Nayeli is proofreading a $75$ page manuscript. Fil\'emon is line editing a $40$ page manuscript. $\newline$ Since proofreading is faster, Nayeli completes $11$ pages an hour, but Fil\'emon only completes $4$ pages an hour. $\newline$ When Nayeli and Fil\'emon have the same number of pages left, how many pages will that be? $\newline$ pages $\newline$ Show Calculator

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Grade 7

Solve proportions: word problems

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Q. Nayeli is proofreading a

75

page manuscript. Fil\'emon is line editing a

40

page manuscript.

\newline

Since proofreading is faster, Nayeli completes

11

pages an hour, but Fil\'emon only completes

4

pages an hour.

\newline

When Nayeli and Fil\'emon have the same number of pages left, how many pages will that be?

\newline

pages

\newline

Show Calculator

Calculate Rates: Determine the rate at which Nayeli and Filémon are completing their tasks. $\newline$ Nayeli's rate = $11$ pages per hour $\newline$ Filémon's rate = $4$ pages per hour
Set Up Equation: Set up an equation to find out after how many hours they will have the same number of pages left. $\newline$ Let $x$ be the number of hours they both work until they have the same number of pages left. $\newline$ Nayeli's remaining pages = Total pages - Pages completed by Nayeli $\newline$ Nayeli's remaining pages = $75 - 11x$ $\newline$ Filémon's remaining pages = Total pages - Pages completed by Filémon $\newline$ Filémon's remaining pages = $40 - 4x$
Solve Equation: Since they have the same number of pages left after $x$ hours, we can set the expressions for their remaining pages equal to each other. $75 - 11x = 40 - 4x$
Find Value of x: Solve for x by first adding $11x$ to both sides of the equation and then subtracting $40$ from both sides. $\newline$ $75 - 11x + 11x = 40 - 4x + 11x$ $\newline$ $75 = 40 + 7x$ $\newline$ $75 - 40 = 7x$ $\newline$ $35 = 7x$
Calculate Remaining Pages: Divide both sides of the equation by $7$ to solve for $x$ . $\newline$ $\frac{35}{7} = \frac{7x}{7}$ $\newline$ $5 = x$
Confirm Solution: Now that we know they will have the same number of pages left after $5$ hours, we can calculate how many pages that will be. $\newline$ Nayeli's remaining pages after $5$ hours = $75 - 11 \times 5$ $\newline$ Nayeli's remaining pages after $5$ hours = $75 - 55$ $\newline$ Nayeli's remaining pages after $5$ hours = $20$
Confirm Solution: Now that we know they will have the same number of pages left after $5$ hours, we can calculate how many pages that will be. $\newline$ Nayeli's remaining pages after $5$ hours = $75 - 11 \times 5$ $\newline$ Nayeli's remaining pages after $5$ hours = $75 - 55$ $\newline$ Nayeli's remaining pages after $5$ hours = $20$ To confirm our solution, we check Fill&# $233$ ;mon's remaining pages after $5$ hours. $\newline$ Fill&# $233$ ;mon's remaining pages after $5$ hours = $40 - 4 \times 5$ $\newline$ Fill&# $233$ ;mon's remaining pages after $5$ hours = $40 - 20$ $\newline$ Fill&# $233$ ;mon's remaining pages after $5$ hours = $20$