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Chapter 7: Problem 6

Complete each statement. If \(\frac{x}{y}=\frac{3}{8},\) then \(\frac{y}{x}=\frac{?}{?}\)

Short Answer

Expert verified
The reciprocal of \(\frac{3}{8}\) is \(\frac{8}{3}\), hence \(\frac{y}{x}=\frac{8}{3}\)

Step by step solution

01

Identify the reciprocal

The reciprocal of a fraction, \(\frac{x}{y}\) is obtained by swapping the numerator and the denominator. In this case, the reciprocal fraction of \(\frac{x}{y}\) is given by \(\frac{y}{x}\)
02

Calculate the reciprocal of the given fraction

Given that \(\frac{x}{y}=\frac{3}{8}\), the reciprocal is obtained by swapping the numerator and the denominator. This yields: \(\frac{y}{x}=\frac{8}{3}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Fractions
Fractions are a way to represent parts of a whole. A fraction consists of two parts: the numerator and the denominator. The numerator is the top number and tells how many parts we have. The denominator, on the bottom, tells how many equal parts the whole is divided into. For example, in the fraction \( \frac{3}{8} \), 3 is the numerator, and 8 is the denominator. This means we have 3 parts out of a total of 8.
Numerator and Denominator
The numerator and denominator hold crucial roles in fractions. The numerator indicates the number of equal parts being considered, while the denominator shows the total number of equal parts in a whole.
When swapping the numerator and denominator, you create a reciprocal. The reciprocal is a type of inverse. For example, the reciprocal of \( \frac{3}{8} \) is \( \frac{8}{3} \), where 8 is now the numerator and 3 is the denominator.
Ratio and Proportion
Ratios are closely related to fractions. They compare two quantities, showing how many times one value contains another. Fractions, like ratios, can show proportions by expressing one value as part of another. For example, \( \frac{3}{8} \) is a fraction but can also be seen as a ratio, illustrating how 3 parts compare to 8 parts in total.
Proportions are equations stating that two ratios are equal. When we say \( \frac{x}{y} = \frac{3}{8} \), we imply that \( x \) is to \( y \) as 3 is to 8.
Mathematical Reasoning
Using mathematical reasoning can help solve problems involving fractions. It involves logical thinking and understanding mathematical properties. When asked to find a reciprocal, we reason that we need to switch the numerator and the denominator. Thus, if \( \frac{x}{y} = \frac{3}{8} \), then \( \frac{y}{x} = \frac{8}{3} \).
Problem-solving often relies on understanding relationships, like those in fractions, and applying known methods to find solutions efficiently.

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