Problem 122 Which is greater, \(\frac{2}{3}\... [FREE SOLUTION]

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Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 4: Problem 122

Which is greater, $\frac{2}{3}$ of $\frac{1}{4}$ or $\frac{1}{4}$ of $\frac{2}{3} ?$

Short Answer

Expert verified

The two expressions are equal because $\frac{2}{3}\times\frac{1}{4}=\frac{1}{4}\times\frac{2}{3}=\frac{1}{6}$.

Step by step solution

Understand the Problem

Determine which of the two expressions $\frac{2}{3} \times \frac{1}{4}$ or $\frac{1}{4} \times \frac{2}{3}$ is greater.

Evaluate $ \frac{2}{3} \times \frac{1}{4} $

Multiply the fractions to find the value of $ \frac{2}{3} \times \frac{1}{4} $: \[ \frac{2}{3} \times \frac{1}{4} = \frac{2 \times 1}{3 \times 4} = \frac{2}{12} = \frac{1}{6}. \]

Evaluate $ \frac{1}{4} \times \frac{2}{3} $

Multiply the fractions to find the value of $ \frac{1}{4} \times \frac{2}{3} $: \[ \frac{1}{4} \times \frac{2}{3} = \frac{1 \times 2}{4 \times 3} = \frac{2}{12} = \frac{1}{6}. \]

Compare the Results

Compare the results from both calculations: $ \frac{1}{6} $ and $ \frac{1}{6} $. Since $ \frac{1}{6} = \frac{1}{6}$, the two expressions are equal.

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Key Concepts

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

fractions

Fractions represent parts of a whole. A fraction consists of a numerator (the top number) and a denominator (the bottom number). The numerator tells us how many parts we have, and the denominator tells us how many equal parts the whole is divided into. For example, in the fraction $\frac{1}{4}$, the number 1 is the numerator, and 4 is the denominator, meaning the whole is divided into 4 equal parts, and we have one of those parts. Fractions are useful in representing quantities less than one, especially in cases where division or parts of an entity are involved.

multiplication

Multiplying fractions is straightforward and essential in many mathematical problems. To multiply fractions, you multiply the numerators together and the denominators together. For example, to find $\frac{2}{3} \times \frac{1}{4}$, you multiply the numerators (2 * 1) to get 2 and the denominators (3 * 4) to get 12, resulting in $\frac{2}{12}$, which simplifies to $\frac{1}{6}$. Always simplify your fractions if possible by finding the greatest common divisor.

comparisons

Comparing fractions helps determine which quantity is greater or if they are equivalent. When comparing fractions, it can be useful to have common denominators; alternatively, you can convert the fractions to decimals or compare their cross-products. In our exercise, you saw that multiplying $\frac{2}{3}$ by $\frac{1}{4}$ and $\frac{1}{4}$ by $\frac{2}{3}$ resulted in the same value鈥擻(\frac{1}{6}\). Thus, these two fractions are equal.

equal fractions

Equal fractions, as demonstrated in the exercise, are fractions that represent the same part of a whole even if they look different. The fractions $\frac{1}{6}$ and $\frac{1}{6}$ obtained from different multiplications tell us that the two expressions are equal. To identify equal fractions, you can simplify the fractions or find a common denominator to compare them easily. In this case, $\frac{2}{3} \times \frac{1}{4} = \frac{1}{4} \times \frac{2}{3} = \frac{2}{12} = \frac{1}{6}$, showing they are indeed equal fractions.

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91影视

Which is greater, \(\frac{2}{3}\) of \(\frac{1}{4}\) or \(\frac{1}{4}\) of \(\frac{2}{3} ?\)

Short Answer

Step by step solution

Understand the Problem

Evaluate \( \frac{2}{3} \times \frac{1}{4} \)

Evaluate \( \frac{1}{4} \times \frac{2}{3} \)

Compare the Results

Key Concepts

fractions

multiplication

comparisons

equal fractions

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