Question

Find the rate in miles per hour. 5 1/3 miles in 7 1/2 minutes

Answer 1

The answer is:  " $42\frac{2}{3}$ miles per hour " .

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Step-by-step explanation:

This is a "unit rate" problem.

"Find the rate in miles per "single [unit] hour."

 {GIven:  "5 1/3 mi per 7 1/2 min."}.

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Using a technique unknown as "dimension analysis" ;

let us convert the given value; in "mi / min" ;

 to its value in "mi/hr" (our answer):

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→    " 5 1/3  mi        60 min                  ,    

      ________  *   ______   =   ?      mi / hr ?  "

       7 1/2 min           1 hr

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Note:  The units of "min" ["minute(s)"] ; cancel out to "1" ;

               →  {since:  "min/min = 1 " ;

  and we have:  " [ (5  1/3 * 60) / (7 1/2) ] mi / hr. " ;

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   →   " 5  1/3 =  [ (3*5) + 1 ] / 3 " ;

                                  = " (15 + 1) /3 " ;

                                  = " 16 / 3 " ;  → (as an "improper fraction") ;

   →   " 7  1/2 =  [ (2*7) + 1 ] / 2 " ;

                                  =  " (14 + 1) / 2) " ;

                                  = "  15 / 2 " ;   → (as an "improper fraction") ;

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So we have:

   →   " $(\frac{16}{3}*\frac{60}{1})$ ÷ $\frac{15}{2}$ " ;

Note: Dividing by a value is the same value as multiplying by the reciprocal of that value.

 In this case:  The reciprocal of:  " $(\frac{15}{2}$$)$ " ;  is:  " $(\frac{2}{15})$ " ;

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So, we can rewrite the expression:

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   →   " $(\frac{16}{3}*\frac{60}{1}) * \frac{2}{15}$ " ;

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     =  " $\frac{(16*60*2)}{(3*1*15)}$ " ;

   

 →   "    ( 16 * 60 * 2)      "

             (3 * 15)    

 →   Take note of the "60" in the numerator;  and the "15" in the denominator ;

  " (60 ÷ 15 = 4) " ;  

 so we can change the "60" to a "15" and cancel out the "15" ; since:

  " (15 ÷ 15 = 1 ) " ;  

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And rewrite:

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   →   "    ( 16 * 4 * 2)      "  

                    3       ;

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$= \frac{(16 * 4 * 2)}{3} = \frac{(64*2)}{3} = \frac{128}{3} =$  (128 ÷ 3) =  " $42\frac{2}{3}$ " .

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The answer is:  " $42\frac{2}{3}$ miles per hour " .

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Hope this is helpful!

 Best of luck to you!

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