Points to Remember:
- The concept of equivalent rational numbers.
- The process of finding equivalent rational numbers by multiplying the numerator and denominator by the same non-zero integer.
- Understanding the sign conventions in rational numbers.
Introduction:
A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Equivalent rational numbers represent the same value but have different numerators and denominators. They are obtained by multiplying or dividing both the numerator and denominator by the same non-zero integer. This question requires us to find an equivalent rational number for 3, with a denominator of -35. This is a factual and mathematical problem requiring a procedural approach.
Body:
Finding the Equivalent Rational Number:
To express 3 as an equivalent rational number with a denominator of -35, we need to determine the factor by which we must multiply the current denominator (implicitly 1) to obtain -35. This factor is -35/1 = -35.
Since we must multiply the denominator by -35 to get -35, we must also multiply the numerator by -35 to maintain the equivalence of the rational number.
Therefore:
3/1 * (-35/-35) = -105/-35
Thus, the equivalent rational number is -105/-35. This can also be simplified to 105/35, or further simplified to 3. However, the question specifically asks for a denominator of -35, so -105/-35 is the correct answer in its requested form.
Verification:
We can verify this by simplifying -105/-35:
-105/-35 = 3
This confirms that -105/-35 is indeed an equivalent rational number to 3.
Conclusion:
In conclusion, the equivalent rational number of 3 with a denominator of -35 is -105/-35. This was achieved by multiplying both the numerator and denominator of the fraction 3/1 by -35. The process highlights the fundamental principle of maintaining equivalence in rational numbers by multiplying or dividing both the numerator and denominator by the same non-zero integer. This understanding is crucial for various mathematical operations and applications. The ability to manipulate rational numbers in this way is essential for further mathematical development and problem-solving. The solution emphasizes the importance of precise and methodical application of mathematical rules.
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