Magie can type $5$ letters in $20$ min - and Suying can type $2$ letters in $10$ min. Working together and assuming each letter is the same length. How many minutes will, to take them to type $45$ letters?

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Grade 8

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Q. Magie can type

5

letters in

20

min - and Suying can type

2

letters in

10

min. Working together and assuming each letter is the same length. How many minutes will, to take them to type

45

letters?

Given: Given: $\newline$ Manny can type $6$ letters in $20$ minutes. $\newline$ Find Manny's typing rate per minute. $\newline$ The rates are given by the reciprocal of their respective times. $\newline$ Manny's\_rate = $\frac{6 \text{ letters}}{20 \text{ minutes}} = \frac{3 \text{ letters}}{10 \text{ minutes}}$ .
Find Manny's rate: Given: $\newline$ Magie can type $5$ letters in $20$ minutes. $\newline$ Find Magie's typing rate per minute. $\newline$ The rates are given by the reciprocal of their respective times. $\newline$ Magie's\_rate $= \frac{5 \text{ letters}}{20 \text{ minutes}} = \frac{1 \text{ letter}}{4 \text{ minutes}}$ .
Find Magie's rate: Given: $\newline$ Suying can type $2$ letters in $10$ minutes. $\newline$ Find Suying's typing rate per minute. $\newline$ The rates are given by the reciprocal of their respective times. $\newline$ Suying's\_rate = $\frac{2 \text{ letters}}{10 \text{ minutes}} = \frac{1 \text{ letter}}{5 \text{ minutes}}$ .
Find Suying's rate: Manny's rate = $\frac{3}{10}$ $\newline$ Magie's rate = $\frac{1}{4}$ $\newline$ Suying's rate = $\frac{1}{5}$ $\newline$ Find the combined rate. $\newline$ Combined\_rate = Manny's\_rate + Magie's rate + Suying's rate $\newline$ = $\left(\frac{3}{10}\right) + \left(\frac{1}{4}\right) + \left(\frac{1}{5}\right)$ $\newline$ = $\left(\frac{6}{20}\right) + \left(\frac{5}{20}\right) + \left(\frac{4}{20}\right)$ $\newline$ = $\left(6 + 5 + 4\right) / 20$ $\newline$ = $\frac{15}{20}$ $\newline$ = $\frac{3}{4}$ letters per minute.
Calculate Combined Rate: Combined rate = $\frac{3}{4}$ letters per minute. $\newline$ Find the number of minutes it would take them to type $45$ letters together. $\newline$ $T = \frac{\text{Total letters}}{\text{Combined rate}}$ $\newline$ $T = \frac{45 \text{ letters}}{(\frac{3}{4} \text{ letters per minute})}$ $\newline$ $T = 45 \times (\frac{4}{3}) \text{ minutes}$ $\newline$ $T = \frac{180}{3} \text{ minutes}$ $\newline$ $T = 60 \text{ minutes}.$