Let us consider #4/15#. If we multiply numerator and denominator by same number, fraction remains same. Similarly if we multiply numerators and denominators in fractions #1/12# and #3/8# by same number, value fractions does not change.
Note one thing more. We can add, subtract or comare these three fractions, only if their denominators are same. Hence purpose of finding Least Common Denominator is to identify a common denominator , to which each of the denominators viz., #15,12# and #8# can be raised.
We can raise #15# to #30,45,60,75,90,105,color(red)120,135,...# by multiplying it by #2,3,4,5,6,7,8,9,....# and so on.
Similarly #12# can be raised to #24,36,48,60,72,84,96,108,color(red)120,132,144,....#
and #8# to #16,24,32,40,48,56,64,72,80,88,96,104,112,color(red)120,128,136,....#
and which is common denominator. This is #120#. In fact if we write more multiples, we may find that we too have as common denominators #240,360,480,...# and #120# is only the #color(red)("Least Common Denominator.")#
Now raising denominators of each fraction to #120#, we have
#4/15=(4xx8)/(15xx8)=32/120#
#1/12=(1xx10)/(12xx10)=10/120# and
#3/8=(3xx15)/(8xx15)=45/120#
Now, we can easily compare them and #45/120>32/120>10/120# i.e. #3/8>4/15>1/12#
or add them #4/15+1/12+3/8=32/120+10/120+45/120=(32+10+45)/120=87/120#
or for example #4/15-1/12+3/8=32/120-10/120+45/120=(32-10+45)/120=67/120#
Observe that #color(red)("Least Common Denominator")# is same as #color(red)("Least Common Multiple")" of denominators."#
