Problem 30 - Fundamentals of Mathematics

Question

For the following problems, determine if the pairs of fractions are equivalent. $$ \frac{5}{28}, \frac{20}{112} $$

Step-by-Step Solution

Answer

Yes, the pairs of fractions are equivalent.

Step 1: Understand Equivalence of Fractions

Two fractions are equivalent if they simplify to the same fraction or if their cross-products are equal. We'll use cross-multiplication to check for equivalence.

Step 2: Cross-Multiply the Fractions

We have the fractions \( \frac{5}{28} \) and \( \frac{20}{112} \). Cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction and equating it to the product of the numerator of the second fraction by the denominator of the first fraction. Thus, we calculate: \( 5 \times 112 \) and \( 20 \times 28 \).

Step 3: Calculate the Cross-Products

Compute \( 5 \times 112 = 560 \) and \( 20 \times 28 = 560 \). Both products are equal, which indicates that the fractions are equivalent.

Step 4: Conclusion

Since both cross-products are equal, \( 560 = 560 \), the fractions \( \frac{5}{28} \) and \( \frac{20}{112} \) are equivalent.

Key Concepts

Cross-Multiplication

Cross-multiplication is a simple yet powerful tool to find out if two fractions are equivalent. It involves multiplying the numerator (top number) of one fraction by the denominator (bottom number) of the other fraction, and then comparing the products.
If the products are equal, the fractions are equivalent!

• Take the fraction \( \frac{5}{28} \) and \( \frac{20}{112} \).
• For cross-multiplication, calculate \(5 \times 112\) and \(20 \times 28\).
• Both yield 560, thus they confirm the equivalence of both fractions.

Cross-multiplication is especially handy if simplifying the fraction directly looks complicated. It's a quick check that avoids dividing and reducing fractions right from the start.

Simplifying Fractions

Simplifying fractions is all about making a fraction as simple as possible without changing its value.
This usually means the numerator and denominator have no common factors other than 1.

• Take \( \frac{20}{112} \). Finding the greatest common divisor, \(4\), divides both 20 and 112.
• Divide 20 by 4 to get 5, and 112 by 4 to get 28, simplifying it to \( \frac{5}{28} \).

Upon simplifying, we see fractions that look different are indeed the same. It helps in understanding equivalent fractions simply by reducing them to their simplest form. Simplifying makes the math straightforward and easier to handle.

Mathematical Equivalence

Mathematical equivalence is about having two expressions or values that represent the same quantity.
In fractions, this can mean they may look different but hold equal value.

• Understanding equivalence involves both cross-multiplying and simplifying as demonstrated in the fractions \( \frac{5}{28} \) and \( \frac{20}{112} \).
• Their value is equal even though how they are presented initially may vary.

Recognizing equivalent fractions are crucial in many math contexts. They provide different ways of viewing the same number, helping to solve problems in fractions seamlessly. Equivalence embraces the idea that appearance doesn't necessarily replicate truth in value. It's a fundamental part of understanding numbers better.