Question 127562
If a number is added to the numerator of 3/4 and twice as much is added to the denominator, the result is 4/7. Find the number.
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Let the number to be added = x
:
{{{((x+3))/((2x+7))}}} = {{{4/7}}}
Cross multiply, and solve for x 
4(2x+7) = 7(x+3)
8x + 28 = 7x + 21
8x - 7x = 21 - 28
x = -7
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Check solution in original equation, substitute -7 for x
{{{((-7+3))/((2(-7)+7))}}} = 
{{{(-4)/((-14+7))}}} = 
{{{(-4)/(-7)}}} = {{{4/7}}}; minus into a minus is a plus
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2. The sum of a number an its reciprocal is 13/6. Find the numbers.
x + {{{1/x}}} = {{{13/6}}}
Multiply equation by 6x to get rid of the denominators
6x(x) + 6x*{{{1/x}}} = 6x*{{{13/6}}}
6x^2 + 6 = 13x
:
6x^2 - 13x + 6 = 0; a quadratic equation
:
After a little slogging, we find we can factor this to:
(2x - 3)(3x - 2) = 0
:
2x = 3
x = {{{3/2}}}
and
3x = 2
x = {{{2/3}}}
:
Check solution using x = 3/2:
{{{3/2}}} + {{{2/3}}} = {{{13/6}}}
{{{9/6}}} + {{{4/6}}} = {{{13/6}}}; confirms our solution
Note that the 2nd solution is the reciprocal of the 1st
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3. An inlet pipe can fill an empty swimming pool in 5 hours, and another inlet pipe can fill the pool in 4 hours . How long will it take both pipes to fill the pool?
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A typical shared work problem:
Let t = time required when both pipes are open
Let the full pool = 1
:
{{{t/5}}} + {{{t/4}}} = 1
Multiply equation by a common multiple, 20 will work
20*{{{t/5}}} + 20*{{{t/4}}} = 20(1)
Cancel out the denominators and you have:
4t + 5t = 20
:
9t = 20
t = {{{20/9}}}
t = 2.22 hrs or 2 hrs & 13.3 min
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Check solution using 2.22 for t:
{{{2.22/5}}} + {{{2.22/4}}} = 1
.444 + .555 = .999 ~ 1; confirms our solution
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Could you follow the procedures in all these problems? Let me know if you have any questions