Question

1. sophie can type 128 words in 3 minutes. how many minutes will it take her to type 576 words? * 1 point

your answer

Explanation:

Step1: Find the rate of typing

The rate of typing is words per minute. First, we know that in 3 minutes, Sophie types 128 words. But we can also think about how many times 128 words fit into 576 words. Let's find the ratio of 576 to 128.
$\frac{576}{128} = 4.5$ Wait, no, actually, let's do it step by step. The time taken is proportional to the number of words. So if $t_1 = 3$ minutes for $w_1 = 128$ words, and we need to find $t_2$ for $w_2 = 576$ words. Since the rate is constant, $\frac{w_1}{t_1}=\frac{w_2}{t_2}$. Wait, actually, rate $r=\frac{w}{t}$, so $t=\frac{w}{r}$. First, find the rate $r=\frac{128}{3}$ words per minute? No, wait, no. Wait, if she types 128 words in 3 minutes, then to type 576 words, we can see how many groups of 128 words are in 576 words. So $576\div128 = 4.5$? Wait, no, 128*4 = 512, 128*4.5=576. Then, since each 128 words takes 3 minutes, then the time for 576 words is $4.5\times3$? Wait, no, that's not right. Wait, let's set up a proportion. Let $x$ be the time in minutes to type 576 words. Then $\frac{128}{3}=\frac{576}{x}$. Wait, no, that's inverted. Because if more words, more time. So $\frac{w_1}{t_1}=\frac{w_2}{t_2}$? Wait, no, $w_1$ is 128, $t_1$ is 3; $w_2$ is 576, $t_2$ is $x$. So $\frac{128}{3}=\frac{576}{x}$? No, that would mean $128x = 3\times576$, so $x=\frac{3\times576}{128}$. Let's calculate that. 576 divided by 128: 128*4=512, 576-512=64, 64 is half of 128, so 576/128=4.5. Then 3*4.5=13.5? Wait, no, wait 128*4.5=576, so if 128 words take 3 minutes, then 576 words (which is 4.5 times 128) will take 4.5 times 3 minutes? Wait, no, that's not correct. Wait, no, the rate is words per minute. So rate $r = \frac{128}{3}$ words per minute. Then time $t = \frac{w}{r}=\frac{576}{\frac{128}{3}}=576\times\frac{3}{128}$. Let's compute that. 576 divided by 128: 128*4=512, 576-512=64, so 576=128*4 + 64=128*4 + 128*0.5=128*4.5. So 576*3/128=4.5*3=13.5? Wait, no, 576/128=4.5, then 4.5*3=13.5? Wait, that seems high. Wait, no, wait 128 words in 3 minutes, so 1 minute she types 128/3 ≈42.67 words. Then 576 words would take 576/(128/3)=576*3/128= (576/128)*3=4.5*3=13.5 minutes. Wait, but let's check with another approach. If she types 128 words in 3 minutes, then in 1 minute she types 128/3 words. To type 576 words, time is 576 divided by (128/3)=576*(3/128)= (576*3)/128. 576 divided by 128: 128*4=512, 576-512=64, so 576=128*4 + 64=128*(4 + 0.5)=128*4.5. So 4.5*3=13.5. Yes, that's correct. Wait, but let's do the division: 576 divided by 128. 128*4=512, 576-512=64, 64/128=0.5, so total 4.5. Then 4.5*3=13.5. So the time taken is 13.5 minutes. Wait, but maybe I made a mistake. Wait, 128*3=384, 128*6=768. So 576 is between 3 and 6 minutes? No, wait no, 128 words in 3 minutes, so 1 word takes 3/128 minutes. Then 576 words take 576*(3/128)= (576/128)*3=4.5*3=13.5. Yes, that's correct. So the time is 13.5 minutes, which is 13 and a half minutes, or 27/2 minutes.

Step2: Calculate the time

We can also set up the proportion:

Let \( t \) be the time in minutes to type 576 words.

Since the typing rate is constant, the ratio of words to time is constant. So:

\[ \frac{128}{3} = \frac{576}{t} \]

Wait, no, that's incorrect. The correct proportion is that the time is proportional to the number of words, so:

\[ \frac{\text{Words}_1}{\text{Time}_1} = \frac{\text{Words}_2}{\text{Time}_2} \]

Wait, no, actually, \(\frac{\text{Time}_1}{\text{Words}_1} = \frac{\text{Time}_2}{\text{Words}_2}\) because time per word is constant.

So \(\frac{3}{128} = \frac{t}{576}\)

Then cross - multiply: \(128t = 3\times576\)

Calculate \(3\times576 = 1728\)

Then \(t=\frac{1728}{128}\)

Simplify \(\frac{1728}{128}\): divide numerator and denominator by 16, we get \(\frac{108}{8}\), divide by 4 again, \(\frac{27}{2}=13.5\)

Answer:

13.5 minutes (or \(\frac{27}{2}\) minutes)