Problem 157 - Elementary Algebra

Question

Without solving the problem "44 is \(80 \%\) of what number" think about what the solution might be. Should it be a number that is greater than 44 or less than 44? Explain your reasoning.

Step-by-Step Solution

Answer

The number must be greater than 44 because 100% is larger than 80%.

Step 1: Understanding the Problem

The exercise asks to determine whether the number that represents 100% of which 44 is 80% is greater than or less than 44.

Step 2: Percentage Fundamentals

Percentages involve comparing a part to a whole. Here, 44 is part (80%), and we need to find the whole (100%).

Step 3: Relationship Between Parts and Whole

Since 80% is a part of the whole (100%), the whole must be larger than the part. That means 100% corresponds to a number greater than 44.

Step 4: Applying the Concept to the Exercise

Given that 44 represents 80% of a number, the full value (100%) needs to be larger to allow 44 to represent only 80% of it.

Key Concepts

Percentage Calculation

Percentage problems can seem tricky at first, but once you understand the basics, they become much easier to handle. A percentage represents a part of a whole, expressed as a fraction of 100. For instance, 80% means 80 out of 100. In mathematical terms, this is written as \(\frac{80}{100}\).
When you're dealing with problems like '44 is 80% of what number?' you are essentially trying to find the whole, given a part and the percentage it represents.
The formula you'll use is: \(\text{Whole} = \frac{\text{Part}}{\text{Percentage}} \times 100\).
Plug in the numbers you know: \(\text{Whole} = \frac{44}{80} \times 100 = 55\). Now you can see that the whole number is 55.

Part and Whole Relationship

Understanding the relationship between the part and the whole is crucial in solving percentage problems. Here, 44 is a part, and we need to find the whole. This whole should logically be bigger than the part because the part is a fraction of the whole.
Consider a pie. If you have 80% of a pie (the part) and you want to find out how big the whole pie is, that whole pie will naturally be larger than just 80% of it. So, if 44 is 80% of some number, that number has to be bigger than 44.
Whenever you look at percentage problems, keep this relationship in mind:

• Part < Whole
• Whole is more significant than any fractional part

Elementary Algebra

Elementary algebra plays a significant role in solving percentage problems. Here, you use variables and equations to find unknown values. If you let \(x\) represent the number that 44 is a percentage of, you can set up an equation: \(0.80 \times x = 44\).
Solving for \(x\) involves dividing both sides by 0.80: \(\frac{44}{0.80} = x\). Simplifying this, you get \(x = 55\).
Algebra helps you systematically solve for unknown values by following simple steps:

• Set up your equation based on the problem
• Isolate the variable by performing inverse operations
• Simplify the equation to find your solution

Following these steps, the solution becomes straightforward and manageable.