Problem 31 - Prealgebra

Question

Estimate. Explain which method you used to estimate. $$88 \% \text { of } 72$$

Step-by-Step Solution

Answer

Estimate 88% of 72 by calculating 90% of 72, which is approximately 64.8.

Step 1: Understand the Problem

We need to estimate the value of 88% of 72. This involves finding a percentage of a number, whose precise calculation might be complex mentally.

Step 2: Round to Easy Numbers

To simplify calculations, round 88% to a nearby easy percentage, such as 90%. This approximation will make it simpler to work with while being close to the original percentage.

Step 3: Calculate 90% of 72

To find 90% of 72, calculate 10% first and then multiply it by 9. First, find 10% of 72 by dividing 72 by 10: \[ 72 \div 10 = 7.2 \]Now, multiply 7.2 by 9 to get 90%:\[ 7.2 \times 9 = 64.8 \]

Step 4: Evaluate the Estimation

The calculation of 90% of 72 is 64.8, which serves as an estimation of 88% of 72. Since 90% is slightly more than 88%, this estimation will be slightly higher than the actual value, reflecting our initial simplification decision.

Key Concepts

Understanding Rounding in Estimations

Rounding numbers is a valuable tool in mental math, especially for percentage estimation. By choosing nearby round numbers, we simplify complex calculations, making them faster and easier to perform in our heads. In this example, we round 88% to 90% because 90% is a simpler and more familiar figure. When rounding:

• Choose numbers that make mental calculations straightforward.
• Retain the original number's approximate value.

For example, rounding 88% to 90% is reasonable because it necessitates fewer steps to calculate percentages of whole numbers. By using 90%, we are only slightly skewing the result, allowing for quick and convenient mental computation, while still maintaining an acceptable level of accuracy.

Utilizing Mental Math Strategies

Practicing mental math strategies provides a quick and efficient way to solve numerical problems without needing a calculator. When dealing with percentages, breaking numbers into more manageable parts is key. Start by calculating a simpler percentage, such as 10%, and then build upon it to find more complex percentages like 90% or even 88%. Here’s how:

• Determine 10% of a number by shifting the decimal point one place to the left. For example, 10% of 72 is 7.2.
• Multiply the 10% result by an appropriate factor – in our case, 9 – to find 90%.

These steps illustrate that by dividing the task into smaller, more digestible parts, and manipulating numbers strategically, complex percentage calculations can become significantly more approachable and less intimidating.

Demystifying Percentage Calculation

Percentage calculation is a fundamental math skill that can help in everyday decisions, like how much you're saving during a sale, or what interest rates mean. By understanding how percentages relate to whole numbers, you make estimations and calculations easier. Think of percentages as a part of 100:

• To find a percentage of a number: divide the number by 100 and then multiply by the percentage desired.
• For instance, to find 88% of 72, you’d follow this process: First, simplify your calculations by choosing a close round number.

Once you've rounded and made calculations based on simpler figures like 90% instead of 88%, you achieve an estimated value that's close yet easily attained. As you practice these steps, percentage calculations can become second nature.