/11 of the collection is red). Blue marbles to total marbles = 7 to 11 (7/11 of the collection is blue). Notice that the sum of the part-to-whole fractions must equal one whole: 4/11 + 7/11 = 11/11 = 1.

Real-World Scaling Word Problems

Distinguishing part-to-part from part-to-whole is the key to solving multi-step scaling problems.

Problem: A punch recipe mixes pineapple juice and sparkling water in a
ratio of 2 : 5. If the caterer needs to make 35 gallons of total punch,
how many gallons of pineapple juice are needed?

Step 1: Identify ratio type:
        2 : 5 is Pineapple to Water (Part-to-Part).
        Total punch is 35 gallons (The Whole!).

Step 2: Find total parts in ratio:
        2 parts pineapple + 5 parts water = 7 total parts.

Step 3: Determine the value of 1 ratio part:
        35 total gallons / 7 total parts = 5 gallons per part.

Step 4: Scale the requested part (pineapple):
        2 parts pineapple * 5 gallons/part = 10 gallons of pineapple juice.

Exercises

Exercise 1. A bag contains 8 blue pens, 5 black pens, and 7 red pens. Write: (a) the part-to-part ratio of blue pens to black pens; (b) the part-to-whole ratio of red pens to total pens in simplest form.

Exercise 2. In a school choir, the ratio of boys to girls is 3 to 4. If there are 42 students in the choir, how many are girls?

Exercise 3. A paint mixture requires 3 parts yellow tint for every 5 parts blue tint. What fraction of the total mixture is blue tint, and what percent does this represent?

Solutions

Solution to Exercise 1. First, find total pens: 8 + 5 + 7 = 20 pens. For (a) part-to-part ratio of blue pens to black pens: 8 to 5 (or 8:5). For (b) part-to-whole ratio of red pens to total pens: 7 to 20 (or 7/20). Since 7 is prime and does not divide 20, 7/20 is in simplest form.

Solution to Exercise 2. The ratio of boys to girls is 3:4. The total number of ratio parts is 3 + 4 = 7 parts. The choir has 42 students. Find the value of 1 ratio part: 42 students / 7 parts = 6 students per part. Girls represent 4 parts: 4 * 6 = 24 girls. (Boys represent 3 * 6 = 18 boys; check: 18 + 24 = 42 students).

Solution to Exercise 3. The ratio of yellow tint to blue tint is 3:5. The total parts in the mixture is 3 + 5 = 8 parts. Blue tint represents 5 out of the 8 parts. The fraction of the mixture that is blue is 5/8. To convert 5/8 to a percent: 5 / 8 = 0.625; multiplying by 100 yields 62.5 percent.