Problem 29 - Prealgebra

Question

Find the following quotients. $$2 \frac{1}{2} \cdot\left(3 \frac{2}{5} \div 4\right)$$

Step-by-Step Solution

Answer

The quotient is \(2 \frac{1}{8}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers to improper fractions. For \(2 \frac{1}{2}\), multiply the whole number 2 by 2 (the denominator) and add 1 (the numerator), resulting in \(\frac{5}{2}\). For \(3 \frac{2}{5}\), multiply the whole number 3 by 5 and add 2, resulting in \(\frac{17}{5}\). So, the expression becomes \(\frac{5}{2} \cdot \left(\frac{17}{5} \div 4\right)\).

Step 2: Convert Division to Multiplication by Reciprocal

In the expression \(\frac{17}{5} \div 4\), convert the division to multiplication by the reciprocal. So, \(\frac{17}{5} \div 4\) is equivalent to \(\frac{17}{5} \cdot \frac{1}{4}\).

Step 3: Multiply the Fractions Inside the Parentheses

Multiply the fractions inside the parentheses: \(\frac{17}{5} \cdot \frac{1}{4} = \frac{17 \cdot 1}{5 \cdot 4} = \frac{17}{20}\). The expression is now \(\frac{5}{2} \cdot \frac{17}{20}\).

Step 4: Multiply All Fractions Together

Finally, multiply \(\frac{5}{2}\) by \(\frac{17}{20}\). Perform the multiplication: \(\frac{5 \cdot 17}{2 \cdot 20} = \frac{85}{40}\).

Step 5: Simplify the Fraction

Simplify \(\frac{85}{40}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 5. This yields \(\frac{17}{8}\). Now, convert \(\frac{17}{8}\) to a mixed number: 17 divided by 8 is 2 with a remainder of 1, so it’s \(2 \frac{1}{8}\).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are called "mixed" because they mix both of these components.
For example, in the exercise, you see expressions like \(2 \frac{1}{2}\) and \(3 \frac{2}{5}\).
To convert a mixed number to an improper fraction, perform the following steps:

• Multiply the whole number by the denominator of the fraction.
• Add the result to the numerator of the fraction.
• The sum becomes the new numerator, while the denominator remains the same.

So, \(2 \frac{1}{2}\) becomes \(\frac{5}{2}\) and \(3 \frac{2}{5}\) becomes \(\frac{17}{5}\). This conversion is particularly useful in simplifying calculations involving multiplication or division.

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator.
These types of fractions are called "improper" because traditionally, fractions are thought of as parts of a whole, and in an improper fraction, we have more parts than a whole.
For instance, when we converted the mixed numbers from earlier, we ended up with improper fractions: \(\frac{5}{2}\) and \(\frac{17}{5}\).
Improper fractions might look odd but they are more useful for computational purposes, especially for operations like multiplication and division, as they avoid the need for mixed number arithmetic.

Reciprocals

Reciprocals are a fundamental concept when it comes to dividing fractions.
The reciprocal of a number is simply 1 divided by that number. For instance, the reciprocal of 4 is \(\frac{1}{4}\).
In the step-by-step solution, we see an application of reciprocals where the division process is converted to a multiplication problem.

• To find the reciprocal of a fraction like \(\frac{a}{b}\), simply swap the numerator and the denominator to get \(\frac{b}{a}\).

This is why \(\frac{17}{5} \div 4\) was changed to \(\frac{17}{5} \cdot \frac{1}{4}\). Multiplying by a reciprocal is how we divide by fractions in mathematics.

Simplifying Fractions

Simplifying fractions is about making them easier to read and work with by reducing them to their simplest form.
This involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
In the final step of the exercise, the fraction \(\frac{85}{40}\) became simplified to \(\frac{17}{8}\) by dividing both parts by 5, their GCD.

• Once simplified, fractions can also sometimes be converted into mixed numbers for easier interpretation.
• In this exercise, \(\frac{17}{8}\) was expressed as \(2 \frac{1}{8}\), which helps visualize how much more than 2 the value is.

Simplifying fractions can be especially useful in ensuring that solutions are clean and easily understood.