Problem 7 - Prealgebra

Question

Estimate. Explain which method you used to estimate. $$\frac{1}{2} \% \text { of } 396$$

Step-by-Step Solution

Answer

The estimated result is 2.

Step 1: Convert Percentage to Decimal

To estimate, first convert the percentage into a decimal. The given percentage is \( \frac{1}{2}\% \), which is equivalent to \( \frac{1}{2 \times 100} = \frac{1}{200} = 0.005 \).

Step 2: Apply Decimal to the Number

Now that we have the decimal equivalent of the percentage, we multiply this by 396 to estimate the percentage of the number. So, calculate \( 0.005 \times 396 \).

Step 3: Estimate the Product

To simplify the calculation, consider rounding 396 to 400, which is a close approximation. Calculate \( 0.005 \times 400 \).

Step 4: Perform Multiplication

Multiply the decimal \( 0.005 \) by the rounded number 400: \( 0.005 \times 400 = 2 \). This gives us the estimated result.

Key Concepts

Decimal Conversion

Converting percentages to decimals is an essential mathematical skill, especially when solving real-world problems. To convert a percentage to a decimal:

• Understand that a percentage represents a part per hundred. For example, 45% is the same as 45 out of 100 or 45/100.
• To convert, divide the percentage by 100. This shifts the decimal point two places to the left.
• For fractional percentages, like \(\frac{1}{2}\%\), first deal with the fraction and then divide by 100. \(\frac{1}{2}\%\) is \(\frac{1}{2} \times \frac{1}{100} = \frac{1}{200} = 0.005\).

This conversion allows you to use straightforward multiplication to find percentages, simplifying calculations considerably.

Multiplication Estimation

Estimating products helps to quickly find an approximate answer, which is useful when an exact number isn't necessary or when making quick decisions.

• One common method is adjusting numbers to round numbers (like multiples of ten) that are easier to work with.
• In the problem \(0.005 \times 396\), you can save time by approximating 396 to 400. This small adjustment simplifies the multiplication significantly.
• The multiplication thus becomes \(0.005 \times 400\), a much simpler computation that yields \(2\) as the estimated product.

Using estimation in multiplication is a handy tool to maintain efficiency without the need for precise calculations all the time.

Rounding Numbers

Rounding is a mathematical process used to simplify numbers, making them easier to work with in estimation and mental calculations.

• To round a number, look at the digit immediately to the right of the desired rounding place. If it's 5 or more, round up the number; if it's less than 5, round down.
• In this context, rounding 396 to the nearest hundred gives 400, because 396 is closer to 400 than 300.
• This not only simplifies your multiplication step but also provides a clearer approximation without unnecessary complexity.

Rounding is a key technique in estimation tasks, avoiding over-complicated calculations while still achieving a result that's reliable enough for practical purposes.