smc-6213-10-Algebraic Fractions

Functions, 2ADV EQ-Bank 06

Simplify \(\dfrac{x^2}{x^2-2 x-15}-\dfrac{x}{x+3}\). (2 marks)

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\(\dfrac{5 x}{(x+3)(x-5)}\)

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\(\dfrac{x^2}{x^2-2 x-15}-\dfrac{x}{x+3}\)	\(=\dfrac{x^2}{(x+3)(x-5)}-\dfrac{x}{x+3}\)
	\(=\dfrac{x^2-x(x-5)}{(x+3)(x-5)}\)
	\(=\dfrac{x^2-x^2+5 x}{(x+3)(x-5)}\)
	\(=\dfrac{5 x}{(x+3)(x-5)}\)

Functions, 2ADV EQ-Bank 05

Simplify \(\dfrac{p+1}{q-q^3} \ ÷ \ \dfrac{p^3+p^2}{q^2-q}\). (2 marks)

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\(\dfrac{1}{p^2(1+q)}\)

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\(\dfrac{p+1}{q-q^3} \ ÷ \ \dfrac{p^3+p^2}{q^2-q}\)	\(=\dfrac{p+1}{q\left(1-q^2\right)} \times \dfrac{q\left(1-q\right)}{p^2(p+1)}\)
	\(=\dfrac{(1-q)}{(1+q)(1-q) p^2}\)
	\(=\dfrac{1}{p^2(1+q)}\)

Functions, 2ADV EQ-Bank 04

Simplify \(\dfrac{a^3 b-a b^3}{a^2+2 a b+b^2}\). (2 marks)

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\(\dfrac{a b(a-b)}{a+b}\)

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\(\dfrac{a^3 b-a b^3}{a^2+2 a b+b^2}\)	\(=\dfrac{a b\left(a^2-b^2\right)}{(a+b)^2}\)
	\(=\dfrac{a b(a+b)(a-b)}{(a+b)^2}\)
	\(=\dfrac{a b(a-b)}{a+b}\)

Functions, 2ADV F1 EQ-Bank 10

Fully simplify the expression \(\dfrac{4}{x^2-9}-\dfrac{2x+1}{x+3}\) (3 marks)

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\(\dfrac{-(2x-7)(x+1)}{(x^2-9)}\)

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\(\dfrac{4}{x^2-9}-\dfrac{2x+1}{x+3}\)	\(=\dfrac{4-(2x+1)(x-3)}{(x+3)(x-3)}\)
	\(=\dfrac{4-(2x^2-5x-3)}{(x+3)(x-3)}\)
	\(=\dfrac{-(2x^2-5x-7)}{(x+3)(x-3)}\)
	\(=\dfrac{-(2x-7)(x+1)}{(x^2-9)}\)

Functions, 2ADV F1 2021 HSC 11

Solve `x+(x-1)/2 = 9`. (2 marks)

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`19/3`

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`x+(x-1)/2`	`=9`
`2x + x-1`	`=18`
`3x`	`=19`
`x`	`=19/3`

Functions, 2ADV F1 SM-Bank 56

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

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`3x(x-3)`

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`(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)`

Functions, 2ADV F1 SM-Bank 55

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

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`-8p`

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

Functions, 2ADV F1 SM-Bank 54

Find the reciprocal of `1/a + 1/b -c/(ab)`. (2 marks)

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`(ab)/(a+b-c)`

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`1/a + 1/b -c/(ab)`	`=b/(ab)+a/(ab)-c/(ab)`
	`=(b+a-c)/(ab)`

`text(Reciprocal of)\ \ x = x^(-1)`

`:.\ text(Reciprocal of)\ \ (b+a-c)/(ab)=((b+a-c)/(ab))^(-1)=(ab)/(a+b-c)`

Functions, 2ADV F1 EQ-Bank 22

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)

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What does the reciprocal of this fraction represent? (1 mark)

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`(a + b)/(ab)`
`text(The reciprocal represents the number of hours it would)`
`text(take to fill one bucket, with A and B working together.)`

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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.)`

Functions, 2ADV F1 EQ-Bank 21

Simplify `(p/q)^3 ÷ (pq^(-2))`. (2 marks)

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`(p^2)/q`

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`(p/q)^3 ÷ (pq^(-2))`	`= (p^3)/(q^3) ÷ p/(q^2)`
	`= (p^3)/(q^3) xx (q^2)/p`
	`= (p^2)/q`

Functions, 2ADV F1 2005 HSC 1d

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

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`(8x-13)/10`

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

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

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

`= (8x-13)/10`

Functions, 2ADV F1 2004 HSC 1c

Solve `(x-5)/3-(x+1)/4 = 5`. (2 marks)

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`83`

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`(x-5)/3-(x+1)/4`	`= 5`
`12((x-5)/3)-12((x+1)/4)`	`= 12 xx 5`
`4x-20-3x-3`	`= 60`
`x-23`	`= 60`
`:. x`	`= 83`

Functions, 2ADV F1 2008 HSC 1c

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

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

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

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

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

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