Problem 36 - Fundamentals of Mathematics

Question

For the following problems, determine if the pairs of fractions are equivalent. $$\frac{5}{7}, \frac{15}{21}$$

Step-by-Step Solution

Answer

The fractions are equivalent.

Step 1: Cross-Multiply the Fractions

To determine if the fractions \( \frac{5}{7} \) and \( \frac{15}{21} \) are equivalent, start by cross-multiplying. Multiply the numerator of the first fraction by the denominator of the second fraction and the numerator of the second fraction by the denominator of the first. Calculate: \[ 5 \times 21 = 105 \] and \[ 15 \times 7 = 105 \]

Step 2: Compare the Cross-Products

Next, compare the values obtained from the cross-multiplication. If the cross-products are equal, then the fractions are equivalent. In this case, both products are 105, indicating the fractions are equivalent.

Key Concepts

Cross-Multiplication

Cross-multiplication is a simple and effective method to determine if two fractions are equivalent without having to reduce them to their lowest terms. It involves the following steps:

• Take the numerator of the first fraction and multiply it by the denominator of the second fraction.
• Then, multiply the numerator of the second fraction by the denominator of the first fraction.

For example, given the fractions \( \frac{5}{7} \) and \( \frac{15}{21} \), cross-multiplication involves calculating \( 5 \times 21 \) and \( 15 \times 7 \). If both these calculations result in the same product, then the fractions are equivalent. In this case, both results are 105, confirming their equivalence. Cross-multiplication is a handy tool for students as it bypasses the need to adjust fractions to a common denominator, providing a quick equivalency check.

Comparing Fractions

Comparing fractions is essential when you need to know which fraction is larger or if they are the same, especially when they have different numerators or denominators. There are several methods, but cross-multiplication is often the fastest for checking equivalence.
When comparing fractions like \( \frac{5}{7} \) and \( \frac{15}{21} \), one could use cross-multiplication to check for equivalency. If the cross-products are equal, then the fractions are the same. Another way to compare fractions is to convert them to decimal form or find a common denominator. However, these methods can be more complex, especially when dealing with large numbers.
Cross-multiplication simplifies the process by quickly revealing equivalency without additional steps, such as reducing fractions or altering their form.

Fractions

Fractions represent parts of a whole and are composed of two numbers: the numerator and the denominator. The numerator is above the line and indicates how many parts we are considering, while the denominator below the line shows into how many parts the whole is divided.
Equivalent fractions, like \( \frac{5}{7} \) and \( \frac{15}{21} \), represent the same amount even if their numbers look different. You can create equivalent fractions by multiplying or dividing the numerator and the denominator by the same number.
Understanding fractions also involves knowing that they can represent values larger than one (where the numerator is greater than the denominator) or less than one, and recognizing that improper fractions can be converted into mixed numbers. Mastering fractions is a fundamental math skill, useful in everyday tasks and all levels of mathematics.