Problem 102 - Fundamentals of Mathematics

Question

Find \(8 \frac{3}{10}\) of \(16 \frac{2}{3}\).

Step-by-Step Solution

Answer

The answer is 138 \( \frac{1}{3} \).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers \( 8 \frac{3}{10} \)and \( 16 \frac{2}{3} \)into improper fractions.For \( 8 \frac{3}{10} \):Multiply 8 by 10 and add 3:\( 8 \times 10 + 3 = 80 + 3 = 83 \)So, \( 8 \frac{3}{10} = \frac{83}{10} \).For \( 16 \frac{2}{3} \):Multiply 16 by 3 and add 2:\( 16 \times 3 + 2 = 48 + 2 = 50 \)So, \( 16 \frac{2}{3} = \frac{50}{3} \).

Step 2: Multiply the Improper Fractions

To find \( 8 \frac{3}{10} \)of\( 16 \frac{2}{3} \),multiply the corresponding improper fractions\( \frac{83}{10} \)and\( \frac{50}{3} \).The multiplication of fractions involves multiplying the numerators together and the denominators together:\[ \frac{83 \times 50}{10 \times 3} = \frac{4150}{30} \].

Step 3: Simplify the Resulting Fraction

Simplify the fraction\( \frac{4150}{30} \).First, find the greatest common divisor (GCD) of 4150 and 30, which is 10.Divide both the numerator and the denominator by 10:\[ \frac{4150}{30} = \frac{415}{3} \].

Step 4: Convert the Simplified Fraction to a Mixed Number

Convert the improper fraction\( \frac{415}{3} \)back to a mixed number.Divide 415 by 3:- 415 divided by 3 is 138 with a remainder of 1.Thus, \( \frac{415}{3} = 138 \frac{1}{3} \).

Key Concepts

Mixed Numbers

Mixed numbers are numbers that have both a whole number and a fractional part. For example, in the mixed number \(8 \frac{3}{10}\), 8 is the whole number, and \(\frac{3}{10}\) is the fractional part. These are useful when expressing quantities that are more than a whole but not entirely another whole number.

To convert mixed numbers into improper fractions (a fraction where the numerator is larger than the denominator), you multiply the whole number by the denominator of the fractional part, add the numerator, and then place it over the original denominator. So for \(8 \frac{3}{10}\), you do the following:

• Multiply the whole number 8 by the denominator 10: \(8 \times 10 = 80\).
• Add the numerator 3 to get: \(80 + 3 = 83\).
• Therefore, \(8 \frac{3}{10}\) becomes \(\frac{83}{10}\).

Improper Fractions

An improper fraction is a type of fraction where the numerator, the number above the fraction line, is larger than or equal to the denominator, the number below the fraction line. This can sometimes look less intuitive, but it can simplify operations such as adding, subtracting, and multiplying fractions. For instance, instead of using \(16 \frac{2}{3}\), you can write it as \(\frac{50}{3}\).

Conversion involves:

• Multiplying the whole number part by the denominator, e.g., \(16 \times 3 = 48\).
• Adding the numerator: \(48 + 2 = 50\).
• Thus, it becomes \(\frac{50}{3}\).

Improper fractions are particularly advantageous when performing multiplication or division, as they eliminate the need for managing two separate parts of a mixed number.

Greatest Common Divisor (GCD)

The Greatest Common Divisor, or GCD, is the largest number that can exactly divide both the numerator and the denominator of a fraction. The GCD is crucial for simplifying fractions, making numbers more manageable and easier to interpret.

To simplify a fraction like \(\frac{4150}{30}\), both numbers must be divisible by their GCD, which in this case is 10. Here's how you apply it:

• Divide the numerator 4150 by 10: \(4150 \div 10 = 415\).
• Divide the denominator 30 by 10: \(30 \div 10 = 3\).
• So, \(\frac{4150}{30}\) simplifies to \(\frac{415}{3}\).

This step makes it easier to handle the fraction in future calculations.

Multiplication of Fractions

When multiplying fractions, the operation is straightforward: multiply the numerators with each other and the denominators with each other. Consider the improper fractions \(\frac{83}{10}\) and \(\frac{50}{3}\). You find the product by:

• Multiplying the numerators: \(83 \times 50 = 4150\).
• Multiplying the denominators: \(10 \times 3 = 30\).
• Form the new fraction: \(\frac{4150}{30}\).

After multiplying, you often need to simplify the result, as discussed in the section on the Greatest Common Divisor. Multiplying fractions is a process that maintains the relationship between the parts, making it a powerful tool in many math scenarios.