REDUCTION OF FRACTIONS

Excerpts

REDUCTION OF FRACTIONS
To reduce a fraction to its lowest terms, divide the terms of the fraction by their greatest common factor.
REDUCTION OF FRACTIONS
a^2bx-b^3xa^2bx-ab^2x = bx(a^2-b^2) ÷bx(a-b)abx(a-b) ÷bx(a-b) = a+ba
REDUCTION OF FRACTIONS
To find an entire or mixed number equivalent to a given fraction, divide the numerator by the denominator.
REDUCTION OF FRACTIONS
ac-bc-dc = a - b -dc
REDUCTION OF FRACTIONS
To change fractions to equivalent fractions having a common denominator, multiply the terms of each fraction by such a number as will make its denominator equal to the L.C.M. of the given denominators.

Equations

REDUCTION OF FRACTIONS
\frac{5a^2b \div 5ab}{10ab^2 \div 5ab} = \frac{a}{2b}
Dividing the numerator and denominator of 5a^2b/(10ab^2) by their common factor 5ab gives the lowest-terms form a/(2b).
REDUCTION OF FRACTIONS
\frac{a^2bx-b^3x}{a^2bx-ab^2x} = \frac{bx(a^2-b^2) \div bx(a-b)}{abx(a-b) \div bx(a-b)} = \frac{a+b}{a}
The fraction (a^2bx - b^3x)/(a^2bx - ab^2x) reduces to (a+b)/a after removing the common factor bx(a-b) from numerator and denominator.
REDUCTION OF FRACTIONS
\frac{ac-bc-d}{c} = a - b -\frac{d}{c}
A fraction whose numerator is divided by its denominator equals a mixed expression: the whole part a - b plus the remaining fraction -d/c.
REDUCTION OF FRACTIONS
b + \frac{a}{c} = \frac{bc + a}{c}
An entire expression b plus a fraction equals a single fraction with denominator c, obtained by multiplying b by c and adding a.
REDUCTION OF FRACTIONS
b - \frac{a - x}{c} = \frac{bc - a + x}{c}
An entire expression b minus a fraction equals a single fraction with denominator c, obtained by multiplying b by c and subtracting a - x.

Problems

Exercise 42

Exercise 42, problem 1
$\frac{\displaystyle 15x^3y^2z}{\displaystyle 40x^2y^2z^2}$.
Printed answer:
- $\displaystyle \frac{3x}{8z}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*x/(8*z)
Exercise 42, problem 10
$\displaystyle \frac{a^2xy-x^3y^3}{ax^2-x^3y}$
Printed answer:
- $\displaystyle \frac{y(a+xy)}{x}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes y*(a+x*y)/x
Exercise 42, problem 11
$\displaystyle \frac{a^4-b^4}{(a^2+2ab+b^2)(a^2+b^2)}$
Printed answer:
- $\displaystyle \frac{a-b}{a+b}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a-b)/(a+b)
Exercise 42, problem 12
$\displaystyle \frac{x^8 -y^8}{(x^4+y^4)(x^4-2x^2y^2+y^4)}$
Printed answer:
- $\displaystyle \frac{x^2+y^2}{x^2-y^2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x**2+y**2)/(x**2-y**2)
Exercise 42, problem 13
$\displaystyle \frac{a^2x^2-16a^2}{ax^2+9ax+20a}$
Printed answer:
- $\displaystyle \frac{a(x - 4)}{x + 5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a*(x-4)/(x+5)
Exercise 42, problem 14
$\displaystyle \frac{a^4-14a^2-51}{a^4-2a^2-15}$
Printed answer:
- $\displaystyle \frac{a^2 - 17}{a^2 - 5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a**2-17)/(a**2-5)
Exercise 42, problem 15
$\displaystyle \frac{3+4x+x^2}{6+5x+x^2}$
Printed answer:
- $\displaystyle \frac{1 + x}{2 + x}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (1+x)/(2+x)
Exercise 42, problem 16
$\displaystyle \frac{a^2-a^2b^2}{(a-ab)^2}$
Printed answer:
- $\displaystyle \frac{1 + b}{1 - b}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (1+b)/(1-b)
Exercise 42, problem 17
At two-thirds of a cent apiece, what $b$ apples cost?
Printed answer:
- $\displaystyle \frac{2}{3}b$, or $\displaystyle \frac{2b}{3}$ cts.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 2*b/3
Exercise 42, problem 18
James is $x$ times as old as George. If James is $a$ years old, how old is George?
Printed answer:
- $\displaystyle \frac{a}{x}$ years.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {G: a/x}
Exercise 42, problem 2
$\frac{\displaystyle 12x^5y^2z^3}{\displaystyle 30xy^3z^3}$.
Printed answer:
- $\displaystyle \frac{2x^4}{5y}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x**4/(5*y)
Exercise 42, problem 3
$\displaystyle \frac{x^2-7x+12}{x^3+x-20}$.
Printed answer:
- $\displaystyle \frac{x-3}{x+5}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem (x-3)/(x+5)
Exercise 42, problem 4
$\displaystyle \frac{x^2+9x+18}{x^2-2x-15}$.
Printed answer:
- $\displaystyle \frac{x+6}{x-5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x+6)/(x-5)
Exercise 42, problem 5
$\displaystyle \frac{a^6+a^4}{a^4-1}$
Printed answer:
- $\displaystyle \frac{a^4}{a^2-1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**4/(a**2-1)
Exercise 42, problem 6
$\displaystyle \frac{x^5-x^2}{x^6-1}$
Printed answer:
- $\displaystyle \frac{x^2}{x^3+1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2/(x**3+1)
Exercise 42, problem 7
$\displaystyle \frac{mx - ny + nx - my}{ax-2by+2bx-ay}$
Printed answer:
- $\displaystyle \frac{m+n}{a+2b}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (m+n)/(a+2*b)
Exercise 42, problem 8
$\displaystyle \frac{ac-bd+ad-bc}{ax-2by+2ay-bx}$
Printed answer:
- $\displaystyle \frac{c+d}{x+2y}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (c+d)/(x+2*y)
Exercise 42, problem 9
$\displaystyle \frac{ac^2d-a^3d^3}{a^3c+a^4d}$
Printed answer:
- $\displaystyle \frac{d(c-ad)}{a^2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes d*(c-a*d)/a**2

Exercise 43

Exercise 43, problem 1
$ \displaystyle \frac{bx - cx + m}{x}$
Printed answer:
- $\displaystyle b-c+\frac{m}{x}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes b - c + m/x
Exercise 43, problem 10
$\displaystyle \frac{a^3 - b^3}{a + b}$
Printed answer:
- $\displaystyle a^2-ab+b^2-\frac{2b^3}{a+b}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 - a*b + b**2 - 2*b**3/(a + b)
Exercise 43, problem 11
$\displaystyle \frac{8a^3}{2a - 1}$
Printed answer:
- $\displaystyle 4a^2+2a+1+\frac{1}{2a-1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*a**2 + 2*a + 1 + 1/(2*a - 1)
Exercise 43, problem 12
$\displaystyle \frac{27 x^3}{3x + 1}$
Printed answer:
- $\displaystyle 9x^2-3x+1-\frac{1}{3x+1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 9*x**2 - 3*x + 1 - 1/(3*x + 1)
Exercise 43, problem 13
$\displaystyle \frac{3x^3 + 8x^2 + 2}{x^2 + 2x - 1}$
Printed answer:
- $\displaystyle 3x+2-\frac{x-4}{x^2+2x-1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*x + 2 - (x - 4)/(x**2 + 2*x - 1)
Exercise 43, problem 14
$\displaystyle \frac{2a^3 + 3a^2 + 10a - 4}{a^2 + 3a + 2}$
Printed answer:
- $\displaystyle 2a-3+\frac{15a+2}{a^2+3a+2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a - 3 + (15*a + 2)/(a**2 + 3*a + 2)
Exercise 43, problem 15
$\displaystyle \frac{2a^4 + 6a^3 - 6a + 2}{a^2 +a - 1}$
Printed answer:
- $ 2a^2+4a-2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a**2 + 4*a - 2
Exercise 43, problem 16
$\displaystyle \frac{3x^4 - 5x^3 + x - 1}{x^2 - x - 1}$
Printed answer:
- $3x^2-2x+1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*x**2 - 2*x + 1
Exercise 43, problem 17
$ \displaystyle x + y - \frac{xy}{x - y}$
Printed answer:
- $\displaystyle \frac{x^2-y^2-xy}{x-y}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x**2 - y**2 - x*y)/(x - y)
Exercise 43, problem 18
$\displaystyle a - b - \frac{2 ab}{a + b}$
Printed answer:
- $\displaystyle \frac{a^2-2ab-b^2}{a+b}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a**2 - 2*a*b - b**2)/(a + b)
Exercise 43, problem 19
$\displaystyle -c + d + \frac{c^3 + d^3}{c^2 + cd + d^2}$
Printed answer:
- $\displaystyle \frac{2d^3}{c^2+cd+d^2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*d**3/(c**2 + c*d + d**2)
Exercise 43, problem 2
$ \displaystyle \frac{mn + an + x}{n}$
Printed answer:
- $\displaystyle m+a+\frac{x}{n}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes m + a + x/n
Exercise 43, problem 20
$\displaystyle \frac{x^2y + xy^2}{x - y} + x^2 - y^2$
Printed answer:
- $\displaystyle \frac{x^3+y^3}{x-y}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x**3 + y**3)/(x - y)
Exercise 43, problem 21
$\displaystyle \frac{x+2}{3x - 1} - x - 2$
Printed answer:
- $\displaystyle \frac{4-4x-3x^2}{3x-1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (4 - 4*x - 3*x**2)/(3*x - 1)
Exercise 43, problem 22
$\displaystyle 2a - 1 - \frac{a-2}{a+3}$
Printed answer:
- $\displaystyle \frac{2a^2+4a-1}{a+3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (2*a**2 + 4*a - 1)/(a + 3)
Exercise 43, problem 23
$\displaystyle a^2 - 3a + 2 - \frac{a + 3}{2a^2 - 1}$
Printed answer:
- $\displaystyle \frac{2a^4-6a^3+3a^2+2a-5}{2a^2-1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (2*a**4 - 6*a**3 + 3*a**2 + 2*a - 5)/(2*a**2 - 1)
Exercise 43, problem 24
$\displaystyle 2 x^2 + x - 3 - \frac{x - 2}{3 x^2 + 1}$
Printed answer:
- $\displaystyle \frac{6x^4+3x^3-7x^2-1}{3x^2+1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (6*x**4 + 3*x**3 - 7*x**2 - 1)/(3*x**2 + 1)
Exercise 43, problem 25
$\displaystyle \frac{3x + 2}{x^2 + x + 2} - 1$.
Printed answer:
- $\displaystyle \frac{4x-x^2}{x^2-x+2}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem (4*x - x**2)/(x**2 - x + 2)
Exercise 43, problem 26
$\displaystyle 1 - \frac{2a^2 + 3}{a^2 - 2a + 3}$.
Printed answer:
- $\displaystyle -\frac{a^2+2a}{a^2-2a+3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -(a**2 + 2*a)/(a**2 - 2*a + 3)
Exercise 43, problem 27
$\displaystyle \frac{a^2}{a^2 - a + 3} - a^2 - a - 1$.
Printed answer:
- $\displaystyle -\frac{a^4+2a^2+2a+3}{a^2-a+3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -(a**4 + 2*a**2 + 2*a + 3)/(a**2 - a + 3)
Exercise 43, problem 28
$\displaystyle \frac{x^4}{x^2 + x - 1} - x^2 + x - 1$.
Printed answer:
- $\displaystyle \frac{x^2-2x+1}{x^2+x-1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (x**2 - 2*x + 1)/(x**2 + x - 1)
Exercise 43, problem 29
$\displaystyle \frac{a^2}{2}, \frac{xy}{3}, \frac{5ab^2}{4}.$
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 43, problem 3
$\displaystyle \frac{x^2 + y^2 + 3}{x - y}$
Printed answer:
- $\displaystyle x+y+\frac{3}{x-y}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem x + y + 3/(x - y)
Exercise 43, problem 30
$\displaystyle \frac{x^3}{3}, \frac{am}{2}, \frac{3x^2y}{5}.$
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 43, problem 31
$\displaystyle \frac{7x^2}{3y}, \frac{5xy}{2a}, \frac{x^3}{ay}.$
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 43, problem 32
$\displaystyle \frac{2a^2}{7m}, \frac{5ab}{3n}, \frac{b^4}{mn}.$
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 43, problem 33
$\displaystyle \frac{5}{y+x}, \frac{3x}{x-y}, \frac{2ab}{x^2-y^2}.$
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 43, problem 34
$\displaystyle \frac{2}{3+a}, \frac{5a}{a-3}, \frac{2ax}{a^2-9}.$
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 43, problem 35
$\displaystyle \frac{b^2}{a^2-ab}, \frac{a^2b}{a^2-b^2}.$
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 43, problem 36
$\displaystyle \frac{m^3}{nm+n^2}, \frac{m^2n}{m^2-n^2}.$
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: the record may be misread
Exercise 43, problem 37
$\displaystyle \frac{2}{x^4-y^4}, \frac{3}{x^2 + xy - 2y^2}.$
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 43, problem 38
$\displaystyle \frac{5}{a^8-b^8}, \frac{2}{a^4 - 4a^2b^2 + 3b^4}.$
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 43, problem 39
$\displaystyle \frac{a-1}{a^2-2a-15}, \frac{a+2}{a^2+a-6}, \frac{a+3}{a^2-7a+10}.$
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 43, problem 4
$\displaystyle \frac{a^3 + b^3 + 3}{x - y}$
Printed answer:
- $\displaystyle a^2-ab+b^2-\frac{2}{a+b}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem a**2 - a*b + b**2 - 2/(a + b)
Exercise 43, problem 40
$\displaystyle \frac{x+1}{x^2-x-6}, \frac{x-2}{x^2+6x+8}, \frac{x+2}{x^2+x-12}.$
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 43, problem 41
$\displaystyle \frac{x^2+3x+2}{x^2-x-6}, \frac{x^2-x}{x^2+5x+4}, \frac{x^2-6x+9}{x^2+x-12}.$
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 43, problem 42
If $x$ quarts of milk cost $m$ cents, what will one pint cost?
Printed answer:
- $\displaystyle \frac{m}{2x}$ ct.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {p: m/(2*x)}
Exercise 43, problem 43
Express four consecutive numbers of which $a$ is the largest.
Printed answer:
- $a-3$, $a-2$, $a-1$, $a$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {n1: a - 3, n2: a - 2, n3: a - 1, n4: a}
Exercise 43, problem 44
What is the next even number above $2m$?
Printed answer:
- $2m+2$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 2*m + 2
Exercise 43, problem 45
What is the next odd number below $4a+1$?
Printed answer:
- $4a-1$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 4*a - 1
Exercise 43, problem 5
$\displaystyle \frac{6 a^3 b^2 c - 9 a^2 b^2 + 3 c}{3 a^2 b}$
Printed answer:
- $\displaystyle 2abc-3b+\frac{c}{a^2b}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a*b*c - 3*b + c/(a**2*b)
Exercise 43, problem 6
$\displaystyle \frac{6 x^3 y^2 + 10 x^2 y^2 z - 2 m}{2 x y^2}$
Printed answer:
- $\displaystyle 3x^2+5xz-\frac{m}{xy^2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*x**2 + 5*x*z - m/(x*y**2)
Exercise 43, problem 7
$\displaystyle \frac{x^3 + 2 x^2 - 2 x + 1}{x^2 - x - 1}$
Printed answer:
- $\displaystyle x+3+\frac{2x+4}{x^2-x-1}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x + 3 + (2*x + 4)/(x**2 - x - 1)
Exercise 43, problem 8
$\displaystyle \frac{2 a^3 + a^2 - 2a + 1}{a^2 + a - 2}$
Printed answer:
- $\displaystyle 2a-1+\frac{3a-1}{a^2+a-2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a - 1 + (3*a - 1)/(a**2 + a - 2)
Exercise 43, problem 9
$\displaystyle \frac{x^3 + y^3}{x - y}$
Printed answer:
- $\displaystyle x^2+xy+y^2+\frac{2y^3}{x-y}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + x*y + y**2 + 2*y**3/(x - y)