Estimate $0.23 \times 61.62$ by rounding each number to the place of its leftmost nonzero digit.
We are tasked with estimating the product of two numbers, \(0.23\) and \(61.62\), by rounding each factor to the place of its leftmost nonzero digit. This process helps us quickly arrive at a reasonable approximation for multiplication operations. The leftmost nonzero digit of a number is the first digit starting from the left that is not zero. We need to identify these digits for both factors and round accordingly.
For the number \(0.23\), the digits are \(0\), \(2\), and \(3\). The leftmost nonzero digit is the \(2\), which is in the tenths place. To round this number to the tenths place, we look at the digit to its right, which is \(3\). Since \(3 < 5\), we round down, which keeps the tenths place digit as \(2\). Thus, \(0.23\) rounds to \(0.2\).
For the number \(61.62\), the digits are \(6, 1, 6, 2\). The leftmost nonzero digit is the \(6\), which is in the tens place. To round this number to the tens place, we look at the digit to its right, which is \(1\). Since \(1 < 5\), we round down, which keeps the tens place digit as \(6\) and replaces the other digits with zeros (for the integer part). Thus, \(61.62\) rounds to \(60\).
Now that we have rounded each factor, we multiply the rounded values together:
\[0.2 \times 60\]
To compute this, we can rewrite \(0.2\) as \(\frac{2}{10}\):
\[\frac{2}{10} \times 60 = 2 \times 6 = 12\]
The estimated product of \(0.23 \times 61.62\) is therefore \(12\).
In everyday shopping, if you are quickly checking if you have enough money, you might use estimation to simplify complex prices. For instance, if you want to buy 0.23 kilograms of an item priced at \$61.62 per kilogram, rounding the values allows for a fast mental estimate. By calculating @@MATH_TOKEN_34@@, you can quickly estimate that the cost will be around \$12, which helps in making quick budgeting decisions without needing precise arithmetic on the spot.