- Calculate the numerator: \(9 - 3(-4) = 9 + 12 = 21\).
- Calculate the denominator: \(5 - 4(-7) = 5 + 28 = 33\).
- Simplify the resulting fraction: \(\frac{21}{33} = \frac{7}{11}\).
- The final answer is \(\boxed{\frac{7}{11}}\).
Explanation
- Understanding the Problem
The expression provided is \(\frac{9-3(-4)}{5-4(-7)}\). To evaluate this, we must follow the order of operations, which dictates that multiplication is performed before subtraction. We will treat the numerator and the denominator as separate groups, simplify each using standard arithmetic rules for integers, and then divide the results.
- Evaluating the Numerator
First, focus on the numerator: \(9 - 3(-4)\). According to the order of operations, calculate the multiplication \(3 \times (-4)\) first, which is \(-12\). Substituting this back into the expression, we get \(9 - (-12)\). Subtracting a negative number is equivalent to adding its positive counterpart, so \(9 + 12 = 21\). The value of the numerator is 21.
- Evaluating the Denominator
Next, focus on the denominator: \(5 - 4(-7)\). Following the order of operations, calculate the multiplication \(4 \times (-7)\) first, which is \(-28\). Substituting this back into the expression, we get \(5 - (-28)\). Again, subtracting a negative number is the same as addition, giving \(5 + 28 = 33\). The value of the denominator is 33.
- Final Simplification and Conclusion
Now, combine the simplified numerator and denominator to form the fraction: \(\frac{21}{33}\). To simplify this fraction, find the greatest common divisor of both the numerator and the denominator. Both 21 and 33 are divisible by 3. Thus, dividing both terms by 3 yields \(\frac{21 \div 3}{33 \div 3} = \frac{7}{11}\). The final simplified value of the expression is \(\frac{7}{11}\).
Examples
In practical life, simplifying algebraic expressions is essential in fields like finance and engineering. For example, if you are calculating a ratio of budget allocations, such as \(\frac{900 - 300(4)}{500 - 400(7)}\), you are performing similar arithmetic operations. Mastering the order of operations ensures that your financial projections or structural engineering calculations remain accurate, preventing errors that could arise from incorrect sequence of operations.