Question 533726: The denominator of a fraction is an integer that is one greater than the numerator. A second fraction is the reciprocal of the integer that is one less than the numerator of the first fraction. If the product of two fractions is 3/8, identify the first fraction.
Answer by fcabanski(1391)   (Show Source):
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Read the words, and write exactly what they say.
The denominator of a fraction is an integer that is one greater than the numerator.
We don't know what the numerator is, so let's call it x. The denominator is one greater than the numerator, so it's x+1. Thus this fraction is x/(x+1). Both x and x+1 must be integers. Otherwise the denominator won't be an integer.
A second fraction is the reciprocal of the integer that is one less than the numerator of the first fraction.
The numerator of the first fraction is x. An integer one less than x is x-1. The reciprocal is 1 over the thing. So this fraction is 1/(x+1).
If the product of two fractions is 3/8, identify the first fraction.
Product means multiply.
x/(x+1) * 1/(x-1) = 3/8
Multiply numerator by numerator and denominator by denominator.
x*1 / (x+1) * (x-1) = 3/8
The numerator is x. The denominator is a binomial multiplication. So use FOIL.
(x-1) * (x+1) =
F - first times first: x*x=x^2
O - outside times outside: 1*x = x
I - inside times inside: -1*x=-x
L - last times last: -1*1=-1
Add those terms: x^2 + x - x - 1 = x^2 - 1
Thus the fraction is x/(x^2-1)=3/8
Cross multiple whenever one fraction equals another. 8 * x = 3 (x^2-1)
Multiply 3 by both terms in the ()
8x = 3x^2 - 3
Bring everything to one side of the equation.
0 = 3x^2 - 8x - 3 Remember any equation can be turned around because we can multiply both sides by -1 and then add each side to both sides.
-1*0 = -1 * (3x^2 - 8x - 3) = -3x^2 + 8x + 3
Add 3x^2 to both sides, subtract 8x from both sides, subtract 3 from both sides.
0 + 3x^2 -8x - 3 = -3x^2 + 3x^2 -8x +8x -3 + 3
3x^2 - 8x - 3 = 0
Reverse foil to get two terms, and then set each equal to 0.
(3x )(x ) we need terms that when one is multiplied by 3 and then they are added, are -8, and when multiplied together = -3. Factors of -3 are 3 and -1, -3 and 1. Try 1 and -3.
(3x+1)(x-3)=0 FOIL proves that works. Set each = to 0 and solve for x.
3x+1=0 subtract 1 from both sides. 3x+1-1=0-1 ---> 3x = -1
Divide by 3 ---> x=-1/3 That won't work, because x has to be an integer.
x-3 = 0 add 3 to both sides. x-3+3=0+3 ---> x=3.
X=3 is the answer. Check by substituting it into the original equation we built.
3/4 * 1/2 = 3/8
3*1 = 3 (the numerator) and 4*2=8 (the denominator).
3/8=3/8 HURRAH!!!!!!!!!
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