Understanding the Problem: Finding the Other Decimal

The question asks us to find one of two decimal numbers when their product is given, along with the value of the other decimal number. We are told that the product of two decimals is 0.768, and one of the decimal numbers is 1.6. We need to determine the value of the second decimal number.

Setting up the Equation for Decimal Product

Let the two decimal numbers be represented by variables, say \(a\) and \(b\). We are given:

• Product of the two decimals: \(a \times b = 0.768\)
• One of the decimal numbers: Let's say \(a = 1.6\)

We need to find the value of the other decimal number, which is \(b\). Substituting the known value into the equation, we get:

\(1.6 \times b = 0.768\)

To find \(b\), we need to perform division. We will divide the product (0.768) by the known decimal number (1.6).

\(b = \frac{0.768}{1.6}\)

Performing Decimal Division

To divide by a decimal number, it's easiest to convert the divisor into a whole number. We can do this by multiplying both the numerator (dividend) and the denominator (divisor) by a power of 10. In this case, the divisor is 1.6, which has one decimal place. Multiplying by 10 will make it a whole number (16).

Multiply both parts of the fraction by 10:

\(b = \frac{0.768 \times 10}{1.6 \times 10} = \frac{7.68}{16}\)

Now we need to divide 7.68 by 16. We can perform long division.

Step-by-Step Long Division

We are dividing 7.68 by 16.

Step 1: Set up the long division. Place the decimal point in the quotient directly above the decimal point in the dividend (7.68).

Step 2: Divide 7 by 16. 16 does not go into 7. Write 0 in the quotient before the decimal point.

Step 3: Consider the first two digits after the decimal point, 76. Divide 76 by 16. \(16 \times 4 = 64\). \(16 \times 5 = 80\). So, 16 goes into 76 four times. Write 4 in the quotient after the decimal point.

Step 4: Subtract \(16 \times 4\) from 76. \(76 - 64 = 12\).

Step 5: Bring down the next digit, 8, to make the number 128.

Step 6: Divide 128 by 16. \(16 \times 8 = 128\). So, 16 goes into 128 eight times. Write 8 in the quotient after the 4.

Step 7: Subtract \(16 \times 8\) from 128. \(128 - 128 = 0\).

The division is complete as the remainder is 0.

The result of the division \(7.68 \div 16\) is 0.48.

Conclusion: The Other Decimal Number

The division shows that \(0.768 \div 1.6 = 0.48\).

Therefore, the other decimal number is 0.48.

We can check this by multiplying the two decimal numbers: \(1.6 \times 0.48\).

\(1.6 \times 0.48 = 0.768\)

This matches the product given in the question, confirming our answer is correct.

Revision Table: Key Concepts

Additional Information: Related Decimal Operations

Understanding decimal operations is crucial for solving various math problems. Here's a quick look at other related operations:

• Decimal Addition and Subtraction: Align the decimal points and then add or subtract as with whole numbers. Place the decimal point in the answer in the same column.
• Decimal Multiplication: Multiply the numbers as if they were whole numbers. Count the total number of decimal places in the numbers being multiplied. Place the decimal point in the product so it has that many decimal places.
• Converting Decimals to Fractions: A decimal can be written as a fraction with a denominator that is a power of 10 (10, 100, 1000, etc.), depending on the number of decimal places. For example, 0.48 is \(48/100\).

Practice with these operations helps build confidence in working with decimal numbers.