Answered

Write the following expanded decimal form as a standard decimal:

$1 \times 1+8 \times \frac{1}{10}+2 \times \frac{1}{100}$

Verified Answer :

  • Identify the place value for each term in the expression: \(1\) in the ones place, \(8\) in the tenths place, and \(2\) in the hundredths place.
  • Convert each product to decimal form: \(1 \times 1 = 1\), \(8 \times 0.1 = 0.8\), and \(2 \times 0.01 = 0.02\).
  • Add the results together: \(1 + 0.8 + 0.02 = 1.82\).
  • State the final decimal value: \(\boxed{1.82}\).

Explanation

  1. Analyzing the Expression and Place Values

The expression is given as the sum of three products: \(1 \times 1 + 8 \times \frac{1}{10} + 2 \times \frac{1}{100}\). This is a base-10 expansion where each term represents a digit at a specific place value. We identify these values as:

  • The term \(1 \times 1 = 1\) represents the ones place (\(10^0\)).
  • The term \(8 \times \frac{1}{10} = 0.8\) represents the tenths place (\(10^{-1}\)).
  • The term \(2 \times \frac{1}{100} = 0.02\) represents the hundredths place (\(10^{-2}\)).
  1. Calculating the Final Decimal Value

Now that we have identified the value of each part, we add them together to find the combined decimal number. Performing the arithmetic:
\[1 + 0.8 + 0.02 = 1.82\]
Thus, the sum of the components leads us directly to the decimal representation.

  1. Conclusion

By summing the values of each place, we have determined that \(1 \times 1 + 8 \times \frac{1}{10} + 2 \times \frac{1}{100} = 1.82\). The final result is \(\boxed{1.82}\).

Examples

Understanding the expanded form of decimals is crucial for tasks like counting money or measuring lengths in different units. For example, if you have 1 dollar, 8 dimes, and 2 pennies, the total value can be expressed as \(1 \times \$1 + 8 \times \$0.10 + 2 \times \$0.01\). This calculation simplifies to \(\$1 + \$0.80 + \$0.02 = \$1.82\), showing how decimal place values allow us to combine different currency units into a single, manageable amount.