COMPLEX FRACTIONS

Excerpts

COMPLEX FRACTIONS
A **complex fraction** is one which has a fraction in one or both of its terms.
COMPLEX FRACTIONS
*To simplify a complex fraction, multiply each term of the complex fraction by the L.C.M. of the denominators of the fractions in the terms, and reduce.*
COMPLEX FRACTIONS
What is a simple fraction? What is the effect of multiplying a simple fraction by its denominator or by any multiple of its denominator?

Equations

COMPLEX FRACTIONS
\begin{array}{ccccc} a &+& \frac{ax}{b} & \times & abd \\ \cline{1-3} \frac{c}{d} &+& \frac{x}{ab} & \times & abd \end{array} = \frac{ a^2bd + a^2dx }{ abc + dx }
Multiplying the numerator and the denominator of the complex fraction (a + ax/b)/(c/d + x/(ab)) by abd clears the inner fractions and gives (a^2bd + a^2dx)/(abc + dx).
COMPLEX FRACTIONS
\begin{array}{ccccc} \frac{1}{1-x} &-& \frac{1}{1+x} & \times & (1-x)(1+x) \\ \cline{1-3} \frac{1}{1-x} &+& \frac{1}{1+x} & \times & (1-x)(1+x) \end{array} = \frac{1 + x - 1 + x}{1 + x + 1 - x} = \frac{2x}{2} = x
Multiplying the numerator and the denominator of the complex fraction by (1-x)(1+x) clears the inner fractions, and the simplified value is x.

Problems

Exercise 51

Exercise 51, problem 1
$\displaystyle \frac{1+\frac{a}{b}}{\frac{x}{b}-1}$.
Printed answer:
- $\displaystyle \frac{b+a}{x-b}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (b + a)/(x - b)
Exercise 51, problem 10
$\frac{1+a+a^2}{1+\frac{1}{a}+\frac{1}{a^2}}$.
Printed answer:
- $a^2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2
Exercise 51, problem 11
$\displaystyle \frac{x-3-\frac{2}{x-4}}{x-1+\frac{2}{x-4}}$.
Printed answer:
- $\displaystyle \frac{x-5}{x-3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x - 5)/(x - 3)
Exercise 51, problem 12
$\displaystyle \frac{\frac{a}{a+b}+\frac{a}{a-b}}{\frac{2a}{a^2-b^2}}$.
Printed answer:
- $a$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a
Exercise 51, problem 13
$\displaystyle \frac{\frac{a+1}{a-1}+\frac{a-1}{a+1}}{\frac{a+1}{a-1}-\frac{a-1}{a+1}}$.
Printed answer:
- $\displaystyle \frac{a^2+1}{2a}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a**2 + 1)/(2*a)
Exercise 51, problem 14
$\displaystyle \frac{\frac{a^3+b^3}{a^2-b^2}}{\frac{a^2-ab+b^2}{a-b}}$.
Printed answer:
- $1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1
Exercise 51, problem 15
$\displaystyle 1-\frac{1}{1+\frac{2}{a-2}}$.
Printed answer:
- $\displaystyle \frac{2}{a}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2/a
Exercise 51, problem 16
$\displaystyle \frac{\frac{a+2b}{b}-\frac{a}{a+b}}{\frac{a+2b}{a+b}+\frac{a}{b}}$.
Printed answer:
- $l$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: FLAG-PARSE 1
Exercise 51, problem 17
How many pounds of pepper can be bought for $y$ dollars, if $x$ pounds cost $2x$ dimes?
Printed answer:
- $5y$ lbs.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem {p: 5*y}
Exercise 51, problem 18a
How many inches in $y$ feet? How many yards?
Printed answer:
- 12y in.
unverified: no computed check settled this one (yet)
How it was checked
- evaluate: the record may be misread 12*y
Exercise 51, problem 18b
How many inches in $y$ feet? How many yards?
Printed answer:
- $\displaystyle \frac{y}{3}$ yds.
unverified: no computed check settled this one (yet)
How it was checked
- evaluate: the printed answer does not match the problem y/3
Exercise 51, problem 19
A man bought $x$ pounds of beef at $x$ cents a pound, and handed the butcher a $y$-dollar bill. How many cents change should he receive?
Printed answer:
- $100y-x^2$ cts.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {c: 100*y - x**2}
Exercise 51, problem 2
$\displaystyle \frac{x+\frac{a}{y}}{m-\frac{b}{y}}$.
Printed answer:
- $\displaystyle \frac{xy+a}{my-b}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x*y + a)/(m*y - b)
Exercise 51, problem 20
$x$ times $I$ is how many times $m$?
Printed answer:
- $\displaystyle \frac{lx}{m}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {k: I*x/m}
Exercise 51, problem 3
$\displaystyle \frac{x}{a+\frac{b}{c}}$.
Printed answer:
- $\displaystyle \frac{cx}{ac+b}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes c*x/(a*c + b)
Exercise 51, problem 4
$\displaystyle \frac{x-\frac{1}{x^2}}{1-\frac{1}{x}}$.
Printed answer:
- $\displaystyle \frac{x^2+x+1}{x}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x**2 + x + 1)/x
Exercise 51, problem 5
$\displaystyle \frac{\frac{a}{b}-\frac{b}{a}}{a-b}$.
Printed answer:
- $\displaystyle \frac{a+b}{ab}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a + b)/(a*b)
Exercise 51, problem 6
$\displaystyle \frac{1+m^3}{1+\frac{1}{m}}$.
Printed answer:
- $m(m^2-m+1)$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes m*(m**2 - m + 1)
Exercise 51, problem 7
$\displaystyle \frac{\frac{x}{y}-\frac{z}{x}}{\frac{a}{y}-\frac{b}{x}}$.
Printed answer:
- $\displaystyle \frac{x^2-yz}{ax-by}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x**2 - y*z)/(a*x - b*y)
Exercise 51, problem 8
$\displaystyle \frac{\frac{ab}{c}-3d}{3c-\frac{ab}{d}}$.
Printed answer:
- $\displaystyle -\frac{d}{c}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -d/c
Exercise 51, problem 9
$\displaystyle \frac{1+\frac{1}{a-1}}{1-\frac{1}{a+1}}$.
Printed answer:
- $\displaystyle \frac{a+1}{a-1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a + 1)/(a - 1)

Exercise 52

Exercise 52, problem 1
Divide $x^3 - 19x + 6x^7 + 20 + 8x^2 + 16x^4-x^6-11x^5$ by $x^2 + 4-3x+2x^3$.
Printed answer:
- $3x^4-2x^3-x+5$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*x**4 - 2*x**3 - x + 5
Exercise 52, problem 10
$\displaystyle \frac{x - 3}{x - 2} + \frac{2 (1 - x)}{x^2 - 6x + 8} - \frac{x - 1}{x - 4}$.
Printed answer:
- $\displaystyle -\frac{6}{x-4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -6/(x - 4)
Exercise 52, problem 11
$\displaystyle \frac{1}{(2 - x)(3 - x)} - \frac{2}{(x - 1)(x - 3)} + \frac{1}{(x - 1)(x - 2)}$.
Printed answer:
- $0$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 0
Exercise 52, problem 12
$\displaystyle \frac{x^2 + xy}{x - y} \times \frac{x^3 - 3x^2y + 3xy^2 - y^3}{x^2 - y^2} \div \frac{2xy - 2y^2}{3}$.
Printed answer:
- $\displaystyle \frac{3x}{2y}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*x/(2*y)
Exercise 52, problem 13
$\displaystyle \left( \frac{1}{m} + \frac{1}{n} \right) (a + b) - \left( \frac{a + b}{m} \right) - \left( \frac{a - b}{n} \right)$.
Printed answer:
- $\displaystyle \frac{2b}{n}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*b/n
Exercise 52, problem 14
$\displaystyle \left( x - \frac{1}{1 + \frac{2}{x - 1}} \right) \div \frac{x + \frac{1}{x}}{\frac{1}{x} + 1}$.
Printed answer:
- $1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1
Exercise 52, problem 15
$\displaystyle \frac{\frac{a}{b} + \frac{b}{a} - 1}{\frac{a^2}{b^2} + \frac{a}{b} + 1} \times \frac{1 + \frac{b}{a}}{a - b} \div \frac{1 + \frac{b^3}{a^3}}{\frac{a^2}{b} - \frac{b^2}{a}}$.
Printed answer:
- $1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1
Exercise 52, problem 16
Prove that $3(a + b)(a + c)(b + c) = (a + b + c)^3 - (a^3 + b^3 + c^3)$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 52, problem 17
How many sevenths in $6xy$?
Printed answer:
- $42xy$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 42*x*y
Exercise 52, problem 18
Divide $\displaystyle 2x^5 - \frac{x^4}{12} + 2\frac{3}{4}x^3 - 2x^2 - x$ by $3x^2 + x$.
Printed answer:
- $\displaystyle \frac{2x^3}{3} - \frac{x^2}{4} + x - 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x**3/3 - x**2/4 + x - 1
Exercise 52, problem 2
Find two numbers whose sum is 100 and whose difference is 10.
Printed answer:
- $45$; $55$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 45, y: 55}
Exercise 52, problem 3
Find the G.C.F. of $x^4-1$, $x^5+ x^4-x^3-x^2 + x + 1$, and $x^2 - x - 2$.
Printed answer:
- $x+1$.
verified: the printed answer passed a computed check
How it was checked
- hcf: passes x + 1
Exercise 52, problem 4a
Factor $x^{10} - 14x^5y + 49y^2$, $a^9-b^9$, $3a^2-3a-216$.
Printed answer:
- $(x^5-7y)^2$
verified: the printed answer passed a computed check
How it was checked
- factor: passes (x**5 - 7*y)**2
Exercise 52, problem 4b
Factor $x^{10} - 14x^5y + 49y^2$, $a^9-b^9$, $3a^2-3a-216$.
Printed answer:
- $(a-b)(a^2+ab+b^2)(a^6+a^3b^3+b^6)$
verified: the printed answer passed a computed check
How it was checked
- factor: passes (a - b)*(a**2 + a*b + b**2)*(a**6 + a**3*b**3 + b**6)
Exercise 52, problem 4c
Factor $x^{10} - 14x^5y + 49y^2$, $a^9-b^9$, $3a^2-3a-216$.
Printed answer:
- $3(a-9)(a+8)$.
verified: the printed answer passed a computed check
How it was checked
- factor: passes 3*(a - 9)*(a + 8)
Exercise 52, problem 5
Reduce $\displaystyle \frac{(2a+2b)(a^2-b^2)}{(a^2+2ab+b^2)(a-b)}$ to lowest terms.
Printed answer:
- $2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2
Exercise 52, problem 6
Factor $\displaystyle \frac{16x^4}{a^4b^4}-\frac{c^4}{81y^4}$.
Printed answer:
- $\displaystyle \left(\frac{4x^2}{a^2b^2}+\frac{c^2}{9y^2}\right) \left(\frac{2x}{ab}+\frac{c}{3y}\right) \left(\frac{2x}{ab}-\frac{c}{3y}\right)$.
verified: the printed answer passed a computed check
How it was checked
- factor: passes (4*x**2/(a**2*b**2) + c**2/(9*y**2))*(2*x/(a*b) + c/(3*y))*(2*x/(a*b) - c/(3*y))
Exercise 52, problem 7
If $x$ and $y$ stand for the digits of a number of two places, what will represent the number?
Printed answer:
- $10x+y$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 10*x + y
Exercise 52, problem 8
Find the L.C.M. of $a^2-5a + 6$, $a^2-16$, $a^2-9$, $a^2- 7a + 12$, and $a^2 - 4$.
Printed answer:
- $(a^2-16)(a^2-9)(a^2-4)$.
verified: the printed answer passed a computed check
How it was checked
- lcm: passes (a**2 - 16)*(a**2 - 9)*(a**2 - 4)
Exercise 52, problem 9
If one picture costs $a$ cents, how many can be bought for $x$ dollars?
Printed answer:
- $\displaystyle \frac{100x}{a}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 100*x/a