Rate of first survey = 7/80
Rate of second survey = 9/63

% Response rate increase (9/63-7/80)/(7/80) = 63%

Answer is E = 63%

I wrote down (9/63) - (7/80) and then cross multiplied to get 279/5040 which I reduced down to about 6/112 so it looks like there was a 5% difference. Answer B!

case 1: (7/80)=x/100
x=9%

case 2: (9/63)=y/100
y=14%

So percent increase is =(y-x)=(14-9)%=5%

Answer is B

Answer = E = 63%

\(\frac{\frac{9}{63} - \frac{7}{80}}{\frac{7}{80}} * 100\)

\(= \frac{31}{560} * \frac{80}{7} * 100 = \frac{3100}{49} \approx{63%}\)

Official Solution:

A survey was sent to 80 customers, 7 of whom responded. Then the survey was redesigned and sent to another 63 customers, 9 of whom responded. By approximately what percent did the response rate increase from the original survey to the redesigned survey?

A. 2%
B. 5%
C. 14%
D. 28%
E. 63%

To calculate the percent increase of any value, we use this formula: \(\frac{\text{New} - \text{Old}}{\text{Old}}\). In other words, we subtract the original value \(\text{Old}\) from the increased value \(\text{New}\), then divide the result by the original value \(\text{Old}\). Even though the values themselves may be written using percents, we must be sure to divide by the original value.

In this problem, the original response rate is \(\frac{7}{80}\). The new response rate is \(\frac{9}{63}\), or \(\frac{1}{7}\). The difference is \(\frac{1}{7} - \frac{7}{80}\). We convert to common denominators: \(\frac{80}{560} - \frac{49}{560} = \frac{31}{560}\). Now, we divide by the original value, \(\frac{7}{80}\). Doing so is the same as multiplying by \(\frac{80}{7}\). This gives us \(\frac{31}{560}*\frac{80}{7} = \frac{31}{49}\). Estimating, we can see that this fraction is slightly more than \(\frac{30}{50}\), which is \(\frac{60}{100}\) or 60%. Thus, the correct answer must be (E): 63%.

Note that the response rates may be written as percents. For instance, the original response rate \(= \frac{7}{80} = 0.0875 = 8.75%\). Likewise, the new response rate \(= \frac{9}{63} = \frac{1}{7} \approx 14.3%\). However, if you write these rates this way, do not simply take the difference! That difference, approximately 5 or 6%, does NOT represent the percent increase in the response rate. You must divide by the original response rate.

Answer: E.
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One can use some approximation in such questions:

Initial Response rate = (7/80)* 100 = 7*5/4 = about 9%
Redesigned Response rate = (9/63)* 100 = (1/7)*100 = 14.28% (You should remember these fraction to percent equivalents)

Increase is of about 5 on 9 which is certainly more than 50%. Only (E) satisfies.

Answer (E)
_________________

We can use the percent change formula, which is (New - Old)/Old x 100, to determine the percent increase in the response rate:

[(9/63) - (7/80)]/(7/80) x 100 =

[(1/7) - (7/80)]/(7/80) x 100 =

Multiplying the left side by 560/560, we have:

(80 - 49)/49 x 100 =

31/49 x 100

We see that 31/49 is roughly 31/50 = 62/100 = 62%

Alternate Solution:

We can approximate the initial response rate of 7/80 as roughly 7/77, or 1/11. The new response rate is 9 /63, or 1/7. We can use the percent change formula, which is (New - Old)/Old x 100, to determine the percent increase in the response rate:

(1/7 - 1/11)/1/11 x 100 =

Multiplying the expression by 77/77, we obtain:

(11 - 7)/7 x 100 =

4/7 x 100

We see that 4/7 is slightly more than ½, so 4/7 x 100 will be slightly greater than 50%.

Answer: E
_________________

Are you doing this math in your head?

It is not hard.

\(\frac{\frac{9}{63} - \frac{7}{80}}{\frac{7}{80}} = \frac{\frac{9}{63}}{\frac{7}{80}} - 1 = \frac{9*80}{63*7} - 1 = 80/49 - 1 = 31/49\)

In percentage terms, it is a bit more than \(\frac{31}{50}* 100 = 62%\)
_________________

Because the answers are so far apart, we can do some very rough approximation.

Lets convert each survey to be out of roughly 240

first: multiply by 3/3 to get 21 / 240
second: multiply each by ~3.9/~3.9 to get somewhere in the 32 to 38 range /240

Even if we aren't great at our estimation and we pick 32, we get an increase of:

(32-21) /21 -> much closer to Option E than Option B.

If we are superb at estimation and pick more accurate numbers, we would multiply the second survey by ~3.8/~3.8, and get ~34 as the number of respondents, then see (34-21)/21 is almost exactly Option E.