Problem 78 - Fundamentals of Mathematics

Question

Perform each multiplication and division. $$\frac{2}{3} \div \frac{15}{7} \cdot \frac{5}{6}$$

Step-by-Step Solution

Answer

The result is \( \frac{7}{27} \).

Step 1: Understanding the Problem

We need to perform a division and a multiplication with fractions: \( \frac{2}{3} \div \frac{15}{7} \cdot \frac{5}{6} \). Let's divide first, then multiply.

Step 2: Division of Fractions

To divide by a fraction, multiply by the reciprocal of that fraction. Thus, \( \frac{2}{3} \div \frac{15}{7} = \frac{2}{3} \times \frac{7}{15} \).

Step 3: Multiply the First Two Fractions

Multiply the fractions: \( \frac{2}{3} \times \frac{7}{15} = \frac{2 \cdot 7}{3 \cdot 15} = \frac{14}{45} \).

Step 4: Multiplication of the Result by Another Fraction

Multiply \( \frac{14}{45} \) by \( \frac{5}{6} \). Our expression is now \( \frac{14}{45} \times \frac{5}{6} \).

Step 5: Multiply the Numerators and Denominators

Perform the multiplication: \( \frac{14 \cdot 5}{45 \cdot 6} = \frac{70}{270} \).

Step 6: Simplifying the Fraction

Simplify \( \frac{70}{270} \) by finding the greatest common divisor (GCD) of the numerator and denominator. The GCD is 10, so divide both by 10: \( \frac{70 \div 10}{270 \div 10} = \frac{7}{27} \).

Key Concepts

Simplification of Fractions

Simplifying fractions is a fundamental skill in math that makes working with fractions much easier. When you simplify a fraction, you are finding an equivalent fraction where the numerator and denominator have no common factors other than 1. This is also known as reducing the fraction to its simplest form.
To simplify a fraction, such as \( \frac{70}{270} \), follow these steps:

• Identify any common factors between the numerator (70) and denominator (270).
• Find the greatest common divisor (GCD) of these numbers.
• Divide both the numerator and denominator by the GCD to get the simplest form.

In our example, the GCD is 10, so we divide both 70 and 270 by 10, resulting in \( \frac{7}{27} \). It's always worth the effort to simplify fractions because it keeps your calculations straightforward and easier to read.

Reciprocal of a Fraction

Understanding the concept of the reciprocal is essential when dividing fractions. The reciprocal of a fraction is simply flipping the numerator and the denominator. For example, the reciprocal of \( \frac{15}{7} \) is \( \frac{7}{15} \).
Here are some key points about reciprocals:

• Exclusive for non-zero numbers. Zero does not have a reciprocal because division by zero is undefined.
• When you multiply a number by its reciprocal, you always get 1. For instance, \( \frac{15}{7} \times \frac{7}{15} = 1 \).
• In division problems involving fractions, replace the division operation with multiplication by the reciprocal of the divisor.

This process is crucial when dividing fractions as it turns a complex problem into a simpler multiplication task. In our exercise, to divide \( \frac{2}{3} \) by \( \frac{15}{7} \), we multiply \( \frac{2}{3} \times \frac{7}{15} \).

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. This concept is an essential tool in simplifying fractions.
To find the GCD of two numbers, such as 70 and 270, follow these steps:

• List the factors of each number.
• Identify the largest factor common to both lists.

For example, the factors of 70 are 1, 2, 5, 7, 10, 14, 35, and 70. The factors of 270 include 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, 135, and 270.
The largest common factor here is 10, making it the GCD. Once you find the GCD, you can use it to simplify fractions by dividing both the numerator and the denominator by this number. Understanding the GCD concept not only helps in simplifying fractions but also in many other areas of math, like finding least common multiples (LCM) and solving problems involving rational numbers.