Solving the Decimal Division Problem

The problem asks us to calculate the value of \(0.0437 \div 1.9\), given the information that \(23 \times 19 = 437\). This initial multiplication fact provides a crucial hint for solving the division problem without needing to perform long division on the decimals.

Understanding the Relationship Between the Numbers

We are given the multiplication fact:

\(23 \times 19 = 437\)

From this multiplication fact, we can derive corresponding division facts:

• \(437 \div 19 = 23\)
• \(437 \div 23 = 19\)

Now, let's look at the division problem we need to solve: \(0.0437 \div 1.9\). Notice that the numbers involved, \(0.0437\) and \(1.9\), are related to \(437\) and \(19\) from the given fact, just with different decimal point positions.

Solving the Division with Decimals

We know that \(437 \div 19 = 23\). The division problem is \(0.0437 \div 1.9\).

Let's compare the numbers:

• The dividend \(0.0437\) is \(437 \times 0.0001\) (because \(437\) is moved 4 places to the right to get \(0.0437\)).
• The divisor \(1.9\) is \(19 \times 0.1\) (because \(19\) is moved 1 place to the right to get \(1.9\)).

So the problem is essentially \((437 \times 0.0001) \div (19 \times 0.1)\).

We can rewrite this as:

\(\frac{437 \times 0.0001}{19 \times 0.1} = \frac{437}{19} \times \frac{0.0001}{0.1}\)

We already know that \(\frac{437}{19} = 23\). Now, let's evaluate the decimal part:

\(\frac{0.0001}{0.1}\)

To divide decimals, we can make the divisor a whole number by multiplying both the numerator and the denominator by the same power of 10. Multiply by 10:

\(\frac{0.0001 \times 10}{0.1 \times 10} = \frac{0.001}{1} = 0.001\)

Alternatively, consider the number of decimal places:

• \(0.0001\) has 4 decimal places.
• \(0.1\) has 1 decimal place.

When dividing, subtract the number of decimal places in the divisor from the number of decimal places in the dividend: \(4 - 1 = 3\). The result should have 3 decimal places, based on dividing \(1\) by \(10\). \(1 \div 10 = 0.1\). No, this is not the correct way to think about dividing the decimal factors. Let's go back to the basic division rule.

When you divide a number with \(n\) decimal places by a number with \(m\) decimal places, the quotient will have \(n-m\) decimal places, provided the whole number division gives the correct sequence of digits.

In our case, \(0.0437\) has \(n=4\) decimal places.

\(1.9\) has \(m=1\) decimal place.

The whole number division is \(437 \div 19 = 23\). The result will have the digits 23.

The number of decimal places in the quotient should be \(n - m = 4 - 1 = 3\).

So, we take the digits 23 and place the decimal point such that there are 3 decimal places. We need to add a zero before the 23 and place the decimal point before that: \(0.023\).

Let's verify this by multiplying the quotient by the divisor:

\(0.023 \times 1.9\)

Ignore the decimal points and multiply: \(23 \times 19 = 437\).

Now, count the total number of decimal places in the factors:

• \(0.023\) has 3 decimal places.
• \(1.9\) has 1 decimal place.
• Total decimal places in the product = \(3 + 1 = 4\).

Place the decimal point in 437 so it has 4 decimal places: \(0.0437\).

This matches the original dividend, \(0.0437\).

Therefore, \(0.0437 \div 1.9 = 0.023\).

Let's look at the options provided:

• 0.0023
• 2.3
• 0.023
• 0.23

Our calculated answer is \(0.023\), which matches one of the options.

Final Answer

Based on the calculation and the relationship derived from the given multiplication, the value of \(0.0437 \div 1.9\) is \(0.023\).

Revision Table: Decimal Division

Additional Information: Working with Decimals

Understanding how to work with decimals is fundamental in mathematics. Here are some additional points:

• Place Value: Each digit in a decimal number has a place value based on powers of 10. For example, in 0.023, the 2 is in the hundredths place (\(10^{-2}\)) and the 3 is in the thousandths place (\(10^{-3}\)).
• Converting to Fractions: Decimals can be easily converted to fractions, which can sometimes simplify operations. For example, \(0.0437 = \frac{437}{10000}\) and \(1.9 = \frac{19}{10}\). The division becomes \(\frac{437}{10000} \div \frac{19}{10} = \frac{437}{10000} \times \frac{10}{19} = \frac{437 \times 10}{10000 \times 19} = \frac{4370}{190000}\). Since \(437 \div 19 = 23\), this is \(\frac{23 \times 190}{1000 \times 190} = \frac{23}{1000} = 0.023\). This confirms our answer.
• Powers of 10: Multiplying a decimal by \(10^n\) moves the decimal point \(n\) places to the right. Dividing a decimal by \(10^n\) moves the decimal point \(n\) places to the left.