60 Is 15 Of What Number

To determine thenumber for which 60 represents 15%, we need to solve the equation: 60 = 0.15 × x. This requires finding the original value when 15% of it equals 60. The solution involves understanding percentages and basic algebra. Let's break this down step by step.

Introduction Understanding percentages is fundamental in mathematics and everyday life, from calculating discounts to interpreting statistics. When we encounter a statement like "60 is 15% of what number?" it presents a classic percentage problem. This article will guide you through solving it systematically, explaining the underlying concepts, and providing practical examples. The answer is not just a number; it's a gateway to mastering proportional reasoning.

Steps to Solve

1. Translate the Statement: The phrase "60 is 15% of what number?" translates mathematically to: 60 = 0.15 × x, where x is the unknown number we seek.
2. Isolate the Variable: To find x, we need to undo the multiplication by 0.15. The inverse operation of multiplication is division. Therefore, divide both sides of the equation by 0.15.
3. Perform the Calculation: Calculate 60 divided by 0.15.
   • 60 ÷ 0.15 = 60 ÷ (15/100) = 60 × (100/15) = (60 × 100) ÷ 15 = 6000 ÷ 15 = 400.
   • Alternatively, moving the decimal point: 60 ÷ 0.15 is the same as 6000 ÷ 15, yielding 400.
4. Verify the Solution: Substitute x = 400 back into the original equation to check.
   • 0.15 × 400 = 60. This confirms the solution is correct.

Scientific Explanation The relationship between a part, the whole, and the percentage is defined by the formula: Part = (Percentage / 100) × Whole. Rearranging this formula to solve for the whole gives: Whole = (Part / (Percentage / 100)) = (Part × 100) / Percentage. Applying this to our problem:

• Part = 60
• Percentage = 15
• Whole = (60 × 100) / 15 = 6000 / 15 = 400. This formula works because a percentage represents a fraction of 100. Dividing the part by the percentage fraction (percentage/100) isolates the whole value. The inverse operation (division) effectively "undoes" the multiplication applied to the whole by the percentage fraction.

Practical Examples

1. Shopping Discount: A shirt is on sale for 15% off, resulting in a final price of $60. What was the original price?
   • Solution: Original Price = $60 ÷ 0.15 = $400. The shirt originally cost $400.
2. Population Statistic: A survey shows that 15% of a town's population is over 60 years old, and there are 60 such residents. What is the total population?
   • Solution: Total Population = 60 ÷ 0.15 = 400. The town has a population of 400.
3. Investment Return: An investment grew by 15%, resulting in a gain of $60. What was the initial investment amount?
   • Solution: Initial Investment = $60 ÷ 0.15 = $400. The investor started with $400.

Frequently Asked Questions (FAQ)

• Q: Why do we divide by 0.15 and not multiply?
  • A: The equation 60 = 0.15 × x means 60 is the result of multiplying 0.15 by x. To find x, we reverse the multiplication by performing its inverse operation, division.
• Q: Can I use fractions instead of decimals?
  • A: Absolutely. 15% is equivalent to 15/100 or 3/20. The equation becomes 60 = (3/20) × x. Solving for x: x = 60 × (20/3) = 400. The result is the same.
• Q: What if the percentage was different, say 20%?
  • A: The method remains identical. For example, "60 is 20% of what number?" would be solved as: 60 = 0.20 × x → x = 60 ÷ 0.20 = 300. The whole number changes based on the percentage given.
• Q: Is there a shortcut formula?
  • A: Yes. The formula Whole = (Part × 100) / Percentage is the direct shortcut derived from the core concept. Plug in the values: Whole = (60 × 100) / 15 = 400.

Conclusion Solving "60 is 15% of what number?" yields the answer: 400. This process demonstrates the power of translating word problems into mathematical equations and applying inverse operations. Understanding how to find the whole when given a part and its percentage is a vital skill applicable across numerous real-world scenarios, from finance to demographics. Mastering this foundational concept unlocks the ability to tackle more complex proportional reasoning problems with confidence. Remember the key formula: Whole = (Part × 100) / Percentage.

Continuing the discussion onpercentage calculations, let's explore a practical scenario that highlights the formula's versatility beyond the initial examples.

Scenario: Calculating Original Value with Multiple Discounts

Imagine a customer purchasing a laptop originally priced at $800. The store offers a 15% discount for first-time buyers and an additional 10% discount for using a store credit card. The customer pays $600 after both discounts. What was the laptop's original price?

While this involves multiple discounts, the core principle remains: finding the whole (original price) given parts (discounted prices) and their combined percentage impact. The calculation would involve first finding the price after the first discount (15% off $800 = $680), then applying the second discount (10% off $680 = $612). However, this scenario illustrates how the fundamental formula Whole = (Part × 100) / Percentage is the bedrock for solving such problems, even when they appear more complex. The key is identifying the relevant "part" (the final discounted price) and the specific percentage representing the reduction applied to the whole.

Addressing a Common Misconception

The FAQ question "Why do we divide by 0.15 and not multiply?" is crucial. It underscores the fundamental mathematical principle of inverse operations. Multiplication scales a quantity up (or down, if the multiplier is less than 1), while division reverses that scaling. Understanding this inverse relationship is not just about solving this specific equation; it's a critical skill for algebra, physics, finance, and countless other fields where relationships between quantities need to be manipulated.

The Formula's Broader Significance

The formula Whole = (Part × 100) / Percentage is more than a computational tool; it's a conceptual bridge. It transforms a word problem describing a relationship between a part and a whole into a solvable mathematical equation. This skill – translating real-world situations into mathematical models and applying the correct operations – is foundational to quantitative reasoning. Whether determining the original price of an item, calculating population growth, analyzing investment returns, or interpreting survey data, the ability to find the whole when given a part and its percentage is indispensable.

Conclusion

The process of solving "60 is 15% of what number?" exemplifies a core mathematical competency: using inverse operations to isolate the unknown variable within a proportional relationship. The derived formula Whole = (Part × 100) / Percentage provides a direct and efficient method for this task. Its application extends far beyond simple examples, forming a vital component of problem-solving in diverse real-world contexts, from commerce and demographics to finance and science. Mastering this foundational concept empowers individuals to decipher numerical relationships, make informed decisions based on data, and tackle increasingly complex proportional reasoning challenges with confidence and precision.