# Mary can read 22 pages in 30 minutes. How long would it take her to read a 100 page book? Give...

## Question:

Mary can read 22 pages in 30 minutes.

How long would it take her to read a 100 page book? Give answer in hours and minutes and round to the nearest minute, if needed.

## Proportions and Variation:

We regularly encounter situations in which two values are directly and proportionally associated. An example is the case where the number of pages read in a book is directly proportional to the time taken to read the book. Therefore, the more time we spend reading the book, the more pages we will read.

{eq}\eqalign{ & {\text{In this specific case}}{\text{, we have two proportional values }}\,x\,\left( {pages{\text{ }}read} \right){\text{ and }} \cr & y\,\left( {hours} \right){\text{ that have a variation in directly proportional form}}{\text{. }} \cr & {\text{So we have:}} \cr & \,\,\,\,{x_1} = 22\,pages \cr & \,\,\,\,{y_1} = 0.5\,hours \cr & \,\,\,\,{x_2} = 100\,pages \cr & \,\,\,\,{y_2} = ?\,\,hours \cr & {\text{Since}}{\text{, }}x{\text{ and }}y{\text{ vary directly}}{\text{, then}}{\text{, when }}x{\text{ increases it also }} \cr & {\text{increases }}y{\text{. For this reason}}{\text{, it must be satisfied that:}} \cr & \,\,\,\,\frac{{{y_2}}}{{{x_2}}} = \frac{{{y_1}}}{{{x_1}}} \cr & {\text{So if we do cross - multiplying:}} \cr & \,\,\,\,{y_2} \cdot {x_1} = {y_1} \cdot {x_2} \cr & {\text{Now}}{\text{, solving for }}\,{y_2}{\text{:}} \cr & \,\,\,\,{y_2} = \frac{{{y_1} \cdot {x_2}}}{{{x_1}}} \cr & {\text{So}}{\text{, substituting the given values:}} \cr & \,\,\,\,{y_2} = \frac{{0.5 \times 100}}{{22}} = 2.27\,hours \cr & \,\,\,\,2.27\,hours = 2\,hours\,{\text{ and }}\,16\,\min \cr & {\text{Therefore}}{\text{, it will take her }}\boxed{2\,{\text{hours and 16}}\,{\text{min}}}{\text{ to read 100 pages}}{\text{.}} \cr} {/eq}