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31 Dec 2023, 01:07
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After completing typing \(\frac{3}{4}\) of a document, Lana recognizes that at her present speed, she'll finish the entire document in just \(\frac{2}{3}\) of the allocated time. By what percentage should she decrease her typing speed to ensure she finishes precisely when intended?
A. \(33 \frac{1}{3}\%\)
B. \(50\%\)
C. \(66 \frac{2}{3}\%\)
D. \(75\%\)
E. \(80\%\)
A. \(33 \frac{1}{3}\%\)
B. \(50\%\)
C. \(66 \frac{2}{3}\%\)
D. \(75\%\)
E. \(80\%\)
31 Dec 2023, 01:07
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Official Solution:
After completing typing \(\frac{3}{4}\) of a document, Lana recognizes that at her present speed, she'll finish the entire document in just \(\frac{2}{3}\) of the allocated time. By what percentage should she decrease her typing speed to ensure she finishes precisely when intended?
A. \(33 \frac{1}{3}\%\)
B. \(50\%\)
C. \(66 \frac{2}{3}\%\)
D. \(75\%\)
E. \(80\%\)
This question can be solved using pure algebra, but an easier approach is to assume numbers. Let's say the document consists of 12 pages (this number is chosen as it is the least common multiple of the denominators 4 and 3 given in the question) and the allocated time is 12 minutes.
Thus, Lana typed \(12*\frac{3}{4} = 9\) pages and realized that instead of taking 12 minutes, she would complete the entire document in only \(12*\frac{2}{3} = 8\) minutes, meaning she spent \(8*\frac{9}{12} = 6\) minutes typing those 9 pages. Her current typing rate is then \(\frac{9}{6} = 1.5\) pages per minute.
She has 3 pages of the document left to finish and wants to do this in the remaining 12 - 6 = 6 minutes. Therefore, she needs to reduce her typing rate to \(\frac{3}{6} = 0.5\) pages per minute.
A reduction to a third, from 1.5 to 0.5, is equivalent to a \(66 \frac{2}{3}\%\) decrease. Alternatively, using a formula, we calculate \(\frac{1.5 - 0.5}{1.5}*100\% = \frac{1}{1.5}*100\% = \frac{2}{3}*100\% = 66 \frac{2}{3}\%\).
Answer: C
After completing typing \(\frac{3}{4}\) of a document, Lana recognizes that at her present speed, she'll finish the entire document in just \(\frac{2}{3}\) of the allocated time. By what percentage should she decrease her typing speed to ensure she finishes precisely when intended?
A. \(33 \frac{1}{3}\%\)
B. \(50\%\)
C. \(66 \frac{2}{3}\%\)
D. \(75\%\)
E. \(80\%\)
This question can be solved using pure algebra, but an easier approach is to assume numbers. Let's say the document consists of 12 pages (this number is chosen as it is the least common multiple of the denominators 4 and 3 given in the question) and the allocated time is 12 minutes.
Thus, Lana typed \(12*\frac{3}{4} = 9\) pages and realized that instead of taking 12 minutes, she would complete the entire document in only \(12*\frac{2}{3} = 8\) minutes, meaning she spent \(8*\frac{9}{12} = 6\) minutes typing those 9 pages. Her current typing rate is then \(\frac{9}{6} = 1.5\) pages per minute.
She has 3 pages of the document left to finish and wants to do this in the remaining 12 - 6 = 6 minutes. Therefore, she needs to reduce her typing rate to \(\frac{3}{6} = 0.5\) pages per minute.
A reduction to a third, from 1.5 to 0.5, is equivalent to a \(66 \frac{2}{3}\%\) decrease. Alternatively, using a formula, we calculate \(\frac{1.5 - 0.5}{1.5}*100\% = \frac{1}{1.5}*100\% = \frac{2}{3}*100\% = 66 \frac{2}{3}\%\).
Answer: C
12 Jul 2024, 08:11
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I believe that this is a high-quality question and I agree with the explanation provided.
