M07-23

Question

Which of the following expressions has the greatest value?

A.  0.456

B.  \frac{7}{16}

C.  \frac{6}{13}

D.  \frac{9}{19}

E.  \sqrt{0.17}

Bunuel · Accepted Answer

Official Solution:

Which of the following expressions has the greatest value?

A.  0.456

B.  \frac{7}{16}

C.  \frac{6}{13}

D.  \frac{9}{19}

E.  \sqrt{0.17}

Note that each option is very close to 1/2. To compare them, we can consider their distance from 1/2. To simplify this, let's convert the denominators of each option to even numbers, making it easier to calculate the difference from 1/2.
 
A.  0.456 = \frac{456}{1000} = \frac{500}{1000} - \frac{44}{1000} = \frac{1}{2} - \frac{11}{250} 
 
B.  \frac{7}{16} = \frac{8}{16} - \frac{1}{16} = \frac{1}{2} - \frac{1}{16} 
 
C.  \frac{6}{13} = \frac{12}{26} = \frac{13}{26} - \frac{1}{26} = \frac{1}{2} - \frac{1}{26}  
 
D.  \frac{9}{19} = \frac{18}{38} = \frac{19}{38} - \frac{1}{38} = \frac{1}{2} - \frac{1}{38} 
 
E.  \sqrt{0.17}  is closer to 0.4 than it is to 0.5. Therefore, it must be less than  0.45 , which can be represented as  0.45 = \frac{45}{100} = \frac{50}{100} - \frac{5}{100} = \frac{1}{2} - \frac{1}{20} 
 
Among options B, C, D, and E, option D has the smallest distance from 1/2 ( \frac{1}{38} < \frac{1}{26} < \frac{1}{20} < \frac{1}{16} ), so it must be the largest among these four. It's also clear that  \frac{1}{38}  is smaller than  \frac{11}{250} , which is approximately  \frac{1}{25} . Therefore, option D must have the greatest value.

Answer: D

Bunuel · Answer

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Prasesh · Answer

Bunuel can you suggest some more problems of similar types. Thanks!

sssdd1234 · Answer

I think this is a  high-quality  question and I agree with explanation.

SergejK · Answer

Hi  Bunuel , one question:

sqrt(0.17)=sqrt(17/100)=sqrt(17)/10 if we square the whole expression, we should get 17/100. where did I go wrong?

Bunuel · Answer

What is your question?  \sqrt{0.17}= \sqrt{\frac{17}{100}} =\frac{\sqrt{17}}{10} . If you square this you get 17/100.

SergejK · Answer

Oh now I see it. Sorry, wrongly squared the whole expression and thought I will land at the same expression. Should make longer breaks.

dnguyen1203 · Answer

I think this is a  high-quality  question and I agree with explanation.

Maulik_Atri · Answer

This is a great question that’s helpful for learning.

Maulik_Atri · Answer

This is a great question that’s helpful for learning.

bondi123 · Answer

I like the solution - it’s helpful.

Paattaa · Answer

This is a great question that’s helpful for learning.

Chrisharley · Answer

I like the solution - it’s helpful.

agrasan · Answer

I like the solution - it’s helpful.

CapedCrusader09 · Answer

I like the solution - it’s helpful.

gemministorm · Answer

I was thinking of this way, if all were in fraction (quick division speeded things a bit)

7/16 -> 1/(16/7) -> 1/2.28
6/13 -> 1/(13/6) -> 1/2.16
9/19 -> 1/(19/9) -> 1/2.11

largest is the one with smallest denominator, hence 9/19.
for other 2 options, a little more adjustments has to be done to bring in this form to compare.

Goldenfuture · Answer

A. 0.456
B. 7/16 = 8/16-1/16
 = 0.5 - 1/16

C. 6/13 = 6.5/13-.5/13
= .5 - .5/13 ( note .5/13 = 1/26 lesser than 1/16 so B is out)

D = 9/19 = 9.5/19 - .5/19 (.5/19 is less than .5/13 - C is out)
 
E. √0.17 ~ .45*.4 =0.16+.02= .18 - out by good margin - option A was .45+ already 
Now we are left b/w A&D .5/19 is like .5/20 = 2.5 from 100 = 0.025 - this means D will be highest

M07-23

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