Which linear equation can be derived from this proportion?
$\frac{12}{5+8}=\frac{7}{5+1}$
A. $12 x+96=7 x+7$
B. $12 x+1=7 x+8$
C. $12 x+12=7 x+56$
D. $12 x+8=7 x+1$
Answer (1)
Simplify the original proportion.
Rewrite the proportion with a variable: x + 8 12 = x + 1 7 .
Cross-multiply to get 12 ( x + 1 ) = 7 ( x + 8 ) .
Expand and simplify: 12 x + 12 = 7 x + 56 . The answer is 12 x + 12 = 7 x + 56 .
Explanation
Understanding the Problem We are given the proportion 5 + 8 12 = 5 + 1 7 and asked to find an equivalent linear equation from the given options.
The options are: A. 12 x + 96 = 7 x + 7 B. 12 x + 1 = 7 x + 8 C. 12 x + 12 = 7 x + 56 D. 12 x + 8 = 7 x + 1
Simplifying the Proportion First, let's simplify the denominators: 5 + 8 = 13 5 + 1 = 6 So the proportion is 13 12 = 6 7 .
Cross-Multiplication To find an equivalent linear equation, we can introduce a variable x into the original proportion such that the proportion becomes a linear equation when cross-multiplied. Let's assume the original proportion was intended to be x + 8 12 = x + 1 7 .
Now, cross-multiply to eliminate the fractions: 12 ( x + 1 ) = 7 ( x + 8 )
Expanding the Equation Expand both sides of the equation by distributing the constants: 12 x + 12 = 7 x + 56
Identifying the Correct Option Now, let's compare the resulting equation with the given options: A. 12 x + 96 = 7 x + 7 B. 12 x + 1 = 7 x + 8 C. 12 x + 12 = 7 x + 56 D. 12 x + 8 = 7 x + 1
The equation we derived, 12 x + 12 = 7 x + 56 , matches option C.
Final Answer Therefore, the linear equation that can be derived from the proportion is 12 x + 12 = 7 x + 56 .
Examples
Proportions and linear equations are fundamental in various real-life applications. For instance, when scaling recipes, maintaining architectural ratios in building design, or calculating currency exchange rates, understanding how to derive and manipulate linear equations from proportions is crucial. In cooking, if a recipe requires a certain ratio of ingredients, like flour to water, and you want to make a larger batch, you use proportions to scale the quantities correctly. Similarly, architects use proportions to ensure that the dimensions of a building are aesthetically pleasing and structurally sound. Currency exchange also relies on proportions to convert one currency to another based on the current exchange rate.
