Problem 29 - Fundamentals of Mathematics

Question

For the following problems, determine if the pairs of fractions are equivalent. $$ \frac{16}{25}, \frac{49}{75} $$

Step-by-Step Solution

Answer

The fractions \( \frac{16}{25} \) and \( \frac{49}{75} \) are not equivalent.

Step 1: Cross Multiply

To determine if two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \) are equivalent, you can cross multiply and see if the products are equal. Multiply the numerator of the first fraction by the denominator of the second fraction and vice versa. For \( \frac{16}{25} \) and \( \frac{49}{75} \), multiply 16 by 75 and 49 by 25: \[16 \times 75 = 1200 \]\[49 \times 25 = 1225 \]

Step 2: Compare the Cross Products

Now compare the two products obtained in Step 1. If the products are equal, the fractions are equivalent. Here, 1200 ≠ 1225, which means the products are not equal.

Step 3: Conclusion

Since the cross products are not equal, this means that the two fractions \( \frac{16}{25} \) and \( \frac{49}{75} \) are not equivalent.

Key Concepts

Cross Multiplication

Cross multiplication is a handy method to quickly determine if two fractions are equivalent. The process involves multiplying the numerator (top number) of one fraction by the denominator (bottom number) of the other fraction. This gives you a cross product. Do the same for the other numerator and denominator to get the second cross product.
For example, with fractions \( \frac{16}{25} \) and \( \frac{49}{75} \), you would multiply 16 and 75 to get one cross product, then multiply 49 and 25 to get the second one:

• \(16 \times 75 = 1200\)
• \(49 \times 25 = 1225\)

If the two cross products are equal, it shows that the two fractions are equivalent. If not, they are not equivalent.

Comparing Fractions

Comparing fractions can sometimes be tricky, especially when the fractions have different denominators. One way to compare fractions is by using the cross multiplication method. This avoids dealing with converting fractions to have common denominators, simplifying the process.
For instance, in the exercise involving \( \frac{16}{25} \) and \( \frac{49}{75} \), after performing cross multiplication, compare the products:

• \(16 \times 75 = 1200\)
• \(49 \times 25 = 1225\)

The key here is to see if these products are equal. If they aren't, like in this example, it means the fractions are not equivalent. This method is particularly useful when working with large numbers or when one's mental arithmetic might struggle with finding common denominators quickly.

Fraction Equivalence

Understanding fraction equivalence is essential in many areas of math. Two fractions are equivalent if they represent the same part of a whole. For instance, \( \frac{1}{2} \) and \( \frac{2}{4} \) are equivalent because they both represent half of a whole.
In determining equivalence, simply having different numbers doesn't necessarily mean the fractions are unequal. Performing operations like cross multiplication helps ascertain their equality or lack thereof.
In the example \( \frac{16}{25} \) and \( \frac{49}{75} \), after cross-multiplying, we found that 1200 does not equal 1225. This proves the fractions are not equivalent. Recognizing equivalent fractions helps in simplifying calculations and checking the consistency of mathematical relationships. By mastering this concept, tackling more complex problems becomes more intuitive.