Problem 46 - Fundamentals of Mathematics

Question

For the following problems, find each part without using a diagram. $$5 \frac{1}{3}\( of \)9 \frac{3}{4}$$

Step-by-Step Solution

Answer

The result is 52.

Step 1: Convert Mixed Numbers to Improper Fractions

Convert the mixed numbers into improper fractions. To do this, multiply the whole number by the denominator, then add the numerator.For \(5 \frac{1}{3}\): - Multiply 5 by 3 and add 1: \[5 \times 3 + 1 = 16\] This gives you \(\frac{16}{3}\).For \(9 \frac{3}{4}\): - Multiply 9 by 4 and add 3: \[9 \times 4 + 3 = 39\] This gives you \(\frac{39}{4}\).

Step 2: Multiply the Improper Fractions

Now multiply the two improper fractions together. \[\frac{16}{3} \times \frac{39}{4}\]Multiply the numerators: \[16 \times 39 = 624\]Multiply the denominators:\[3 \times 4 = 12\]The result is the fraction:\[\frac{624}{12}\]

Step 3: Simplify the Fraction

To simplify \(\frac{624}{12}\), divide the numerator by the denominator. \[\frac{624}{12} = 52\]This means the multiplication of the two mixed numbers in simplest form is 52.

Key Concepts

Improper Fractions

When working with mixed numbers, it is often useful to convert them into improper fractions for ease of computation, especially in multiplication and division scenarios. A mixed number consists of a whole number and a fraction, such as \(5 \frac{1}{3}\). To convert a mixed number to an improper fraction, you can follow these simple steps:

• Multiply the whole number by the denominator of the fractional part.
• Add the result to the numerator of the fractional part.

For instance, in \(5 \frac{1}{3}\), you multiply the whole number 5 by the denominator 3, giving you 15. Then, add the numerator 1, resulting in a total of 16. Therefore, \(5 \frac{1}{3}\) becomes \(\frac{16}{3}\). This approach transforms any mixed number into an improper fraction, where the numerator is larger than the denominator, simplifying subsequent mathematical operations.

Multiplying Fractions

After converting your mixed numbers to improper fractions, the next step in solving problems like \(5 \frac{1}{3} \text{ of } 9 \frac{3}{4}\) is multiplying these fractions together. Fraction multiplication is straightforward:

• Multiply the numerators of the fractions together to get the new numerator.
• Multiply the denominators of the fractions together to get the new denominator.

For instance, consider multiplying \(\frac{16}{3}\) and \(\frac{39}{4}\). You multiply the numerators: \(16 \times 39 = 624\), and the denominators: \(3 \times 4 = 12\), resulting in the fraction \(\frac{624}{12}\). Multiplying fractions is simpler than adding or subtracting because there is no need for a common denominator—just multiply straight across the numerators and denominators.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form where the numerator and the denominator have no common factors other than 1. This simplification can make the fraction easier to understand or more pleasant to work with in future calculations. To simplify a fraction, follow these steps:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

In the example \(\frac{624}{12}\), you can simplify by dividing both the numerator and the denominator by their GCD, which is 12. When you perform the division, you get \(\frac{624}{12} = 52\), a whole number. Simplification helps present answers in their most reduced form, making them easier to interpret and use further.