What are two fractions equivalent to
2
1
?
The Correct Answer and Explanation is :
To find two fractions equivalent to ( \frac{2}{1} ), we can multiply both the numerator (the top number) and the denominator (the bottom number) by the same non-zero integer. This method works because multiplying by the same number does not change the value of the fraction; it simply changes its representation.
- Finding the first equivalent fraction:
Let’s choose the integer ( 3 ). If we multiply both the numerator and denominator of ( \frac{2}{1} ) by ( 3 ): [
\frac{2 \times 3}{1 \times 3} = \frac{6}{3}
] Thus, ( \frac{6}{3} ) is equivalent to ( \frac{2}{1} ). - Finding the second equivalent fraction:
Now, let’s choose the integer ( 5 ). If we multiply both the numerator and denominator of ( \frac{2}{1} ) by ( 5 ): [
\frac{2 \times 5}{1 \times 5} = \frac{10}{5}
] Hence, ( \frac{10}{5} ) is also equivalent to ( \frac{2}{1} ).
Explanation
An equivalent fraction maintains the same value even if its appearance changes. This is based on the property of fractions, which states that if you multiply or divide both the numerator and denominator of a fraction by the same non-zero number, the fraction’s value remains unchanged. This property is particularly useful in various mathematical applications, including simplifying fractions, finding common denominators, and performing operations like addition or subtraction.
For example, in our case, both ( \frac{6}{3} ) and ( \frac{10}{5} ) simplify back to ( \frac{2}{1} ). To simplify, you can divide the numerator and the denominator by their greatest common divisor (GCD). For ( \frac{6}{3} ), the GCD is ( 3 ), so ( \frac{6 \div 3}{3 \div 3} = \frac{2}{1} ). Similarly, for ( \frac{10}{5} ), the GCD is ( 5 ), resulting in ( \frac{10 \div 5}{5 \div 5} = \frac{2}{1} ).
This understanding of equivalent fractions is foundational in math, allowing for flexibility in solving problems and working with different numerical representations.