60 Is 75 Of What Number

60 is 75% of what number? This is a common percentage problem that requires solving for the unknown value. Understanding how to find the original number when given a percentage and its corresponding part is a fundamental mathematical skill with practical applications in everyday life, such as calculating discounts, interpreting statistics, or understanding financial data. This article will guide you through the step-by-step process to find the solution, explain the underlying mathematical principle, and address common questions.

The Core Equation

The problem states: 60 equals 75% of some unknown number, let's call it x. Mathematically, this is expressed as:

60 = 0.75 × x

Our goal is to isolate x to find its value. This requires manipulating the equation using basic arithmetic operations.

Step-by-Step Solution

1. Isolate the Variable (x): To find x, we need to undo the multiplication by 0.75. The inverse operation of multiplication is division. Therefore, we divide both sides of the equation by 0.75.

   • Equation: 60 = 0.75 × x
   • Divide both sides by 0.75: 60 ÷ 0.75 = x

2. Perform the Division: Calculate 60 divided by 0.75.

   • 60 ÷ 0.75
   • To make division by a decimal easier, multiply both the numerator and the denominator by 100 to eliminate the decimal point:
   • 60 ÷ 0.75 = (60 × 100) ÷ (0.75 × 100) = 6000 ÷ 75

3. Calculate the Result: Divide 6000 by 75.

   • 6000 ÷ 75
   • You can simplify this fraction: 6000 ÷ 75 = 6000 ÷ (25 × 3) = (6000 ÷ 25) ÷ 3 = 240 ÷ 3 = 80
   • Alternatively, using long division: 75 goes into 600 eight times (75 × 8 = 600). Subtract, bring down the 0, 75 goes into 00 zero times. So, 6000 ÷ 75 = 80.

4. State the Answer: Therefore, the number we were looking for is 80.

Verification

It's always wise to verify your solution. Plug the answer back into the original equation:

• 75% of 80 = 0.75 × 80 = 60
• This matches the given part (60), confirming that 80 is indeed the correct number.

The Underlying Principle: Percentages as Fractions

Understanding percentages involves recognizing they represent parts per hundred. 75% literally means 75 per hundred, or the fraction 75/100, which simplifies to 3/4. Therefore, saying 60 is 75% of x is equivalent to saying 60 is 3/4 of x. Solving 60 = (3/4) × x is the same process as solving 60 = 0.75 × x. Dividing both sides by 3/4 (or multiplying by its reciprocal, 4/3) yields the same result: x = 60 × (4/3) = 80.

Why This Matters

Mastering this skill is crucial. It allows you to:

• Calculate Original Prices: If an item is on sale for $60, which is 75% of its original price, you can determine the original price was $80.
• Interpret Data: If 60 students represent 75% of the class, you know the class has 80 students.
• Solve Financial Problems: Understanding interest rates, loan terms, or investment returns often involves finding original values from percentages.

FAQ

• Q: What if the percentage is given as a fraction, like 3/4?
  • A: The process is identical. Solve 60 = (3/4) × x by multiplying both sides by the reciprocal of 3/4, which is 4/3. So, x = 60 × (4/3) = 80.
• Q: What if the percentage is greater than 100%, like 150%?
  • A: The same method applies. For example, if 60 is 150% of x, then 60 = 1.5 × x, and x = 60 ÷ 1.5 = 40.
• Q: What if I only know the percentage and the part, but not the whole?
  • A: This is the exact scenario described here. You use the formula: Whole = (Part ÷ Percentage) × 100. In this case, Whole = (60 ÷ 75) × 100 = 0.8 × 100 = 80.
• Q: Is there a shortcut formula?
  • A: Yes! The formula derived from the process is: Whole = (Part ÷ Percentage) × 100. This is derived from Part = (Percentage/100) × Whole, rearranged to solve for Whole.

Conclusion

Finding the original number when given a part and its percentage is a straightforward process once you understand the core equation and the inverse operations involved. By setting up the equation (Part = (Percentage/100) × Whole), isolating the unknown (Whole), and performing the division, you can efficiently find the answer. In this specific case, 60 is 75% of 80. This fundamental mathematical concept underpins numerous real-world calculations, from shopping discounts to understanding demographic data. Practicing these problems builds confidence and strengthens your quantitative reasoning skills, making you more adept at interpreting numerical information in daily life. Remember the formula Whole = (Part ÷ Percentage) × 100 as a quick reference for similar problems in the future.