Problem 120 - Elementary and Intermediate Algebra

Question

Find the result when \(\frac{7}{8}\) is multiplied by its reciprocal.

Step-by-Step Solution

Answer

\( \frac{7}{8} \times \frac{8}{7} = 1 \)

Step 1: Understanding Reciprocals

The reciprocal of a number is what you multiply it by to get 1. For a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). Hence, the reciprocal of \( \frac{7}{8} \) is \( \frac{8}{7} \).

Step 2: Multiplying Fractions

To multiply fractions, multiply the numerators together and the denominators together. So we multiply \( \frac{7}{8} \) by \( \frac{8}{7} \) by calculating \( \frac{7 \times 8}{8 \times 7} \).

Step 3: Simplifying the Product

The product \( \frac{7 \times 8}{8 \times 7} \) simplifies to \( \frac{56}{56} \). Division of any number by itself is 1, so \( \frac{56}{56} = 1 \).

Key Concepts

Multiplication of Fractions

When multiplying fractions, you are essentially finding a portion of a portion. To do this, follow these simple steps: Multiply the numerators (top numbers) together and the denominators (bottom numbers) together.
For example, in multiplying \( \frac{7}{8} \) by \( \frac{8}{7} \), you multiply the tops — 7 and 8, giving you 56. Then, multiply the bottoms — 8 and 7, also resulting in 56. Hence, the expression becomes \( \frac{56}{56} \).

• This operation can be visualized as taking one fraction and stacking it on the other.
• It’s like multiplying single numbers: you’re finding out how many parts one fraction fits into another.

Ensuring you multiply correctly will lay down the base for accurate simplification later.

Understanding Reciprocals

Reciprocals are a fascinating part of fractions, acting like a mirror image. The reciprocal of a fraction is merely flipping its numerator and denominator. If you have the fraction \( \frac{7}{8} \), its reciprocal is \( \frac{8}{7} \).
Think of reciprocals as the partner that turns the equation into 1 when multiplied together.

• This concept is key in divisions because dividing by a fraction is the same as multiplying by its reciprocal.
• Understanding this symmetry helps you in fraction division and easily transforms any equation.

Recognizing reciprocals simplifies complex fractions and underpins many algebraic maneuvers.

Simplifying Fractions

Simplifying fractions is about making them as straightforward as possible. Once you've multiplied fractions, like in the case of \( \frac{56}{56} \), you want to simplify. Since \( \frac{56}{56} \) means 56 divided by 56, the simplest form is 1.

• In any fraction, if the numerator and denominator are equal, their quotient is always 1.
• It’s like reducing a recipe to the smallest possible portion that retains the same flavor.

Simplifying is crucial as a last step because it makes your fractions easy to interpret and compare, maintaining clarity without sacrificing the actual value. Always look for the common factors to guide your simplification path.