Problem 25 - Algebra 1: Concepts and Skills

Question

Solve the percent problem. 2% of what amount is $20?

Step-by-Step Solution

Answer

The amount in question, or the 'whole', is $1000.

Step 1: Understand the Problem

First, the most important part is to clearly understand what the problem is asking for. The problem presents that 2% of a certain amount is $20. Given this information, we know that the result or 'part' of a certain amount is $20 and that this corresponds to 2% of this original quantity or 'whole'.

Step 2: Rearrange the Formula

We use the formula \(percentage = \frac{part}{whole} \times 100%\). Given we need to find the 'whole', it is more convenient if we rearrange it to \(whole = \frac{part}{percentage} = \frac{part}{2%}\).

Step 3: Substitute Values into the Formula

Next, we substitute the given values into the formula. The 'part' is $20 and the 'percentage' is 2% or 0.02 when expressed as a decimal, hence \(whole = \frac{$20}{0.02}\).

Step 4: Compute for the Whole

Finally, we calculate the value of the whole which is the required amount. So the operation is \(whole = \frac{$20}{0.02} = $1000\).

Key Concepts

Understanding the Percentage Formula

The percentage formula is a crucial tool in solving percent problems. It connects three components: the percentage, the part, and the whole. You can think of the percentage as a fraction of 100, describing how one quantity compares to another.

The basic formula is given by:

• \[ \text{percentage} = \left( \frac{\text{part}}{\text{whole}} \right) \times 100 \]

This equation shows that the part (numerator) is a percentage of the whole (denominator).

To solve problems like "2% of what amount is $20?", it's necessary to rearrange the formula to solve for the unknown (in this case, the whole). By setting the problem backwards, you can solve for any component as long as the other two are known:

• \[ \text{whole} = \frac{\text{part}}{\text{percentage}} \]

Here, substituting the given values allows you to find the desired amount.

Using Algebra in Percent Problems

Algebra helps us manipulate equations to find unknown values. In our percent problem, algebra plays a key role in rearranging the percentage formula.

Initially, the formula reads:\[ \text{percentage} = \left( \frac{\text{part}}{\text{whole}} \right) \times 100 \]

To isolate the 'whole', we use algebraic techniques to rearrange the formula. We aim to express 'whole' in terms of the known quantities 'part' and 'percentage'. This is achieved by:

• Dividing both sides by 100 to leave the fraction alone.
• Flipping the equation to solve for the whole, yielding: \[ \text{whole} = \frac{\text{part}}{\text{percentage}} \]

Thus, algebra simplifies the percentage calculation, making it straightforward to plug in known values and find the missing number.

Through algebra, the seemingly complex problem becomes more manageable, providing a reliable pathway to the solution.

Approaching Problem-Solving Steps for Percent Problems

Solving percent problems involves a systematic approach. Breaking it down into clear steps can make the process much simpler and more effective.

Here’s a guide to tackle such problems:

• Understand the Problem: Begin by identifying what's given and what's needed. In our case, we know 2% of the amount equals \(20, and we need to find the total amount.
• Rearrange the Formula: Adjust the percentage formula to isolate the unknown quantity, which is the whole in this problem. This is done by expressing the whole as \[ \frac{\text{part}}{\text{percentage}} \].
• Substitute Values: Insert the known values into the rearranged formula. Convert the percentage into decimal form (e.g., 2% becomes 0.02).
• Calculate the Result: Perform the arithmetic operation to find the solution. In the example, calculating \[ \frac{20}{0.02} \] gives us the answer of \)1000.

Following these steps consistently ensures accuracy and builds confidence in handling similar problems in the future.