Finding the Whole Given Part & Percent - Sixth Grade Mathematics

Finding the whole when given a part and its corresponding percent reverses standard percent calculations, using division and algebraic proportions to reconstruct complete reference totals.

The Structural Problem: Inverting the Percent Equation

In standard percent problems, you are given the percent and the whole, and you multiply to find the part. In reverse percent problems, you are given the part and its percentage, and you must determine the original 100% total.

Standard Problem:   Percent * Whole = Part    -->  What is 20% of 80? (Find Part)
Reverse Problem:    Part / Percent  = Whole   -->  16 is 20% of what number? (Find Whole)

Two Solution Pathways

There are two primary methods for finding the whole: the Percent Proportion Method and the Algebraic Division Method.

+-------------------------------------------------------------------------+
|                  TWO PATHWAYS TO FIND THE WHOLE                         |
+-------------------+-----------------------+-----------------------------+
| Method            | Formula & Setup       | Example: 18 is 30% of what? |
+-------------------+-----------------------+-----------------------------+
| Method 1: Percent | Part / Whole = % / 100| 18 / W = 30 / 100           |
| Proportion        | Cross-multiply:       | 30 * W = 18 * 100           |
| (Universal)       | % * Whole = Part * 100| 30W = 1,800  -->  W = 60    |
+-------------------+-----------------------+-----------------------------+
| Method 2: Decimal | Whole = Part / Decimal| 30% = 0.30                  |
| Division          | Divide part by the    | Whole = 18 / 0.30           |
| (Algebraic)       | decimal percent       | Whole = 180 / 3 = 60        |
+-------------------+-----------------------+-----------------------------+

The Unitary Method (Finding 1% First)

The Unitary Method is an intuitive mental math strategy that calculates the value of 1% before scaling up to 100%.

The Unitary Protocol for: "24 is 40% of what number?"
Step 1: If 40% = 24, find the value of 1% by dividing by 40:
        1% = 24 / 40 = 0.6.
Step 2: Since the whole is 100%, multiply 1% by 100:
        100% = 0.6 * 100 = 60.
Conclusion: The original whole is 60.

Using Tape Diagrams to Find the Whole

Tape diagrams make reverse percent problems visually intuitive by modeling the whole as a collection of equal percentage blocks.

Problem: A student saved $45, which is 60% of the cost of a bicycle.
What is the total cost of the bicycle?

Step 1: Model 60% as three 20% blocks:
[ 20% ] [ 20% ] [ 20% ]  <-- 3 blocks = 60% = $45
Value of 1 block (20%) = $45 / 3 = $15.

Step 2: The complete whole (100%) consists of five 20% blocks:
[ $15 ] [ $15 ] [ $15 ] [ $15 ] [ $15 ]  =  5 * $15 = $75.
Conclusion: The total cost of the bicycle is $75.

Exercises

Exercise 1. Find the whole number in each case: (a) 28 is 70% of what number? (b) 45 is 15% of what number? (c) 120 is 80% of what number?

Exercise 2. During a basketball season, Marcus made 36 free throws, which represented 75% of all his free throw attempts. How many total free throws did Marcus attempt?

Exercise 3. A cell phone battery charges to 40% in 32 minutes. At this constant rate, how many total minutes are required for a complete 100% charge?

Solutions

Solution to Exercise 1. For (a) 28 is 70% of W: use proportion: 28 / W = 70 / 100; cross-multiply: 70W = 2,800; divide by 70: W = 2,800 / 70 = 40. For (b) 45 is 15% of W: divide by decimal percent: W = 45 / 0.15 = 4,500 / 15 = 300. For (c) 120 is 80% of W: unitary method: 80% = 120; 10% = 120 / 8 = 15; 100% = 15 * 10 = 150.

Solution to Exercise 2. Set up the proportion with 36 as the part and W as the unknown whole attempts: 36 / W = 75 / 100. Simplify 75/100 to 3/4: 36 / W = 3 / 4. Cross-multiply: 3 * W = 36 * 4 = 144. Divide by 3: W = 144 / 3 = 48. Marcus attempted 48 total free throws.

Solution to Exercise 3. The part is 32 minutes, and the percent is 40%. Use the unitary method: 40% takes 32 minutes. Find 10%: divide 32 by 4: 10% takes 8 minutes. Since the full charge is 100%, multiply 10% by 10: 8 minutes * 10 = 80 minutes. A complete 100% charge requires 80 minutes (1 hour 20 minutes).