Problem 38 - Prealgebra

Question

Number Problem Find \(\frac{5}{6}\) of \(2 \frac{4}{15}\).

Step-by-Step Solution

Answer

\(\frac{5}{6}\) of \(2 \frac{4}{15}\) is \(1 \frac{8}{9}\).

Step 1: Convert Mixed Fraction

We begin by converting the mixed fraction \(2 \frac{4}{15}\) into an improper fraction. To do this, multiply the whole number (2) by the denominator (15), resulting in 30. Then add the numerator (4) to get a total of 34. Hence, the improper fraction is \(\frac{34}{15}\).

Step 2: Set Up Multiplication

To find \(\frac{5}{6}\) of \(2 \frac{4}{15}\), we take the improper fraction \(\frac{34}{15}\) and multiply it by \(\frac{5}{6}\). This results in the expression \(\frac{5}{6} \times \frac{34}{15}\).

Step 3: Multiply Numerators and Denominators

Multiply the numerators of the fractions: \(5 \times 34 = 170\), and multiply the denominators: \(6 \times 15 = 90\). Thus, the product is \(\frac{170}{90}\).

Step 4: Simplify the Fraction

Simplify \(\frac{170}{90}\) by finding the greatest common divisor of 170 and 90, which is 10. Divide both the numerator and the denominator by 10 to get \(\frac{17}{9}\).

Step 5: Convert Improper Fraction to Mixed Number

Convert the improper fraction \(\frac{17}{9}\) to a mixed number by dividing 17 by 9, which equals 1 with a remainder of 8. This gives us the mixed number \(1 \frac{8}{9}\).

Key Concepts

Mixed Fractions

A mixed fraction combines a whole number with a fraction. For example, when you see a mixed fraction like \(2 \frac{4}{15}\), it means you have 2 whole parts plus \(\frac{4}{15}\) of another part.
Understanding mixed fractions is crucial as it is often the starting point in fraction problems. They are most commonly used in daily life activities, such as cooking or dividing materials, because they make it clear how much more than a whole there is of something.
To use them in calculations, mixed fractions need to be converted to improper fractions first. This involves multiplying the whole number by the fraction's denominator and then adding the numerator. So, \(2 \frac{4}{15}\) converts to \(\frac{34}{15}\) because \(2 \times 15 = 30\) and \(30 + 4 = 34\).
Remember, turning mixed into improper fractions makes complex calculations easier to handle.

Improper Fractions

An improper fraction has a numerator that's greater than or equal to its denominator. These fractions can be useful in mathematical calculations because they are easier to manipulate in operations like multiplication and division.
Improper fractions look like \(\frac{34}{15}\), where the top number (34) is larger than the bottom number (15). Converting a mixed fraction into an improper fraction is a key step, as it allows for direct calculation, such as in multiplication \(\frac{34}{15} \times \frac{5}{6}\).
While they may not always be practical for direct interpretation without conversion back to mixed numbers, improper fractions allow us to perform mathematical operations more seamlessly. Once calculations are complete, it's often helpful to convert the result back into a mixed number for easier comprehension.

Fraction Multiplication

Multiplying fractions involves multiplying the numerators together for a new numerator and the denominators together for a new denominator. This operation often arises in problems where we need to find a fraction of a fraction or number.
For example, to find \(\frac{5}{6}\) of \(\frac{34}{15}\), multiply the numerators \(5\) and \(34\) to get \(170\), and the denominators \(6\) and \(15\) to get \(90\). The result is the fraction \(\frac{170}{90}\).

• Step 1: Multiply the numerators \(5 \times 34 = 170\).
• Step 2: Multiply the denominators \(6 \times 15 = 90\).
• Step 3: Combine to get \(\frac{170}{90}\).

This straightforward method shows how fractional parts interact and result in a new fraction that represents the combined or scaled part of the original amounts.

Simplifying Fractions

Once you have a fraction, simplifying it is essential for ease of understanding and use. Simplifying involves reducing the fraction to its simplest form, where the numerator and denominator share no common factors other than 1.
Take \(\frac{170}{90}\) as an example. To simplify, find the greatest common divisor (GCD) of the numerator and denominator. Here, the GCD is 10. Dividing both 170 and 90 by 10, we get the simplified fraction \(\frac{17}{9}\).
Checklist for Simplifying:

• Find the GCD of the numerator and denominator.
• Divide both by this GCD.
• Result in the simplest form \(\frac{17}{9}\).

Simplifying fractions is useful for clearer communication and more efficient calculations, making sure the fraction is easily interpreted and used in subsequent steps or contexts.