Levi can shovel the driveway in $3$ hours, but if his sister Mariah helps it would take $2$ . $5$ hours. How long would it take Mariah to shovel the driveway alone? $\newline$ $\_\_\_\_\_$ hours

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Ratios and rates: word problems

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Q. Levi can shovel the driveway in

3

hours, but if his sister Mariah helps it would take

2

5

hours. How long would it take Mariah to shovel the driveway alone?

\newline

\_\_\_\_\_

hours

Denote Levi's Rate: Let's denote the rate at which Levi shovels the driveway as $L$ driveways per hour and Mariah's rate as $M$ driveways per hour. Since Levi can shovel the driveway in $3$ hours, his rate is $L = \frac{1 \text{ driveway}}{3 \text{ hours}} = \frac{1}{3}$ driveway per hour.
Combined Rate Equation: When Levi and Mariah work together, they can shovel the driveway in $2.5$ hours. The combined rate when they work together is $1$ driveway / $2.5$ hours = rac{2}{5} driveway per hour.
Substitute and Solve: The combined rate is also the sum of their individual rates, so we have $L + M = \frac{2}{5}$ . We already know that $L = \frac{1}{3}$ , so we can substitute this into the equation to find $M$ : $\frac{1}{3} + M = \frac{2}{5}$ .
Find Mariah's Rate: To solve for $M$ , we subtract $\frac{1}{3}$ from both sides of the equation: $M = \frac{2}{5} - \frac{1}{3}$ .
Calculate Mariah's Time: To subtract these fractions, we need a common denominator. The least common denominator for $5$ and $3$ is $15$ . Converting both fractions to have a denominator of $15$ gives us: $M = \frac{2\times3}{5\times3} - \frac{1\times5}{3\times5} = \frac{6}{15} - \frac{5}{15}.$
Calculate Mariah's Time: To subtract these fractions, we need a common denominator. The least common denominator for $5$ and $3$ is $15$ . Converting both fractions to have a denominator of $15$ gives us: $M = (2 \times 3)/(5 \times 3) - (1 \times 5)/(3 \times 5) = 6/15 - 5/15$ . Subtracting the fractions gives us $M = (6 - 5) / 15 = 1/15$ . This means Mariah's rate is $1/15$ of a driveway per hour.
Calculate Mariah's Time: To subtract these fractions, we need a common denominator. The least common denominator for $5$ and $3$ is $15$ . Converting both fractions to have a denominator of $15$ gives us: $M = \frac{2\times3}{5\times3} - \frac{1\times5}{3\times5} = \frac{6}{15} - \frac{5}{15}$ . Subtracting the fractions gives us $M = \frac{6 - 5}{15} = \frac{1}{15}$ . This means Mariah's rate is $\frac{1}{15}$ of a driveway per hour. To find out how long it would take Mariah to shovel the driveway alone, we take the reciprocal of her rate. The reciprocal of $\frac{1}{15}$ is $\frac{15}{1}$ , which means it would take Mariah $15$ hours to shovel the driveway alone.