Problem 11 - Prealgebra

Question

Write your answers as proper fractions or mixed numbers, not as improper fractions. Find the following products. (Multiply.) $$\frac{3}{5} \cdot 5 \frac{1}{3}$$

Step-by-Step Solution

Answer

The product is \(3 \frac{1}{5}\).

Step 1: Convert Mixed Number to Improper Fraction

Convert the mixed number \(5 \frac{1}{3}\) to an improper fraction. Multiply the whole number 5 by the denominator 3 and add the numerator 1 to get \(\frac{16}{3}\).

Step 2: Multiply Fractions

Multiply the fractions \(\frac{3}{5}\) and \(\frac{16}{3}\). The multiplication of fractions is done by multiplying the numerators together and the denominators together: \(\frac{3 \cdot 16}{5 \cdot 3} = \frac{48}{15}\).

Step 3: Simplify the Fraction

Simplify \(\frac{48}{15}\) by finding the greatest common factor of 48 and 15, which is 3. Divide both the numerator and the denominator by 3 to get \(\frac{16}{5}\).

Step 4: Convert to Mixed Number

Convert \(\frac{16}{5}\) into a mixed number. Divide 16 by 5 to get 3 with a remainder of 1, so \(\frac{16}{5} = 3 \frac{1}{5}\).

Key Concepts

Converting Mixed Numbers

When you encounter a mixed number in math, like \(5 \frac{1}{3}\), it can be easier to work with if you convert it to an improper fraction. This process involves a couple of straightforward steps that simplify calculations, especially in multiplication and division problems.
When converting a mixed number, such as \(5 \frac{1}{3}\), follow these steps:

• First, multiply the whole number by the denominator of the fractional part. Here, multiply 5 by 3 to get 15.
• Then, add the numerator to this product. Adding 1 to 15 gives you 16.
• This result becomes the new numerator of the improper fraction, with the original denominator remaining the same, resulting in \(\frac{16}{3}\).

Converting mixed numbers makes it easier to perform arithmetic operations because improper fractions can be directly used in multiplication or division without having to worry about whole numbers separately.

Improper Fractions

Improper fractions are fractions where the numerator is larger than the denominator, such as \(\frac{16}{3}\). They are useful in calculations because they simplify the arithmetic process.
Here’s a breakdown of why improper fractions are handy:

• They allow you to perform multiplication and division of fractions without additional steps of converting between improper and proper forms during the calculation.
• When multiplying fractions, you simply multiply straight across: the numerators together and the denominators together.

For the multiplication of \(\frac{3}{5} \cdot \frac{16}{3}\), the result is obtained directly by multiplying the numerators and denominators:\[\frac{3 \times 16}{5 \times 3} = \frac{48}{15}\]This straightforward approach makes improper fractions convenient to use in mathematical operations.

Simplifying Fractions

After performing arithmetic with fractions, you may end up with a fraction that can be simplified, like \(\frac{48}{15}\). Simplifying fractions means reducing them to their simplest form.
The process of simplifying a fraction involves these steps:

• First, find the greatest common factor (GCF) of the numerator and the denominator. For \(48\) and \(15\), the GCF is \(3\).
• Next, divide both the numerator and the denominator by the GCF. So, \(\frac{48}{15}\) becomes \(\frac{16}{5}\) after dividing by \(3\).

Once simplified, if the problem requires the result as a proper fraction or mixed number, convert as necessary. For \(\frac{16}{5}\), convert to a mixed number by dividing the numerator by the denominator. The division of \(16\) by \(5\) gives \(3\) remainder \(1\), resulting in the mixed number \(3 \frac{1}{5}\). Simplification helps in presenting the answer in the neatest form possible and ensures clarity in communication of mathematical results.