Problem 26 - Fundamentals of Mathematics

Question

Find the value of each of the following. $$ 9 \frac{1}{3} \cdot \sqrt{\frac{81}{100}} $$

Step-by-Step Solution

Answer

The value is \(8 \frac{2}{5}\).

Step 1: Convert Mixed Number to Improper Fraction

The given expression involves a mixed number, \(9 \frac{1}{3}\). First, convert this mixed number to an improper fraction. Multiply the whole number 9 by the denominator 3, then add the numerator 1: \[9 \times 3 + 1 = 27 + 1 = 28\]So, \(9 \frac{1}{3} = \frac{28}{3}\).

Step 2: Simplify the Square Root

Next, simplify \(\sqrt{\frac{81}{100}}\). Recall that the square root of a fraction is the quotient of the square roots of the numerator and the denominator:\[\sqrt{\frac{81}{100}} = \frac{\sqrt{81}}{\sqrt{100}} = \frac{9}{10}\]

Step 3: Multiply the Improper Fraction by the Simplified Square Root

Now, multiply the improper fraction by the simplified square root:\[\frac{28}{3} \cdot \frac{9}{10}\]To multiply fractions, multiply the numerators together and the denominators together:\[\frac{28 \times 9}{3 \times 10} = \frac{252}{30}\]

Step 4: Simplify the Resulting Fraction

Simplify \(\frac{252}{30}\) by finding the greatest common divisor (GCD) of 252 and 30, which is 6. Divide both the numerator and the denominator by their GCD:\[\frac{252 \div 6}{30 \div 6} = \frac{42}{5}\]

Step 5: Convert the Fraction to a Mixed Number

Finally, convert \(\frac{42}{5}\) to a mixed number. Divide 42 by 5. The quotient is the whole number, and the remainder is the new numerator:- \(42 \div 5 = 8\) remainder \(2\)- The mixed number is \(8 \frac{2}{5}\).Thus, the final value is \(8 \frac{2}{5}\).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a proper fraction. They are used when a value is greater than a whole but not another complete whole number. For instance, in the mixed number \(9 \frac{1}{3}\), the 9 is the whole part, and \(\frac{1}{3}\) is the fractional part.
Converting mixed numbers to improper fractions simplifies calculations, especially in multiplication or division.

• Multiply the whole number by the fraction's denominator.
• Add the result to the fraction's numerator.
• The sum becomes the numerator of the improper fraction, with the original denominator.

For example, with \(9 \frac{1}{3}\), we calculate \(9 \times 3 + 1 = 28\), thus the mixed number is converted to \(\frac{28}{3}\). This step is crucial for computations.

Improper Fractions

Improper fractions have numerators larger than or equal to their denominators. They are useful for arithmetic operations because they offer a straightforward method of handling whole and fractional quantities together.
For example, \(\frac{28}{3}\) from the problem is an improper fraction representing the mixed number \(9 \frac{1}{3}\).
To understand their structure:

• The numerator (28) tells us how many parts we have.
• The denominator (3) shows the size of each part.

These fractions simplify arithmetic, like in multiplying by fractions, because they eliminate the need for separate handling of whole numbers and fractions.

Simplifying Square Roots

To simplify square roots, especially for fractions, we take the square root of both the numerator and the denominator independently. This is expressed as: \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\).
In our example, we have \(\sqrt{\frac{81}{100}}\).

• Calculate \(\sqrt{81} = 9\).
• Calculate \(\sqrt{100} = 10\).

Therefore, \(\sqrt{\frac{81}{100}} = \frac{9}{10}\). This simplification is necessary before performing any multiplication involving square roots to ensure accuracy and ease of further calculations.

Multiplying Fractions

When multiplying fractions, we multiply straight across: multiply the numerators together to get the new numerator and the denominators together for the new denominator.
Applying this to \(\frac{28}{3} \cdot \frac{9}{10}\):

• Multiply the numerators: \(28 \times 9 = 252\).
• Multiply the denominators: \(3 \times 10 = 30\).

The result is \(\frac{252}{30}\). Once we have the product, simplifying the fraction is important.
This may involve finding the greatest common divisor (GCD) to simplify \(\frac{252}{30}\) to its simplest form, \(\frac{42}{5}\), or converting it further to a mixed number if needed. Multiplying fractions requires this attention to simplification to provide clear and concise results.