Problem 60 - Algebra 1: Concepts and Skills

Question

Decide whether the statement is true or false. Explain. \(\frac{1}{4}=\frac{3}{12}\)

Step-by-Step Solution

Answer

The statement \(\frac{1}{4}=\frac{3}{12}\) is true.

Step 1: Identify given fractions

The fractions given in the problem are \(\frac{1}{4}\) and \(\frac{3}{12}\).

Step 2: Perform cross multiplication

Cross multiplication is performed between two fractions to establish if they're equal. In this case, cross multiply \(\frac{1}{4}\) with \(\frac{3}{12}\) and check if 1*12 equals 4*3.

Step 3: Verification

We find upon cross-multiplication that 1*12 = 12 and 4*3 = 12. Since the products are equal, according to the rule of equivalent fractions, \(\frac{1}{4}\) is equal to \(\frac{3}{12}\).

Key Concepts

Fractions

Fractions represent a part of a whole and are expressed as \ \( \frac{numerator}{denominator} \) \ where the numerator indicates how many parts we have, and the denominator shows into how many parts the whole is divided.
Understanding fractions involves:

• Numerator: Located at the top, representing the number of parts considered.
• Denominator: Found at the bottom, indicating the total segments in the whole.

For example, in the fraction \ \( \frac{1}{4} \), the '1' is the numerator, meaning one part is taken out of the '4' equal parts of the whole.
To consider fractions equivalent, they need to represent the same portion of the whole. Fractions such as \ \( \frac{1}{4} \) and \ \( \frac{3}{12} \) are equivalent because they cover the same amount of the whole albeit with different numerators and denominators.

Cross Multiplication

Cross multiplication is a handy tool used to compare two fractions to verify their equivalence.
Here's the process:

• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Multiply the denominator of the first fraction by the numerator of the second fraction.

For our example, you would multiply 1 (numerator of \ \( \frac{1}{4} \)) with 12 (denominator of \ \( \frac{3}{12} \)), and 4 (denominator of \ \( \frac{1}{4} \)) with 3 (numerator of \ \( \frac{3}{12} \)).
After multiplication, we get 12 both ways: \ \( 1 \times 12 = 12 \) and \ \( 4 \times 3 = 12 \).
The result shows equal products, confirming that these fractions are equivalent. Cross multiplication is a reliable method for comparing fractions quickly, especially when decimals or common denominators are complex to use. It simplifies the process by focusing on simple arithmetic.

Mathematical Verification

Mathematical verification involves proving that two quantities are indeed equal through logical and systematic computation. For equivalent fractions, this verification illustrates the concept and enforces the rule.During verification, check the cross multiplication result:
1. Multiply across and compare: is \ \( 1 \times 12 = 4 \times 3 \)?2. Since both products are 12, the equation holds true.Verification confirms that two fractions indeed represent the same value.
This process ensures students understand why rules in math apply.

• Encourages logical thinking.
• Instills confidence in computation methods.
• Reinforces foundational math concepts.

By following systematic steps, students develop trust in mathematical procedures and confidence in solving similar problems. Verification is about ensuring sound reasoning, transforming abstract math into clear and concrete evidence.