Problem 52 - Elementary Algebra

Question

Determine the reciprocal of the following numbers. $$ 85 $$

Step-by-Step Solution

Answer

The reciprocal of 85 is \( \frac{1}{85} \).

Step 1: Understanding Reciprocals

The reciprocal of a number is 1 divided by that number. It is also known as the multiplicative inverse. For any non-zero number \( a \), the reciprocal is \( \frac{1}{a} \).

Step 2: Write the Number as a Fraction

To find the reciprocal, first express the number as a fraction. The number 85 can be written as \( \frac{85}{1} \).

Step 3: Find the Reciprocal

Take the reciprocal of \( \frac{85}{1} \), which means flipping the fraction. Thus, the reciprocal is \( \frac{1}{85} \).

Step 4: Verify the Reciprocal

Verify by multiplying the original number by its reciprocal. Multiplying 85 by \( \frac{1}{85} \) gives 1, confirming the reciprocal is correct.

Key Concepts

Reciprocals

When we talk about reciprocals, we are diving into a fundamental concept of elementary algebra. The reciprocal of a number is what you get when you generate a flipped fraction. In simpler terms, for any given number, the reciprocal is 1 divided by that number. This concept is pivotal because it represents how numbers "pair up" to equal one when multiplied.
For example, if you take a whole number such as 85, you first express it in fractional form, which is \( \frac{85}{1} \), and then flip the fraction to find its reciprocal, \( \frac{1}{85} \).

• Useful when dividing fractions.
• A reciprocal's role is to return a product of 1 when multiplied with its original number.
• Reciprocals are only defined for non-zero numbers.

Multiplicative Inverse

The concept of a multiplicative inverse plays a crucial role in solving algebraic equations and simplifying expressions. Essentially, it is another name for the reciprocal, and it refers to a number which, when multiplied by the original number, will result in the multiplicative identity, which is 1.

For the number 85, its multiplicative inverse is \( \frac{1}{85} \). The process is the same—invert the fraction representation. When you multiply 85 and \( \frac{1}{85} \), the product is 1: \( 85 \times \frac{1}{85} = 1 \). This property validates the calculation and consistently reassures us of the accuracy of finding reciprocals or inverses.

• In algebra, multiplicative inverses are essential in solving equations like \( ax = b \), where we need to isolate \( x \).
• Helps in transforming division into multiplication for easier computation.

Fractions

Fractions offer a simple representation of numbers that are not whole, expressing parts or portions of a whole. They are composed of two numbers: a numerator (top number) and a denominator (bottom number).

Understanding how fractions work is important when dealing with reciprocals and multiplicative inverses. Any whole number can be expressed as a fraction by placing it over 1, like \( \frac{85}{1} \) for the example of 85. This method allows us to find reciprocals easier by simply flipping the fraction, turning it into \( \frac{1}{85} \).

• Fractions are versatile for simplifying mathematical operations.
• They facilitate operations with decimals, percentages, and ratios.
• One of the foundational elements crucial to mastering arithmetic and algebra.