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John finds that the distance between two landmarks on a map is 6 1/2inches. If the map scale reads 3/4 inch=120 miles, what is the actual distance between the two landmarks?

I know that 6 inches would be 960 miles but im not sure how to calculate the last 1/2 inch. please help.

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I want to know why the knucklehead mapmapkers included a scale that reads "3/4 inch=120 miles" instead of one that read "1 inch = 160 miles" like any sane person would have done. – MJD Sep 18 '12 at 3:23
I suspect that there's no such map - this smells like homework to me. – user22805 Sep 18 '12 at 3:25

We have the ratio $\frac{3/4}{120}=\frac{13/2}{x}$. Solving for $x$ gives $x=1040$ miles.

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If $\frac34$ inch is $120$ miles, then $\frac43\cdot\frac34$ inch is $\frac43\cdot120$ miles. Doing the multiplications, we find that $1$ inch is $160$ miles. Now just multiply $6.5\cdot160$ to get $1040$ miles altogether; there’s no need to deal with the extra half inch separately.

In general, if $a$ inches represents $b$ miles, we can divide both numbers by $a$ to find that $1$ inch represents $\frac{b}a$ miles. Then, given a measurement of $c$ inches, we know right away that it represents $c\cdot\frac{b}a$ miles. (That’s really what I did in the first paragraph: multiplying by $\frac43$ is the same as dividing by $\frac34$, which was the $a$ in that particular case. The $b$ was $120$, and the $\frac{b}a$ was therefore $160$.)

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Thank you so much. I'm doing test prep for my math section of the GED and I haven't done this stuff in a long time. really helpful! – Jenna Sep 18 '12 at 3:20

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