smc-983-40-Algebraic Fractions

JACK v1 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))`

v2 Functions, 2ADV F1 SM-Bank 54

Find the reciprocal of `1/(xy) + 1/y-z/x`. (2 marks)

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`(xy)/(x+y-z)`

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`1/(xy) + 1/y-z/x`	`=1/(xy)+x/(xy)-(zy)/(xy)`
	`=(1+x-zy)/(xy)`

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

`:.\ text(Reciprocal of)\ \ (1+x-zy)/(xy)=((1+x-zy)/(xy))^(-1)=(xy)/(1+x-zy)`

v1 Functions, 2ADV F1 SM-Bank 54

Find the reciprocal of `r/p-1/(pq) + 1/q`. (2 marks)

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`(pq)/(p+q-r)`

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`r/p-1/(pq) + 1/q`	`=(qr)/(pq)-1/(pq)+p/(pq)`
	`=(qr-1+p)/(pq)`

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

`:.\ text(Reciprocal of)\ \ (qr-1+p)/(pq)=((qr-1+p)/(pq))^(-1)=(pq)/(qr-1+p)`

v1 Functions, 2ADV F1 2004 HSC 1c

Solve `(x+4)/5-(x-2)/6 = 4`. (2 marks)

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`38`

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`(x+4)/5-(x-2)/6`	`= 4`
`30((x+4)/5)-30((x-2)/6)`	`= 30 xx 4`
`6x+24-5x+10`	`= 120`
`x+34`	`= 120`
`:. x`	`= 38`

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))`

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))`