A gallery owner is hanging up 444 pictures for an exhibit. She has put up 37 pictures in 9 minutes. If she continues at the same rate, how long will it take her to hang the rest of the pictures? Show two ways to solve this problem using proportions.
Answer (3)
37 pictures in 9 minutes can be written as a ratio of 37:9.
Now 444 is divisible by 37, it's 12. So we can multiply both 37 and 9 from the ratio by 12 to find out how long will it take to hang 444 pictures:
37 * 12 : 9 * 12 444 : 108
Answer: It'll take 108 minutes (1 hour and 48 minutes) to hang 444 pictures.
The student has asked about how long it will take to hang the rest of the pictures if a gallery owner hung 37 pictures in 9 minutes. This is a rate problem that can be solved using proportions.
First Method: Direct Proportion
We can set up the direct proportion as follows:
37 pictures in 9 minutes,
X pictures in Y minutes.
Since there are 444 pictures in total, and the gallery owner has already put up 37, she has 444 - 37 = 407 more pictures to hang. Our proportion will look like this:
37 / 9 = 407 / Y Y = (9 × 407) / 37 Y = 99 minutes
Second Method: Unit Rate
We can also find the rate at which the gallery owner hangs up the pictures:
She hangs up 37 pictures in 9 minutes, so the rate is 37 / 9 pictures per minute. Then, we multiply the number of remaining pictures (407) by the unit rate to find the time:
407 pictures × (9 / 37) minutes/picture = 99 minutes
In both cases, it will take the gallery owner an additional 99 minutes to hang up the remaining 407 pictures.
Using proportion calculations, the gallery owner will take an additional 99 minutes to hang the remaining 407 pictures after hanging 37 pictures in 9 minutes. This conclusion is reached through both direct proportion and unit rate methods. Both methods arrive at the same time required for completing the task.
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