Ruby can assemble $2$ gift baskets by herself in $7$ minutes. Emma can assemble $4$ gift baskets by herself in $15$ minutes. Ruby begins assembling gift baskets at $1:00$ p.m., and Emma begins assembling gift baskets at $1:15$ p.m. If they continue to work at the above rates, at what time will they finish the $54^{\text{th}}$ basket? $\newline$ Choose $1$ answer: $\newline$ (A) $2:30$ p.m. $\newline$ (B) $2:42$ p.m. $\newline$ (C) $2:45$ p.m. $\newline$ (D) $3:00$ p.m.

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Grade 7

Identify equivalent linear expressions: word problems

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Q. Ruby can assemble

2

gift baskets by herself in

7

minutes. Emma can assemble

4

gift baskets by herself in

15

minutes. Ruby begins assembling gift baskets at

1:00

p.m., and Emma begins assembling gift baskets at

1:15

p.m. If they continue to work at the above rates, at what time will they finish the

54^{\text{th}}

basket?

\newline

Choose

1

answer:

\newline

(A)

2:30

p.m.

\newline

(B)

2:42

p.m.

\newline

(C)

2:45

p.m.

\newline

(D)

3:00

p.m.

Determine Ruby's Rate: Determine Ruby's rate of assembling gift baskets. $\newline$ Ruby can assemble $2$ baskets in $7$ minutes, so her rate is: $\newline$ Rate = Number of baskets / Time $\newline$ Rate = $2$ baskets / $7$ minutes
Determine Emma's Rate: Determine Emma's rate of assembling gift baskets. $\newline$ Emma can assemble $4$ baskets in $15$ minutes, so her rate is: $\newline$ Rate = Number of baskets / Time $\newline$ Rate = $\frac{4 \text{ baskets}}{15 \text{ minutes}}$
Calculate Ruby's Baskets: Calculate the number of baskets Ruby can assemble from $1:00$ p.m. to $1:15$ p.m. $\newline$ Since Ruby starts at $1:00$ p.m. and Emma starts at $1:15$ p.m., Ruby will have a $15$ -minute head start. $\newline$ Ruby's rate is $2$ baskets / $7$ minutes, so in $15$ minutes she can assemble: $\newline$ Number of baskets = Rate * Time $\newline$ Number of baskets = $(2/7)$ baskets/minute * $15$ minutes $\newline$ Number of baskets = $1:15$ $0$ $\newline$ Number of baskets = $1:15$ $1$ ... $\newline$ Ruby can assemble $1:15$ $2$ baskets in her first $15$ minutes (since she cannot assemble a fraction of a basket, we only count the whole baskets).
Calculate Combined Rate: Calculate the combined rate of Ruby and Emma. $\newline$ Ruby's rate is $2$ baskets / $7$ minutes, and Emma's rate is $4$ baskets / $15$ minutes. To combine their rates, we add them together: $\newline$ Combined rate = Ruby's rate + Emma's rate $\newline$ Combined rate = $(2/7) + (4/15)$ $\newline$ To add these fractions, we need a common denominator, which is $105$ . $\newline$ Combined rate = $(30/105) + (28/105)$ $\newline$ Combined rate = $(58/105)$ baskets/minute
Calculate Remaining Baskets: Calculate the number of baskets remaining after Ruby's head start. $\newline$ Ruby has already assembled $4$ baskets, so there are $54 - 4 = 50$ baskets left to assemble.
Calculate Time Remaining: Calculate the time it will take for Ruby and Emma to assemble the remaining $50$ baskets. $\newline$ Time $=$ Number of baskets $/$ Combined rate $\newline$ Time $=$ $50$ baskets $/$ $(58/105)$ baskets/minute $\newline$ Time $=$ $50 \times (105/58)$ minutes $\newline$ Time $=$ $=$ $0$ minutes $\newline$ Time $=$ $=$ $2$ minutes
Convert Time to Hours: Convert the time to hours and minutes. $86.2069$ minutes is $1$ hour and $26.2069$ minutes. Since we only need whole minutes, we round down to $26$ minutes.
Add Time to Finish: Add the time to Emma's start time to find the finish time. $\newline$ Emma started at $1:15$ p.m., so adding $1$ hour and $26$ minutes gives us: $\newline$ Finish time = $1:15$ p.m. + $1$ hour and $26$ minutes $\newline$ Finish time = $2:41$ p.m. $\newline$ However, we rounded down the minutes, so the actual finish time will be a minute or two later.
Choose Closest Time: Choose the closest time option that is after $2:41$ p.m. $\newline$ The closest time option after $2:41$ p.m. is (B) $2:42$ p.m.