Question

Michael cycled a distance of 12 5/6miles in 2/3hours. What is his cycling unit rate in miles per hour?A. 4/77mile per hourB. 6 5/9miles per hourC. 12 2/3miles per hourD. 19 1/4miles per hour

Answer 1

Michael cycled a distance of 12 5/6 miles in 2/3 hours.

To calculate the speed you have to divide the distance traveled by the time it takes to travel the said distance:

`Speed=\frac{\text{distance}}{\text{time}}=12(5)/(6)/(2)/(3)`

To be able to make this division, the first step is to express the mixed number 12 5/6 as a fraction.

-First, divide the whole number by 1 to write it as a fraction and add 5/6:

`(12)/(1)+(5)/(6)`

-Second, to add both fractions, you have to express both fractions with the same denominator, which is "6", so you have to multiply the numerator and denominator of 12/1 by 6:

`\begin{gathered} (12\cdot6)/(1\cdot6)+(5)/(6) \\ (72)/(6)+(5)/(6) \end{gathered}`

-Third, add both fractions:

`(72)/(6)+(5)/(6)=(72+5)/(6)=(77)/(6)`

The division is now:

`(77)/(6)/(2)/(3)`

When you divide two fractions, the procedure is to invert the divisor fraction, which means, to determine its reciprocal fraction, and multiply the dividend by the reciprocal fraction of the divisor:

-First, determine the reciprocal fraction of 2/3:

`(2)/(3)\to(3)/(2)`

-Second, multiply 77/6 by 3/2

`(77)/(6)\cdot(3)/(2)=(77\cdot3)/(6\cdot2)=(231)/(12)`

The result is 231/12, both values "231" and "12" are divisible by 3, so you can simplify the fraction by dividing the numerator and denominator by 3:

`(231/3)/(12/3)=(77)/(4)`

To express 77/4 as a mixed number, you have to determine how many "whole numbers" are included in the fraction, to do so simply divide 77 by 4:

`77/4=19.25`

This means that there are 19 "whole numbers" and 1/4 left.

`(77)/(4)=19(1)/(4)`

Michael's speed is 19 1/4 miles/hour