Question 118506
It looks like you have put your entire homework assignment into one question.  This site was not created so that you could get someone to do your homework for you.  You have two different types of problems.  I'll show you how to do one of each.


1.  One half of a number is 3 more than one sixth of the same number. What is the number?


Let the number be x.  One half of the number is then {{{x/2}}}.  One sixth of the number is {{{x/6}}} and 3 more than that is {{{(x/6)+3}}}.  So you can write:


{{{x/2=(x/6)+3}}}


Multiply both sides by 6 to eliminate the denominators:


{{{3x=x+18}}}


Subtract x from both sides to get the variable on the left:


{{{2x=18}}}


Divide both sides by 2:


{{{x=9}}}


Check your answer:

Half of {{{9}}} is {{{9/2}}}
One sixth of {{{9}}} is {{{9/6=3/2}}}
{{{(3/2)+3=(3/2)+(6/2)=9/2}}}, answer checks


6. A cold water faucet could fill a sink in 15 minutes, and a hot water faucet can fill it in 12 minutes. The drain can empty the sink in 25 minutes. If both faucets are on and the drain is open, how long would it take to fill the sink?


If the cold faucet can fill the sink in 15 minutes, it fills {{{1/15}}}th of the sink in 1 minute.  Likewise, the hot faucet fills {{{1/12}}}th of the sink in 1 minute.  The LCD of 15 and 12 is 60, so the cold faucet fills the sink at {{{4/60}}}ths per minute, and the hot at {{{5/60}}}ths per minute.  Together, they fill the sink at {{{(4/60)+(5/60)=9/60}}}ths per minute.  Just as a sanity check, it would take 6 and 2/3 minutes to fill the sink IF the drain were CLOSED.


The drain empties the sink at {{{1/25}}} of the sink per minute.  Now we need an LCD for 25 and 60:

{{{25=5*5}}}
{{{60=2*2*3*5}}}, so the LCD is {{{2*2*3*5*5=300}}}


With both faucets and the drain open fully, the sink is filling at a rate of {{{45/300}}}ths per minute minus {{{12/300}}}ths per minute, or {{{33/300}}}ths per minute.


The sink therefore fills in {{{300/33=100/11}}} minutes exactly, or 9 minutes as a very close approximation (rounded from the repeating decimal 9.09...).  If you require a decimal answer (instead of a fraction), express it as 9 because you shouldn't give an answer that is more precise than the least precise measurement given in the problem.  The time measurements given were all in whole minutes, so your answer should be in whole minutes.


By the way, remember the sanity check.  6 and 2/3 minutes to fill if the drain were closed and 9 minutes with the drain open is a ratio of about 1.4.  45/300 with the drain closed and 33/300 to fill with the drain open is also a ratio of about 1.4.  That quick mental calculation assures us that the answer is reasonable.


Hope that helps.
John