Problem 42 - Prealgebra

Question

Simplify each expression as much as possible. $$15+\frac{13}{14} \div \frac{13}{42}$$

Step-by-Step Solution

Answer

The expression simplifies to 18.

Step 1: Identify the Fraction Division

The expression given is \( 15 + \frac{13}{14} \div \frac{13}{42} \). The first part of simplifying this expression is to focus on the division of the fractions \( \frac{13}{14} \div \frac{13}{42} \).

Step 2: Rewrite Division as Multiplication

Recall that dividing by a fraction is the same as multiplying by the reciprocal of that fraction. Therefore, \( \frac{13}{14} \div \frac{13}{42} \) becomes \( \frac{13}{14} \times \frac{42}{13} \).

Step 3: Simplify by Cancelling Common Factors

Before multiplying the fractions, cancel any common factors. Here, the \( 13 \) in the numerator and denominator will cancel: \( \frac{1}{14} \times \frac{42}{1} \). This leaves us with \( \frac{42}{14} \).

Step 4: Divide to Simplify the Fraction

Divide the numerator by the denominator: \( \frac{42}{14} = 3 \).

Step 5: Combine with the Whole Number

Now that the fraction division is simplified to \( 3 \), combine it with the whole number part of the expression. Thus, the expression now is \( 15 + 3 \).

Step 6: Final Simplification

Add the numbers together: \( 15 + 3 = 18 \).

Key Concepts

Reciprocal

In division of fractions, the concept of the reciprocal is crucial. The reciprocal of a number is simply found by flipping its numerator and denominator. For instance, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).

• To divide fractions, you multiply by the reciprocal of the second fraction.
• This fundamental principle allows us to turn any fraction division into multiplication.

In the original exercise, the division \( \frac{13}{14} \div \frac{13}{42} \) was transformed into a multiplication problem by using the reciprocal. Thus, it turned into \( \frac{13}{14} \times \frac{42}{13} \). This transformation simplifies the fraction division process and sets the stage for easier calculations.
Understanding reciprocals is a key skill for solving more complex mathematical problems. It is also often used across different areas of mathematics, so mastering it is very beneficial.

Fraction Simplification

Fraction simplification is about making the fraction as simple as possible to understand and work with. This involves reducing it to its lowest terms by finding common factors for the numerator and denominator.

• First, identify any common factors between the numerator and the denominator.
• Next, divide both the numerator and the denominator by their greatest common factor.
• Continue simplifying until no more common factors remain.

In our exercise, after flipping and finding the reciprocal, we had \( \frac{13}{14} \times \frac{42}{13} \). Both the numerator and denominator in \( \frac{13}{14} \times \frac{42}{13} \) share a factor of 13, which can be cancelled, resulting in \( \frac{1}{14} \times \frac{42}{1} \). Simplifying this fraction further, \( \frac{42}{14} \), can be divided by 14 in both the numerator and denominator, giving 3.
Reducing fractions simplifies calculations and makes results clearer.

Addition of Fractions

Once fractions are simplified, the next step is often integrating them with whole numbers or other fractions. In the case of mixed operations, it's important to perform calculations in the correct order and form.

• Additions with fractions and whole numbers are straightforward once fractions are simplified.
• Make sure to complete any divisions or multiplications first, as per the order of operations.
• Convert fractions into whole numbers, if feasible, before adding to whole numbers.

In the exercise, after simplifying the division to \( \frac{42}{14} = 3 \), the entire expression became \( 15 + 3 \). Adding these results in 18.
Understanding when and how to incorporate fractions into other operations like addition is key for mastering the manipulation of algebraic expressions.