Finding the Whole from Part and Percent

Lesson 3 of 3: Percent Relationships

Learning Objectives:

• Identify problems where the "whole" is missing
• Find the original amount using division
• Find the whole using proportions
• Solve real-world discount and tax problems

The Three Types of Percent Problems

In any percent situation, there are three values. We usually know two and need to find the third.

Identifying "Find the Whole" Problems

Look for key phrases that signal the Whole is unknown:

• "...is 25% of what number?"
• "...represents 10% of the total"
• "What was the original price?"
• "Before the discount..."

If the question asks for the starting amount or total group, you are finding the Whole.

Practice: Identifying the Unknown

Which value is missing? Part, Whole, or Percent?

1. "What is 20% of 80?"
2. "15 is 30% of what number?"
3. "12 is what percent of 60?"

Pause and think before revealing the answer.

Answers

1. "What is 20% of 80?" → Find Part (Whole is 80)
2. "15 is 30% of what number?" → Find Whole ("what number" follows "of")
3. "12 is what percent of 60?" → Find Percent

Today's Focus: Type #2

Method 1: The Division Method

Recall: Part = Percent × Whole

To find the Whole, we use the inverse operation (division):

Whole = Part ÷ Percent (as decimal)

Example 1: Division Method

Problem: 15 is 25% of what number?

1. Identify: Part = 15, Percent = 25%
2. Convert: 25% = 0.25
3. Divide: Whole = Part ÷ Percent

Check: Is 25% of 60 equal to 15?

Visualizing the Relationship

Problem: 24 is 60% of the original amount.

Thinking: "If 24 is the 60% chunk, the full 100% bar must be larger."

Your Turn: Division Method

Find the whole number.

1. 8 is 20% of what number?
2. 50 is 10% of what number?

Remember: Convert percent to decimal, then divide.

Answers

1. 8 is 20% of what number?

   • Check:

2. 50 is 10% of what number?

   • Check:

Method 2: The Proportion Method

We can use the same template as before:

This time, the Whole (bottom left) is the unknown .

Example 2: Proportion Method

Problem: 12 is 40% of what number?

1. Set up:
2. Cross multiply:
3. Simplify:
4. Divide:

Answer: The number is 30.

Comparing Methods

Problem: Find the whole if 9 is 15%.

Choose the method that makes the most sense to you!

Real-World Application: Discounts

A shirt is on sale for $24. This is 60% of the original price. What was the original price?

• Part: $24 (Sale Price)
• Percent: 60%
• Whole: Unknown Original Price

The original price was $40.

Real-World Application: Tax

Tricky Case: "After 8% tax, the total was $54. Find the original price."

The total ($54) represents 100% (price) + 8% (tax) = 108%.

• Part: $54
• Percent: 108% (or 1.08)
• Whole: Unknown Original Price

The price before tax was $50.

Practice: Word Problems

1. Survey: 45 students said they like pizza. This is 30% of the school. How many students are in the school?

2. Tip: The total bill with a 20% tip was $36. What was the bill before the tip? (Hint: Total = 120%)

Pause and solve.

Answers

1. Survey:

   • There are 150 students.

2. Tip:

   • Percent =
   • The bill was $30.

Summary: Key Takeaways

• Find the Whole when you know the Part and the Percent.
• Division Method:
• Proportion Method:
• Check your answer: The Whole should usually be larger than the Part (unless percent > 100%).

Watch Out For...

Misconception: Multiplying instead of dividing.

• Incorrect: (This finds 25% of 15)
• Correct: (This finds the number 15 is 25% of)

Misconception: Using the wrong percent for total + tax.

• Remember: Total includes the original 100%.
• Use 100% + Tax% (e.g., 108% for 8% tax).

Next Steps

You have now mastered all three percent relationships!

1. Finding the Part ()
2. Finding the Percent ()
3. Finding the Whole ()

Complete the Practice Set to reinforce these skills with mixed problems.