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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^("th ") basket?

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?

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GCF and LCM: word problems

Full solution

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?

Calculate Ruby's Rate: Calculate Ruby's rate of assembling gift baskets. $\newline$ Ruby can assemble $2$ baskets in $7$ minutes. $\newline$ So, Ruby's rate is $\frac{2 \text{ baskets}}{7 \text{ minutes}}$ .
Calculate Emma's Rate: Calculate Emma's rate of assembling gift baskets. $\newline$ Emma can assemble $4$ baskets in $15$ minutes. $\newline$ So, Emma's rate is $\frac{4 \text{ baskets}}{15 \text{ minutes}}$ .
Convert Rates to Baskets: Convert both rates to baskets per minute. $\newline$ Ruby's rate: $\frac{2}{7}$ baskets per minute. $\newline$ Emma's rate: $\frac{4}{15}$ baskets per minute.
Calculate Combined Rate: Calculate the combined rate when both Ruby and Emma are working together. $\newline$ Ruby's rate + Emma's rate = $(\frac{2}{7}) + (\frac{4}{15})$ baskets per minute. $\newline$ To add these fractions, find a common denominator, which is $105$ . $\newline$ Ruby's rate in terms of the common denominator: $(\frac{2}{7})\times(\frac{15}{15}) = \frac{30}{105}$ baskets per minute. $\newline$ Emma's rate in terms of the common denominator: $(\frac{4}{15})\times(\frac{7}{7}) = \frac{28}{105}$ baskets per minute. $\newline$ Combined rate: $(\frac{30}{105}) + (\frac{28}{105}) = \frac{58}{105}$ baskets per minute.
Calculate Ruby's Baskets: Calculate the number of baskets Ruby assembles before Emma starts. $\newline$ Ruby starts at $1:00$ p.m. and Emma starts at $1:15$ p.m., so Ruby works alone for $15$ minutes. $\newline$ Ruby's baskets in $15$ minutes: $(\frac{2}{7}) \times 15 = \frac{30}{7}$ baskets. $\newline$ Since Ruby cannot make a fraction of a basket, we round down to the nearest whole number. $\newline$ Ruby assembles $4$ baskets in $15$ minutes ( $\frac{30}{7} \approx 4.28$ , rounded down to $4$ ).
Subtract Baskets Assembled: Subtract the number of baskets Ruby assembles from the total needed. $\newline$ Total baskets needed: $54$ . $\newline$ Baskets assembled by Ruby: $4$ . $\newline$ Remaining baskets: $54 - 4 = 50$ baskets.
Calculate Time for Remaining Baskets: Calculate the time it takes for Ruby and Emma to assemble the remaining baskets together. $\newline$ Time = Number of baskets / Combined rate. $\newline$ Time = $50$ baskets / $(58/105)$ baskets per minute. $\newline$ Time = $50 \times (105/58)$ minutes. $\newline$ Time $\approx 90.52$ minutes.
Convert Time to Hours: Convert the time to hours and minutes. $\newline$ $90.52$ minutes is $1$ hour and $30.52$ minutes. $\newline$ Since we cannot have a fraction of a minute, we round down to $30$ 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. $\newline$ Finish time = $1:15$ p.m. + $1$ hour and $30$ minutes. $\newline$ Finish time = $2:45$ p.m. ( $1:15$ p.m. + $1:30$ = $2:45$ p.m.).

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