How do you order the following from least to greatest #sqrt14, sqrt(49/36), 3 3/5, 3#?

Question

Eli Assouline · Accepted Answer

#sqrt(49/36), 3, 3 3/5, sqrt14#

The first issue stems from the fact that the values are all in different formats. Converting them to decimals makes comparisons the simplest.

The simplest and most obvious approach is to simply use a calculator, but that is a cop-out that only requires pressing buttons. Instead, let's consider the question carefully.

#sqrt 14 " lies between " sqrt 9 and sqrt 16, and "is 3....."# We need to calculate this one. #sqrt14 = 3.742#

#sqrt(49/36) = 1 13/36#

#3 3/5 = 3.6#

We can now use the original formats to put them in order.

#sqrt(49/36), 3, 3 3/5, sqrt14#

Isaiah Anthony · Answer

#color(red)("Using a 'trick'")#
The order from smallest to largest is: #sqrt(49/36);" " 3;" "3 3/5;" "sqrt(14)# #color(blue)("A bit of a cheat method")# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(magenta)("I have just realised that there is a slight flaw in my logic")# As it happens the values are such that the answer is till correct but this is more by luck! The problem is that #sqrt(49/36)>49/36# ' ~ ~~~~~~~~~~~~~~~~\~~ Original values:#" "sqrt(14),sqrt(49/36),3 3/5,3# #color(white)(.)# #color(green)("There is a trap in the logic I previously applied. Which was:")# #color(brown)("If you magnify the values by some consistent operation you")# #color(brown)("increase the difference but do not change the order.")# #color(magenta)("I squared everything and then compared. Wrong!!!!")# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Solution re-worked")# Consider #sqrt(49/36) = (sqrt(49))/(sqrt(36)) = 7/6# giving #sqrt(14)," "7/6," "3 3/5," "3# #color(magenta)("This is a much better method!")# For comparison, round the numbers to the nearest whole number and multiply the difference by, say, 100. Providing: #sqrt(14),color(white)(.)" "7/6,color(white)(.)" "3 3/5,color(white)(...)" "3# #color(white)(.)374,color(white)(.)" "116,color(white)(.)" "360,color(white)(.)" "300 color(brown)(larr" magnified approximations")# #color(white)(.)uarr" "color(white)(.) uarr" " color(white)(.) uarr" "color(white)(.) uarr# #color(white)(..)4" "1" "3" "2 color(brown)(" "larr" ranking: lowest rank is 1")# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Remember that #7/6-=sqrt(49/36)# So the order lowest to highest is: #sqrt(49/36);" " 3;" "3 3/5;" "sqrt(14)#

Eva Adamson · Answer

undefined The order from least to greatest is: sqrt(14), 3, 3 3/5, sqrt(49/36).