Problem 82 - Prealgebra

Question

Multiply. $$0.15(63)$$

Step-by-Step Solution

Answer

The result of multiplying 0.15 by 63 is 9.45.

Step 1: Convert the Decimal to a Fraction

First, recognize that the decimal \(0.15\) can be converted into a fraction. The decimal \(0.15\) is equal to \(\frac{15}{100}\), because it is 15 hundredths.

Step 2: Simplify the Fraction

The fraction \(\frac{15}{100}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 5. Doing so, we get \(\frac{3}{20}\).

Step 3: Express the Multiplication

We now need to multiply \(\frac{3}{20}\) by the whole number 63. This multiplication is \(\frac{3}{20} \times 63\).

Step 4: Multiply the Fraction by the Whole Number

To multiply \(\frac{3}{20} \times 63\), you can rewrite it as \(\frac{3 \times 63}{20}\). This simplifies to \(\frac{189}{20}\).

Step 5: Simplify the Resulting Fraction

The fraction \(\frac{189}{20}\) is an improper fraction. Divide 189 by 20 to convert it to a mixed number. Doing the division, we get 9 whole parts with a remainder of 9, so \(\frac{189}{20} = 9\frac{9}{20}\).

Step 6: Verify the Solution

Finally, we check that multiplying \(0.15\) by 63 directly gives us 9.45, and since \(9\frac{9}{20}\) is equal to 9.45, our calculations are correct.

Key Concepts

Convert Decimals to Fractions

To convert decimals into fractions, you need to understand the place value of the digits in the decimal number. Consider the decimal 0.15. The "1" is in the tenths place, and the "5" is in the hundredths place. Therefore, 0.15 is equivalent to 15 hundredths, or \(\frac{15}{100}\). This process is straightforward for any decimal:

• Identify the last digit's place value. In 0.15, it's the hundredths place.
• Write the decimal as a fraction. Here, 0.15 becomes \(\frac{15}{100}\).
• If needed, simplify the fraction, which we'll discuss in the next section.

By converting a decimal to a fraction, you make it easier to perform operations such as multiplication and division, especially when dealing with whole numbers.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This is done by finding the greatest common divisor (GCD) of the numerator and the denominator. Looking at \(\frac{15}{100}\), both 15 and 100 can be divided by 5, the GCD:

• Divide 15 by 5 to get 3.
• Divide 100 by 5 to get 20.

So \(\frac{15}{100}\) simplifies to \(\frac{3}{20}\). Simplified fractions are easier to work with and understand. Always check to see if both numerator and denominator can be reduced further by exploring their divisors. This simplification helps not just in calculations but also in understanding proportions and comparing fractions easily.

Improper Fractions

An improper fraction has a numerator larger than its denominator, indicating that the value of the fraction is greater than one. For example, \(\frac{189}{20}\) from our multiplication result is an improper fraction because 189 (numerator) is larger than 20 (denominator).

Improper fractions are handled differently:

• They can be left as they are for certain calculations or when needing exact values.
• Often, they are converted to mixed numbers to better represent their value, especially when used in real-world contexts or additional operations.

Understanding improper fractions is important as they frequently appear in results, especially after multiplying fractions by whole numbers.

Mixed Numbers

When you have an improper fraction like \(\frac{189}{20}\), it's often converted to a mixed number because mixed numbers provide a clearer understanding of size and value. To convert an improper fraction to a mixed number:

• Perform division on the numerator by the denominator. Divide 189 by 20.
• The quotient, 9, is the whole number portion.
• The remainder, 9, becomes the numerator of the fractional part, resulting in \(9\frac{9}{20}\). The denominator remains 20.

Mixed numbers make it easy to visualize and interpret the quantity, combining whole numbers with fractional elements. For instance, 9\(\frac{9}{20}\) tells us the value is 9 and almost half of another whole, giving a more intuitive sense of scale.