Given $p : q = 9 : 5$, evaluate $\frac{15p - 2q}{5p + 16q}$
Answer (1)
Substitute p = 9 k and q = 5 k into the expression 5 p + 16 q 15 p − 2 q .
Simplify the expression to 45 k + 80 k 135 k − 10 k .
Further simplify to 125 k 125 k .
Cancel out the common factor 125 k to get the final answer: 1 .
Explanation
Understanding the Problem We are given the ratio p : q = 9 : 5 and asked to evaluate the expression 5 p + 16 q 15 p − 2 q . This means that for some constant k , p = 9 k and q = 5 k . We can substitute these expressions into the given expression to evaluate it.
Substitution Substitute p = 9 k and q = 5 k into the expression: 5 p + 16 q 15 p − 2 q = 5 ( 9 k ) + 16 ( 5 k ) 15 ( 9 k ) − 2 ( 5 k )
Simplification Simplify the expression: 5 ( 9 k ) + 16 ( 5 k ) 15 ( 9 k ) − 2 ( 5 k ) = 45 k + 80 k 135 k − 10 k = 125 k 125 k
Final Evaluation Since k is a non-zero constant, we can cancel out the common factor 125 k from the numerator and the denominator: 125 k 125 k = 1
Conclusion Therefore, the value of the expression 5 p + 16 q 15 p − 2 q is 1.
Examples
Ratios and proportions are fundamental in various real-life applications. For instance, in cooking, maintaining the correct ratio of ingredients is crucial for the taste and texture of the final dish. Similarly, in construction, the ratio of cement to sand in concrete affects its strength and durability. Understanding how to manipulate and evaluate expressions involving ratios helps in making informed decisions in these practical scenarios.
