Question 118506
1. One half of a number is 3 more than one sixth of the same number. What is the number?
:
Let x = the number:
:
Just write what is says; (the word "is" usually means equal)
"One half of a number is 3 more than one sixth of the same number."
.5x = 3 + {{{1/6}}}x
Usually written like this:
.5x = {{{1/6}}}x + 3
Multiply equation by 6 to get rid of the denominator
6(.5x) = 6*{{{1/6}}}x + 6(3)
3x = x + 18
3x - x = 18
2x = 18
x = 18/2
x = 9
:
Check solution in the original equation:
.5(9) = 3 + {{{1/6}}}*9
4.5 = 3 + 1.5, confirms out solution
:
:
2. The denominator of a certain fraction is three times the numerator. If one is
 added to the numerator and subtracted from the denominator, the result equals
 1/2. Find the original fraction.
:
Let x = the numerator:
It says,"The denominator of a certain fraction is three times the numerator."
Therefore: 3x = the denominator
:
It says,"If one is added to the numerator and subtracted from the denominator, the result equals  1/2."
{{{((x+1))/((3x-1))}}} = {{{1/2}}}
:
Cross multiply and you have:
2(x + 1) = 1(3x - 1)
2x + 2 = 3x - 1
2 + 1 = 3x - 2x
x = 3
:
Check solution in original equation:
{{{((3+1))/((9-1))}}} = {{{1/2}}}; confirms our solution
:
:
In the next 2 problems, I will construct the equation and you can solve it. 
If you have difficulty, you can email me for help.
:
3. The denominator of a fraction is 5 more than the numerator.
{{{x/((x+5))}}} 
If 5 is added to the numerator and 2 is added to the denominator, the value of the fraction is 8/9. 
{{{((x+5))/((x+7))}}} = {{{8/9}}}
Find the original fraction; (you can cross multiply here)
:
:
4. The numerator is one less than the denominator. If two is added to the denominator and subtracted from the numerator, the value of the fraction is 1/2. Find the original fraction.
:
{{{x/((x+1))}}}
Add 2 to the denominator and subtract 2 from the numerator
{{{((x-2))/((x+3))}}} = {{{1/2}}}
Do the same here
:
:
5. Ed can do a job in four days. When Ed and May work together, it would take them 2 1/3 days.
:
Let x = time required for M to do the job alone
Let the completed job = 1
Use the decimal equiv of 1/3: 2.333 days
:
How long would the job take May to do it alone?
A simple equation:
{{{2.333/4}}} + {{{2.333/x}}} = 1
Multiply equation by 4x to get rid of the denominators, resulting in:
2.333x + 4(2.333) = 4x
9.333 = 4x - 2.333
9.333 = 1.67x
x = 9.333/1.667
x = 5.6 days, M's time working alone
:
Check solution in original equation
{{{2.333/4}}} + {{{2.333/5.6}}}} =
 .58325 + .4166 = .9998 ~ 1, confirms our solution
:
:
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?
:
Let x = time for this to take place
Let the full sink = 1
We have 3 elements here. Let filling be + and draining be neg
{{{x/15}}} + {{{x/12}}} - {{{x/25}}} = 1
We want to find a common multiple of all three denominators here
If we multiply the equation by that, it becomes pretty easy, doesn't it.
:
The best I can come up with is 15*12*25 = 4500
4500*{{{x/15}}} + 4500*{{{x/12}}} - 4500*{{{x/25}}} = 4500(1)
cancel out the denominators and we have:
300x + 375x - 180x = 4500
I think that you  can finish up this one now? (It won't be an even number)
:
:
7. Julius can paint a house three times faster than Ruben. Working together,
 they can paint a house in four days. How long would it take the faster
 painter if he works alone?
:
"J can paint a house 3 time faster than R"
We can take this to mean if J's time = x, then R's time is 3x
:
{{{4/x}}} + {{{4/3x}}} = 1
Multiply equation by 3x:
3x*{{{4/x}}} + 3x{{{4/3x}}} = 3x(1)
cancel out the denominators and we have:
3(4) + 4 = 3x
12 + 4 = 3x
3x = 16
x = 16/3
x = 5 {{{1/3}}} days
:
Check our solution using x = 5.333 days
{{{4/5.333}}} + {{{4/3(5.333)}}} = 
  .75  + .25 = 1; confirms our solution
:
:
Dear Student, I have taken a significant amount of time to show you how to do 
these problems. Please, if you have question on what was done here, email me.
Your success in this is my main concern. A