what is the equivalent rational number for -15/36 with (1) denominator 12 (2) numerator -75 ?tell the answer
The Correct Answer and Explanation is:
Solution for (1): Denominator 12
We are given the fraction −1536\frac{-15}{36}36−15 and need to express it with a denominator of 12.
Step 1: Simplify the original fraction −1536\frac{-15}{36}36−15
First, simplify −1536\frac{-15}{36}36−15 by finding the greatest common divisor (GCD) of 15 and 36. The GCD of 15 and 36 is 3.
Step 2: Divide the numerator and denominator by the GCD
Now, simplify the fraction:−1536=−15÷336÷3=−512\frac{-15}{36} = \frac{-15 \div 3}{36 \div 3} = \frac{-5}{12}36−15=36÷3−15÷3=12−5
Thus, −1536\frac{-15}{36}36−15 simplifies to −512\frac{-5}{12}12−5.
Since we need the denominator to be 12 and we already have −512\frac{-5}{12}12−5, the equivalent rational number with denominator 12 is −512\frac{-5}{12}12−5.
Solution for (2): Numerator -75
Now, we want to express −1536\frac{-15}{36}36−15 in a form where the numerator is -75.
Step 1: Set up the equation with the new numerator
We are aiming for a fraction with a numerator of -75. We will start with the simplified form of the original fraction, which is −512\frac{-5}{12}12−5.
Let the equivalent fraction be −75x\frac{-75}{x}x−75.
Step 2: Solve for xxx using cross-multiplication
We know that:−512=−75x\frac{-5}{12} = \frac{-75}{x}12−5=x−75
Now, perform cross-multiplication:−5×x=−75×12-5 \times x = -75 \times 12−5×x=−75×12−5x=−900-5x = -900−5x=−900
Now, solve for xxx:x=−900−5=180x = \frac{-900}{-5} = 180x=−5−900=180
Thus, the equivalent fraction with numerator -75 is −75180\frac{-75}{180}180−75.
Conclusion:
- For a denominator of 12, the equivalent fraction is −512\frac{-5}{12}12−5.
- For a numerator of -75, the equivalent fraction is −75180\frac{-75}{180}180−75.
