Solving Multiplication with Decimals and Powers of Ten

We are given a fundamental multiplication fact: 19 multiplied by 23 equals 437. We need to use this information to find the value of a related expression involving decimals and larger numbers: (190 × 0.023).

Let's break down the expression (190 × 0.023) and see how it relates to the given information (19 × 23 = 437).

We can express 190 as 19 multiplied by 10:

\( 190 = 19 \times 10 \)

We can express 0.023 as 23 multiplied by a decimal. To figure out the decimal, let's think about place values. 0.023 has the digit 2 in the hundredths place and the digit 3 in the thousandths place. This is the same as 23 thousandths, which can be written as 23 multiplied by 0.001:

\( 0.023 = 23 \times 0.001 \)

Alternatively, using powers of ten, 0.001 is \(10^{-3}\):

\( 0.023 = 23 \times 10^{-3} \)

Now, substitute these into the expression we need to evaluate:

\( 190 \times 0.023 = (19 \times 10) \times (23 \times 0.001) \)

Using the associative and commutative properties of multiplication, we can rearrange the terms:

\( (19 \times 10) \times (23 \times 0.001) = (19 \times 23) \times (10 \times 0.001) \)

We know from the problem statement that \( 19 \times 23 = 437 \). Substitute this value:

\( (19 \times 23) \times (10 \times 0.001) = 437 \times (10 \times 0.001) \)

Now, calculate the value of \( 10 \times 0.001 \). Multiplying a decimal by 10 moves the decimal point one place to the right:

\( 10 \times 0.001 = 0.010 = 0.01 \)

So the expression becomes:

\( 437 \times 0.01 \)

Multiplying a number by 0.01 (or \(10^{-2}\) or one-hundredth) is equivalent to dividing by 100. This moves the decimal point two places to the left. The number 437 can be written as 437.0.

\( 437.0 \times 0.01 = 4.37 \)

Therefore, the value of \( (190 \times 0.023) \) is 4.37.

Let's compare this result with the given options:

• 0.0437
• 0.437
• 43.7
• 4.37

Our calculated value, 4.37, matches one of the options.

Here is a summary of the steps:

The final answer is 4.37.

Revision Table: Key Multiplication Concepts

Additional Information: Decimal Multiplication and Place Value

Understanding place value is crucial when multiplying by decimals. In the number 0.023, the digit 2 is in the hundredths place (\(10^{-2}\)) and the digit 3 is in the thousandths place (\(10^{-3}\)).

When you multiply a number by 0.01, you are essentially finding one-hundredth of that number. For example, 1% of 437 is \(437 \times 0.01\), which is 4.37.

Consider the number of decimal places in the factors. In the expression \( 190 \times 0.023 \):

• 190 has 0 decimal places (considered as 190.0)
• 0.023 has 3 decimal places

The product should ideally have \(0 + 3 = 3\) decimal places if we were just multiplying the numbers ignoring the base fact \(19 \times 23\). If we did \(190 \times 23 = 4370\), then \(190 \times 0.023\) would be 4370 with the decimal moved three places to the left, resulting in 4.370 or 4.37.

Using the given information \(19 \times 23 = 437\) makes the calculation simpler by allowing us to factor out powers of 10.

\( 190 \times 0.023 = (19 \times 10) \times (23 \times 0.001) \)

\( = (19 \times 23) \times (10 \times 0.001) \)

\( = 437 \times 0.01 \)

This calculation \(437 \times 0.01\) clearly shows the decimal point moving two places to the left from the original 437, resulting in 4.37.