Problem 123 - Prealgebra

Question

Multiply or divide as indicated. $$\frac{35}{110} \cdot \frac{80}{63} \div \frac{16}{27}$$

Step-by-Step Solution

Answer

The simplified result is \( \frac{5}{6} \).

Step 1: Express as Multiplication

Instead of dividing by a fraction, we multiply by its reciprocal. Thus, the expression \( \frac{35}{110} \cdot \frac{80}{63} \div \frac{16}{27} \) becomes \( \frac{35}{110} \cdot \frac{80}{63} \cdot \frac{27}{16} \).

Step 2: Multiply Numerators and Denominators

Multiply the numerators together and the denominators together: \( (35 \cdot 80 \cdot 27) \) for the numerator and \( (110 \cdot 63 \cdot 16) \) for the denominator.

Step 3: Simplify the Fractions

To simplify, we look for common factors among the numerators and denominators. Simplifying directly: \( \frac{35 \cdot 80 \cdot 27}{110 \cdot 63 \cdot 16} \).- Divide 80 by 16 to get 5.- Divide 110 by 35 to get 2.- Divide 63 by 27 to get 3.After simplifying: \( \frac{1 \cdot 5 \cdot 1}{2 \cdot 1 \cdot 3} \) which simplifies to \( \frac{5}{6} \).

Key Concepts

Reciprocal of a Fraction

Dividing by a fraction might seem a bit tricky at first. However, knowing about reciprocals can make things simpler. The reciprocal of a fraction is just flipping the numerator and the denominator. For example, the reciprocal of \( \frac{16}{27} \) is \( \frac{27}{16} \).

When you divide by a fraction, you actually multiply by its reciprocal. This changes our original problem from - \( \frac{35}{110} \cdot \frac{80}{63} \div \frac{16}{27} \)into - \( \frac{35}{110} \cdot \frac{80}{63} \cdot \frac{27}{16} \).Using reciprocals helps in simplifying multiplication and division problems, preventing confusion.

Simplifying Fractions

Fractions can get complex quickly, but simplifying them makes everything clearer. We simplify fractions by dividing both the numerator and the denominator by their greatest common factor (GCF). Doing this reduces the fraction to its simplest form, making it easy to work with and understand.

In the exercise, after multiplying the numerators and the denominators, we have a large fraction:- \( \frac{35 \cdot 80 \cdot 27}{110 \cdot 63 \cdot 16} \)To simplify, look for numbers that are common to both the numerator and the denominator:- Dividing 80 by 16 gives 5.- Dividing 110 by 35 gives 2.- Dividing 63 by 27 gives 3.This results in: - \( \frac{1 \cdot 5 \cdot 1}{2 \cdot 1 \cdot 3} \)The fraction simplifies to \( \frac{5}{6} \). Always aim to simplify fractions for clear solutions.

Numerators and Denominators

In any fraction, the top part is called the numerator, and the bottom part is the denominator. Understanding these terms helps when you perform operations like multiplication or division with fractions.

Taking our example,- \( \frac{35}{110} \) has a numerator of 35 and a denominator of 110. When multiplying fractions, you multiply all the numerators together and also multiply all the denominators together. So for our problem:- Multiply numerators: \( 35 \cdot 80 \cdot 27 \)- Multiply denominators: \( 110 \cdot 63 \cdot 16 \)Each plays a crucial role in determining the size and position in the fraction. Recognizing numerators and denominators in this way helps in both simplifying fractions and solving problems efficiently.