well, it starts with the assumption that Union Center has more miles of roadway than Amanville, so if we can assume that, how many more times?
yes, I agree, the wording is very poor, is simply asking on what ratio is Union Center greater than Amanville in miles, well, let's dig in and and put both sites and mileage in ratio forms.
[tex]\cfrac{\stackrel{A}{\textit{Amanville's mileage}}}{\underset{U}{\textit{Union's mileage}}}~~ = ~~\cfrac{6^2\cdot 7}{6\cdot 7^3}\implies \cfrac{A}{U}~~ = ~~\cfrac{6}{7^2}\implies 7^2 A~~ = ~~6U \\\\\\ \cfrac{7^2 A}{6}~~ = ~~U \implies \cfrac{49A}{6}=U\implies \cfrac{49}{6}A=U\implies 8\frac{1}{6}A=U[/tex]
now, we can say, well, U is really 49/6 of A, in ration terms, and then we can say that
[tex]\cfrac{49}{6}A=U\implies \cfrac{6+43}{6}A=U\implies \cfrac{6}{6}A+\cfrac{43}{6}A=U \\\\\\ \underset{\textit{\LARGE U}~is~7\frac{1}{6}~\textit{more in miles than }\textit{\LARGE A}}{A+\cfrac{43}{6}A=U\implies A+7\frac{1}{6}A=U}[/tex]
Answer:
8 1/6 times as much roadway
Step-by-step explanation:
The problem statement is giving you two numbers and telling you one of them is some factor larger than the other. It asks for that factor.
The question asks for the value of "k" in ...
Union Center has k times more miles than Amanville
6·7³ = k·(6²·7) . . . . . . using the given numbers of miles
As with all one-step linear equations, the solution is found by dividing by the coefficient of k. You can take advantage of the fact that an exponent signifies repeated multiplication. You can also cancel common factors from the numerator and denominator of a fraction.
[tex]\dfrac{6\cdot7^3}{6^2\cdot7}=k\qquad\text{divide by the coefficient of k}\\\\\dfrac{6\cdot7\cdot7\cdot7}{6\cdot6\cdot7}=k\\\\\dfrac{7\cdot7}{6}=k\qquad\text{cancel $6\cdot7$ from numerator and denominator}\\\\k=\dfrac{49}{6}=8\dfrac{1}{6}[/tex]
Union Center has 8 1/6 times as many roadway miles as Amanville.
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Additional comment
The words "larger" and "more" often signify addition:
However, these same words often show up in ways that should be interpreted as multiplication:
The ambiguity that can be involved here suggests a better wording is ...
6 is 2 times as large as 3.
