Algebraic Fractions

Algebraic Techniques, SMB-074

Worker A picks a bucket of blueberries in `a` hours. Worker B picks a bucket of blueberries in `b` hours.

Write an algebraic expression for the fraction of a bucket of blueberries that could be picked in one hour if A and B worked together. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---
What does the reciprocal of this fraction represent? (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

i. `(a + b)/(ab)`

ii. `text(The reciprocal represents the number of hours it would)`

`text(take to fill one bucket, with A and B working together.)`

Show Worked Solution

i. `text(In one hour:)`

COMMENT: Note that the question asks for “a fraction”.

`text(Worker A picks)\ 1/a\ text(bucket.)`

`text(Worker B picks)\ 1/b\ text(bucket.)`

`:.\ text(Fraction picked in 1 hour working together)`

`= 1/a + 1/b`

`= (a + b)/(ab)`

ii. `text(The reciprocal represents the number of hours it would)`

`text(take to fill one bucket, with A and B working together.)`

Algebraic Techniques, SMB-073

Simplify `(4p-12p^2)/3 xx (6p)/(3p^2-p)`. (3 marks)

Show Answers Only

`-8p`

Show Worked Solution

`(4p-12p^2)/3 xx (6p)/(3p^2-p)`	`= (4p(1-3p))/3 xx (6p)/(p(3p-1))`
	`= (8p(1-3p))/(3p-1)`
	`= (-8p(3p-1))/(3p-1)`
	`=-8p`

Algebraic Techniques, SMB-072

Simplify `(9x^2)/(x+3) -: (3x)/(x^2-9)`. (3 marks)

Show Answers Only

`3x(x-3)`

Show Worked Solution

`(9x^2)/(x+3) -: (3x)/(x^2-9)`	`=(9x^2)/(x+3) xx (x^2-9)/(3x)`
	`=(9x^2)/(x+3) xx ((x-3)(x+3))/(3x)`
	`=3x(x-3)`

Algebraic Techniques, SMB-071

Simplify the expression `4/(y^2-16)+1/(3y-12)` (3 marks)

Show Answers Only

`(y+16)/(3(y^2-16))`

Show Worked Solution

`4/(y^2-16)+1/(3y-12)`	`=4/((y-4)(y+4))+1/(3(y-4))`
	`=12/(3(y-4)(y+4))+(y+4)/(3(y-4)(y+4))`
	`=(y+16)/(3(y-4)(y+4))`

Algebraic Techniques, SMB-070

Simplify the expression `(5a-15)/3xx(6a)/(a^2-9)` (3 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`(10a)/(a+3)`

Show Worked Solution

`(5a-15)/3xx(6a)/(a^2-9)`	`=(5(a-3))/3xx(2xx3xxa)/((a-3)(a+3))`
	`=(10a)/(a+3)`

Algebraic Techniques, SMB-069

Factorise the expression `5/(x^2-1)-3/(x-1)` (3 marks)

Show Answers Only

`(2-3x)/((x-1)(x+1))`

Show Worked Solution

`5/(x^2-1)-3/(x-1)`	`=5/((x+1)(x-1))-3/(x-1)`
	`=5/((x+1)(x-1))-(3(x+1))/((x+1)(x-1))`
	`=(5-3(x+1))/((x-1)(x+1))`
	`=(2-3x)/((x-1)(x+1))`

Algebraic Techniques, SMB-068

Factorise the expression `5/(b^2-b)-2/(b^2-1)` (3 marks)

Show Answers Only

`(3b+5)/(b(b-1)(b+1))`

Show Worked Solution

`5/(b^2-b)-2/(b^2-1)`	`=5/(b(b-1))-2/((b+1)(b-1))`
	`=(5(b+1))/(b(b-1)(b+1))-(2b)/(b(b-1)(b+1))`
	`=(5b+5-2b)/(b(b-1)(b+1))`
	`=(3b+5)/(b(b-1)(b+1))`

Algebraic Techniques, SMB-067

Factorise the expression `(x^2-7x+12)/(x-3)` (2 marks)

Show Answers Only

`x-4`

Show Worked Solution

`(x^2-7x+12)/(x-3)`	`=((x-3)(x-4))/(x-3)`
	`=x-4`

Algebraic Techniques, SMB-053

Simplify the expression `(4y+1)/8-(6-2y)/3` (2 marks)

Show Answers Only

`(28y-45)/24`

Show Worked Solution

`(4y+1)/8-(6-2y)/3`	`=(3(4y+1))/24-(8(6-2y))/24`
	`=(12y+3-48+16y)/24`
	`=(28y-45)/24`

Algebraic Techniques, SMB-052

Simplify the expression `(3a+2)/3-(2a-1)/5` (2 marks)

Show Answers Only

`(9a+13)/15`

Show Worked Solution

`(3a+2)/3-(2a-1)/5`	`=(5(3a+2))/15-(3(2a-1))/15`
	`=(15a+10-6a+3)/15`
	`=(9a+13)/15`

Algebraic Techniques, SMB-051

Simplify the expression `x/4-(x+2)/5` (2 marks)

Show Answers Only

`(x-8)/20`

Show Worked Solution

`x/4-(x+2)/5`	`=(5x)/20-(4(x+2))/20`
	`=(5x-4x-8)/20`
	`=(x-8)/20`

Algebraic Techniques, SMB-050

Simplify the expression `(2p-1)/2+(p+1)/5` (2 marks)

Show Answers Only

`(12p-3)/10`

Show Worked Solution

`(2p-1)/2+(p+1)/5`	`=(5(2p-1))/10+(2(p+1))/10`
	`=(10p-5+2p+2)/10`
	`=(12p-3)/10`

Algebraic Techniques, SMB-049

Simplify the expression `(x-4)/3+(2x+1)/6` (2 marks)

Show Answers Only

`(4x-7)/6`

Show Worked Solution

`(x-4)/3+(2x+1)/6`	`=(2(x-4))/6+(2x+1)/6`
	`=(2x-8+2x+1)/6`
	`=(4x-7)/6`

Algebraic Techniques, SMB-036

Simplify the expression `3/(4x)+2/(5x)` (2 marks)

Show Answers Only

`23/(20x)`

Show Worked Solution

`3/(4x)+2/(5x)`	`=(3xx5)/(20x)+(2xx4)/(20x)`
	`=(15+8)/(20x)`
	`=23/(20x)`

Algebraic Techniques, SMB-035

Simplify the expression `5/(4a)-1/(3a)` (2 marks)

Show Answers Only

`11/(12a)`

Show Worked Solution

`5/(4a)-1/(3a)`	`=(5xx3)/(12a)-(1xx4)/(12a)`
	`=(15-4)/(12a)`
	`=11/(12a)`

Algebraic Techniques, SMB-034

Simplify the expression `3/x-1/(2x)` (2 marks)

Show Answers Only

`5/(2x)`

Show Worked Solution

`3/x-1/(2x)`	`=6/(2x)-1/(2x)`
	`=(6-1)/(2x)`
	`=5/(2x)`

Algebraic Techniques, SMB-033

Simplify the expression `(36p^2q^4)/7 xx 14/(4p^3q)` (2 marks)

Show Answers Only

`(18q^3)/p`

Show Worked Solution

`(36p^2q^4)/7 xx 14/(4p^3q)`	`=(4 xx 9 xxp^2q^4 xx 2xx7)/(7 xx 4 xxp^3q)`
	`=(18q^3)/p`

Algebraic Techniques, SMB-032

Simplify the expression `(x^2y^5)/6 xx 18/(x^2y^2)` (2 marks)

Show Answers Only

`3y^3`

Show Worked Solution

`(x^2y^5)/6 xx 18/(x^2y^2)`	`=(6xx3 xx x^2y^5)/(6 xx x^2y^2)`
	`=3y^3`

Algebraic Techniques, SMB-031

Simplify the expression `(3a^2)/(12b^4) -: (9a^3)/(4b)` (2 marks)

Show Answers Only

`1/(9ab^3)`

Show Worked Solution

`(3a^2)/(12b^4) -: (9a^3)/(4b)`	`=(3a^2)/(12b^4) xx (4b)/(9a^3)`
	`=(3xxa^2 xx 4xxb)/(3 xx 4xxb^4xx3xx3xxa^3`
	`=1/(9ab^3)`

Algebraic Techniques, SMB-030

Simplify the expression `(3xy)/4 xx 12/(4x)` (2 marks)

Show Answers Only

`(9y)/4`

Show Worked Solution

`(3xy)/4 xx 12/(4x)`	`=(3xy xx 4 xx 3)/(4 xx 4x)`
	`=(9y)/4`

Algebraic Techniques, SMB-029

Simplify the expression `(24p^3q^2)/(8p^2q)` (2 marks)

Show Answers Only

`3pq`

Show Worked Solution

`(24p^3q^2)/(8p^2q)`	`=(8xx3xxp^3xxq^2)/(8xxp^2xxq)`
	`=3pq`

Algebraic Techniques, SMB-028

Simplify the expression `(18a^2b)/(3ab^3)` (2 marks)

Show Answers Only

`(6a)/b^2`

Show Worked Solution

`(18a^2b)/(3ab^3)`	`=(3xx6xxa^2xxb)/(3xxaxxb^3)`
	`=(6a)/b^2`

Algebraic Techniques, SMB-027

Simplify the expression `(12y^3)/(4y^2)` (2 marks)

Show Answers Only

`3y`

Show Worked Solution

`(12y^3)/(4y^2)`	`=(4xx3xxy^3)/(4xxy^2)`
	`=3y`

Algebraic Techniques, SMB-012

Simplify the expression `(9h)/2 -: (h)/3` (2 marks)

Show Answers Only

`(27)/2`

Show Worked Solution

`(9h)/2 -: (h)/3`	`=(9h)/2 xx 3/(h)`
	`=(9h xx 3)/(2 xx h)`
	`=(27)/2`

Algebraic Techniques, SMB-011

Simplify the expression `(13x)/15 -: (2x)/5` (2 marks)

Show Answers Only

`(13)/6`

Show Worked Solution

`(13x)/15 -: (2x)/5`	`=(13x)/15 xx 5/(2x)`
	`=(13x xx 5)/(15 xx 2x)`
	`=(13)/6`

Algebraic Techniques, SMB-010

Simplify the expression `(3a)/4 -: (7a)/2` (2 marks)

Show Answers Only

`(3)/14`

Show Worked Solution

`(3a)/4 -: (7a)/2`	`=(3a)/4 xx 2/(7a)`
	`=(3a xx 2)/(4 xx 7a)`
	`=(3)/14`

Algebraic Techniques, SMB-009

Simplify the expression `(3p)/4 xx (8p)/9` (2 marks)

Show Answers Only

`(2p^2)/3`

Show Worked Solution

`(3p)/4 xx (8p)/9`	`=(3p xx 8p)/(4 xx 9)`
	`=(p xx 2p)/3`
	`=(2p^2)/3`

Algebraic Techniques, SMB-008

Simplify the expression `(2a)/7 xx (14a)/3` (2 marks)

Show Answers Only

`(4a^2)/3`

Show Worked Solution

`(2a)/7 xx (14a)/3`	`=(2a xx 14a)/(7 xx 3)`
	`=(2a xx 2a)/3`
	`=(4a^2)/3`

Algebraic Techniques, SMB-007

Simplify the expression `(5)/6 xx c/4` (2 marks)

Show Answers Only

`(5c)/24`

Show Worked Solution

`(5)/6 xx c/4`	`=(5 xx c)/(6 xx 4)`
	`=(5c)/24`

Algebraic Techniques, SMB-006

Simplify the expression `(5x)/9-x/6` (2 marks)

Show Answers Only

`(7x)/18`

Show Worked Solution

`(5x)/9-x/6`	`=(10x)/18-(3x)/18`
	`=(7x)/18`

Algebraic Techniques, SMB-505

Simplify the expression `b/2-b/3` (2 marks)

Show Answers Only

`b/6`

Show Worked Solution

`b/2-b/3`	`=(3b)/6-(2b)/6`
	`=b/6`

Algebraic Techniques, SMB-005

Simplify the expression `(4x)/5-(x)/3` (2 marks)

Show Answers Only

`(7x)/15`

Show Worked Solution

`(4x)/5-(x)/3`	`=(12x)/15-(5x)/15`
	`=(7x)/15`

Algebraic Techniques, SMB-004

Simplify the expression `(3t)/5-(t)/2` (2 marks)

Show Answers Only

`(t)/10`

Show Worked Solution

`(3t)/5-(t)/2`	`=(6t)/10-(5t)/10`
	`=(t)/10`

Algebraic Techniques, SMB-003

Simplify the expression `(3a)/5+(a)/4` (2 marks)

Show Answers Only

`(17a)/20`

Show Worked Solution

`(3a)/5+(a)/4`	`=(12a)/20+(5a)/20`
	`=(17a)/20`

Algebraic Techniques, SMB-002

Simplify the expression `x/4+(2x)/3` (2 marks)

Show Answers Only

`(11x)/12`

Show Worked Solution

`x/4+(2x)/3`	`=(3x)/12+(8x)/12`
	`=(11x)/12`

Algebraic Techniques, SMB-001

Simplify the expression `p/2+p/3` (2 marks)

Show Answers Only

`(5p)/6`

Show Worked Solution

`p/2+p/3`	`=(3p)/6+(2p)/6`
	`=(5p)/6`

Functions, 2ADV F1 2005 HSC 1d

Express `((2x-3))/2-((x-1))/5` as a single fraction in its simplest form. (2 marks)

--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

`(8x-13)/10`

Show Worked Solution

`((2x-3))/2-((x-1))/5`

`= (5(2x-3)-2(x-1))/10`

`= (10x-15-2x + 2)/10`

`= (8x-13)/10`

Functions, 2ADV F1 2008 HSC 1c

Simplify `2/n-1/(n+1)`. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`(n + 2)/(n(n+1))`

Show Worked Solution

`2/n-1/(n+1)`

`= (2(n+1)-1(n))/(n(n+1))`

`= (2n + 2-n)/(n(n+1))`

`= (n+2)/(n(n+1))`