Another way of writing \(5^2\) is
- \(5\times 2\)
- \(5\times 5\)
- \(5+5\)
- \(2\times 2\times 2\times 2\times 2\)
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Another way of writing \(5^2\) is
\(B\)
\(5^2=5\times 5\)
\(\Rightarrow B\)
Fully factorise \(24p^2qr^3+18p^3q^2+6p^2qr^2\). (2 marks)
\(6p^2q(4r^3+3pq+r^2)\)
\(24p^2qr^3+18p^3q^2+6p^2qr^2\) | \(=6p^2q\times 4r^3+6p^2q\times 3pq+6p^2q\times r^2\ \ \ \ (\text{HCF}=6p^2q)\) |
\(=6p^2q(4r^3+3pq+r^2)\) |
Fully factorise \(2mn^2+6m^2n-8mn\). (2 marks)
\(2mn(n+3m-4)\)
\(2mn^2+6m^2n-8mn\) | \(=2mn\times n+2mn\times 3m+2mn\times (-4)\ \ \ \ (\text{HCF}=2mn)\) |
\(=2mn(n+3m-4)\) |
A rectangle has an area of \(2pq^2+4p^2q\). By factorising the expression, find the length of the rectangle if the width is \(2pq\). (2 marks)
\(q+2p\)
\(2pq^2+4pq^2\) | \(=2pq\times q+2pq\times 2p\ \ \ \ (\text{HCF}=2pq)\) |
\(=2pq(q+2p)\) |
\(\therefore\ \text{Length of rectangle}=q+2p\)
A rectangle has an area of \(9a^2-6a\). By factorising the expression, find the width of the rectangle if the length is \(3a\). (2 marks)
\(3a-2\)
\(9a^2-6a\) | \(=3a\times 3a+3a\times (-2)\ \ \ \ (\text{HCF}=3a)\) |
\(=3a(3a-2)\) |
\(\therefore\ \text{Width of rectangle}=3a-2\)
A rectangle has an area of \(9a^2-6a\). By factorising the expression, find the width of the rectangle if the length is \(3a\). (2 marks)
\(3a-2\)
\(9a^2-6a\) | \(=3a\times 3a+3a\times (-2)\ \ \ \ (\text{HCF}=3a)\) |
\(=3a(3a-2)\) |
\(\therefore\ \text{Width of rectangle}=3a-2\)
Fully factorise the following:
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a. \(2x(2x-1)\)
b. \(-3a(2b-1)\)
c. \(-5q(1+2q)\)
a. | \(4x^2-2x\) | \(=2x\times 2x+2x\times (-1)\ \ \ \ (\text{HCF}=2x)\) |
\(=2x(2x-1)\) |
b. | \(-6ab+3a\) | \(=-3a\times 2b-3a\times(-1)\ \ \ \ (\text{HCF}=-3a)\) |
\(=-3a(2b-1)\) |
c. | \(-5q-10q^2\) | \(=-5q\times 1-5q\times 2q\ \ \ \ (\text{HCF}=-5q)\) |
\(=-5q(1+2q)\) |
Which of the following is the correct factorisation of \(7y+14x\).
\(A\)
\(7y+14x\) | \(=7\times y+7\times 2x\ \ \ \ (\text{HCF}=7)\) |
\(=7(y+2x)\) |
\(\Rightarrow A\)
Which of the following is the correct factorisation of \(15m-20\).
\(C\)
\(15m-20\) | \(=5\times 3m-5\times 4\ \ \ \ (\text{HCF}=5)\) |
\(=5(3m-4)\) |
\(\Rightarrow C\)
Which of the following is the correct factorisation of \(3x-12\).
\(D\)
\(3x-12\) | \(=3\times x-3\times 4\ \ \ \ (\text{HCF}=3)\) |
\(=3(x-4)\) |
\(\Rightarrow D\)
State the highest common factor of \(36mn^2\ \text{and}\ 24m^2n\). (1 mark)
\(12mn\)
\(\text{HCF of}\ 36\ \text{and}\ 24=12\)
\(\text{HCF of}\ mn^2\ \text{and}\ m^2n=mn\)
\(\therefore\ \text{HCF of}\ 36mn^2\ \text{and}\ 24m^2n=12mn\)
State the highest common factor of \(27ab^2c\ \text{and}\ 3a^2b^2c\). (1 mark)
\(3ab^2c\)
\(\text{HCF of}\ 27\ \text{and}\ 3=3\)
\(\text{HCF of}\ ab^2c\ \text{and}\ a^2b^2c=ab^2c\)
\(\therefore\ \text{HCF of}\ 27ab^2c\ \text{and}\ 3a^2b^2c=3ab^2c\)
State the highest common factor of \(15xy\ \text{and}\ 20y\). (1 mark)
\(5y\)
\(\text{HCF of}\ 15\ \text{and}\ 20=5\)
\(\text{HCF of}\ xy\ \text{and}\ y=y\)
\(\therefore\ \text{HCF of}\ 15xy\ \text{and}\ 20y=5y\)
List all the factors of \(-6z\). (2 marks)
\( 1\ ,\ 2\ ,\ 3\ ,\ 6\ ,-z\ ,-2z \ ,-3z\ ,-6z\)
\( -1\ , -2\ , -3\ , -6\ ,\ z\ ,\ 2z \ ,\ 3z\ ,\ 6z\)
\(\text{Listing factors in pairs}\)
\((1\ ,-6z) (2\ ,-3z) (3\ ,-2z) (6\ ,-z)\)
\((-1 ,\ 6z) (-2 , 3z) (-3 ,\ 2z) (-6 , z)\)
\(\therefore\ \text{Factors are:}\)
\( 1\ ,\ 2\ ,\ 3\ ,\ 6\ ,-z\ ,-2z \ ,-3z\ ,-6z\)
\( -1\ , -2\ , -3\ , -6\ ,\ z\ ,\ 2z \ ,\ 3z\ ,\ 6z\)
List all the factors of \(-7q\). (2 marks)
\( 1\ ,\ 7\ ,-q\ ,-7q \ , -1\ , -7\ ,\ q\ ,\ 7q\)
\(\text{Listing factors in pairs}\) | \(\longrightarrow (1\ ,-7q) (7\ ,-q) (-1 ,\ 7q) (-7 , q) \) |
\(\therefore\ \text{Factors are:}\)
\( 1\ ,\ 7\ ,-q\ ,-7q \ , -1\ , -7\ ,\ q\ ,\ 7q\)
List all the factors of \(-12y\). (2 marks)
\( 1\ ,\ 2\ ,\ 3\ ,\ 4\ ,\ 6\ ,\ 12\ ,-y\ ,-2y \ ,-3y\ ,-4y\ ,-6y\ ,-12y\)
\( -1\ , -2\ , -3\ , -4\ , -6\ , -12\ ,\ y\ ,\ 2y \ ,\ 3y\ ,\ 4y\ ,\ 6y\ ,\ 12y\)
\(\text{Listing factors in pairs}\)
\((1\ ,-12y) (2\ ,-6y) (3\ ,-4y) (4\ ,-3y) (6\ ,-2y) (12\ ,-y)\)
\((-1 ,\ 12y) (-2 , 6y) (-3 , 4y) (-4 , 3y)(-6 , 2y) (-12 , y)\)
\(\therefore\ \text{Factors are:}\)
\( 1\ ,\ 2\ ,\ 3\ ,\ 4\ ,\ 6\ ,\ 12\ ,-y\ ,-2y \ ,-3y\ ,-4y\ ,-6y\ ,-12y\)
\( -1\ , -2\ , -3\ , -4\ , -6\ , -12\ ,\ y\ ,\ 2y \ ,\ 3y\ ,\ 4y\ ,\ 6y\ ,\ 12y\)
List all the factors of \(10m\). (2 marks)
\( 1\ ,\ 2\ ,\ 5\ ,\ 10\ ,\ m\ ,\ 2m \ ,\ 5m\ ,\ 10m\)
\(-1 , -2 , -5 , -10 , -m , -2m , -5m , -10m\)
\(\text{Listing factors in pairs}\)
\((1\ ,\ 10m) (2\ ,\ 5m) (5\ ,\ 2m) (10\ ,\ m) (-1 ,-10m) (-2 , -5m) (-5 , -2m) (-10,-m)\)
\(\therefore\ \text{Factors are:}\)
\( 1\ ,\ 2\ ,\ 5\ ,\ 10\ ,\ m\ ,\ 2m \ ,\ 5m\ ,\ 10m\)
\(-1 , -2 , -5 , -10 , -m , -2m , -5m , -10m\)
List all the factors of \(2x\). (2 marks)
\(\ 1\ ,\ 2\ ,\ x\ ,\ 2x \ , -1\ , -2 \ , -x\ , -2x\)
\(\text{Listing factors in pairs}\) | \(\longrightarrow (1\ ,\ 2x)\ \ (2\ ,\ x)\ \ (-1\ ,-2x)\ \ (-2\ ,-x)\) |
\(\therefore \text{Factors are:}\ 1\ ,\ 2\ ,\ x\ ,\ 2x \ , -1\ , -2 \ , -x\ , -2x\)
List all the factors of \(4b\). (2 marks)
\(\ 1\ ,\ 2\ ,\ 4\ ,\ b\ ,\ 2b ,\ 4b\ , -1\ , -2\ , -4\ , -b\ , -2b\ , -4b\)
\(\text{Listing factors in pairs}\)
\((1\ ,\ 4b)\ \ (2\ ,\ 2b)\ \ (4\ ,\ b)\)
\((-1\ ,-4b)\ \ (-2\ ,-2b)\ \ (-4\ ,-b)\)
\(\therefore\ \text{Factors are:}\)
\(\ 1\ ,\ 2\ ,\ 4\ ,\ b\ ,\ 2b ,\ 4b ,\ -1\ , -2\ , -4\ , -b\ , -2b\ , -4b\)
Use the rectangle below to prove that \((a+b)^2=a^2+2ab+b^2\). (3 marks)
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\(\text{See worked solution}\)
The triangle below has a base of \((2+6\large m)\) and a perpendicular height of \(\large m\).
Write an algebraic expression and expand it to represent the area of the triangle. (2 marks)
\(x^2+xy\)
\(\text{Area of Triangle}\) | \(=\dfrac{1}{2}\times \text{base} \times \text{height}\) |
\(=\dfrac{1}{2}\times \Big(2+6m\Big)\times m\) | |
\(=\dfrac{m}{2}\Big(2+6m\Big)\) | |
\(=\dfrac{m}{2}\times 2 +\dfrac{m}{2}\times 6m\) | |
\(=m+3m^2\) |
The expansion of \(-2q(q-8)\) is:
\(B\)
\(-2q(q-8)\) | \(=-2q\times q-2q\times (-8)\) |
\(=-2q^2+16q\) |
\(\Rightarrow B\)
The expansion of \(4x(5-y)\) is:
\(D\)
\(4x(5-y)\) | \(=4x\times 5+4x\times (-y)\) |
\(=20x-4xy\) |
\(\Rightarrow D\)
Jimmy works \(\large x\) hours every week and his hourly rate of pay is \((40+\large y)\).
Write an algebraic expression and expand it to represent Jimmy's weekly wage. (2 marks)
\(40x+xy\)
\(x(40+y)\) | \(=x\times 40+x\times y\) |
\(=40x+xy\) |
A number \(\large y\) is halved and 3 is added. The result is then multiplied by 4.
Write an algebraic expression and expand it to represent this information. (2 marks)
\(2y+12\)
\(4\bigg(\dfrac{y}{2}+3\bigg)\) | \(=4\times \dfrac{y}{2}+4\times 3\) |
\(=\dfrac{4y}{2}+12\) | |
\(=2y+12\) |
A number \(\large x\) is doubled and 1 is added. The result is then multiplied by 5.
Write an algebraic expression and expand it to represent this information. (2 marks)
\(10x+5\)
\(5(2x+1)\) | \(=5\times 2x+5\times 1\) |
\(=10x+5\) |
A number \(b\) is tripled and \(5\) is subtracted. The result is then doubled.
Write an algebraic expression and expand it to represent this information. (2 marks)
\(6b-10\)
\(2(3b-5)\) | \(=2\times 3b+2\times (-5)\) |
\(=6b-10\) |
A number \(x\) is added to \(4\) and the result is doubled.
Write an algebraic expression and expand it to represent this information. (2 marks)
\(2x+8\)
\(2(x+4)\) | \(=2\times x+2\times 4\) |
\(=2x+8\) |
Expand and simplify the expression \(2(5-4x)+3(6x-1)\). (2 marks)
\(7+10x\)
\(2(5-4x)+3(6x-1)\) | \(=2\times 5+2\times (-4x)+3\times 6x+3\times (-1)\) |
\(=10-8x+18x-3\) | |
\(=7+10x\) |
Expand and simplify the expression \(3(x-1)+4(2x+3)\). (2 marks)
\(11x+9\)
\(3(x-1)+4(2x+3)\) | \(=3\times x-3\times 1+4\times 2x+4\times 3\) |
\(=3x-3+8x+12\) | |
\(=11x+9\) |
Expand and simplify the expression \(3xy-2x(y-5)\). (2 marks)
\(xy+10x\)
\(3xy-2x(y-5)\) | \(=3xy-2x\times y-2x\times (-5)\) |
\(=3xy-2xy+10x\) | |
\(=xy+10x\) |
Expand and simplify the expression \(2(m+2)+7m\). (2 marks)
\(9m+4\)
\(2(m+2)+7m\) | \(=2\times m+2\times 2+7m\) |
\(=2m+4+7m\) | |
\(=9m+4\) |
Use the distributive law to expand the expression \(9x(x-2y)\). (2 marks)
\(9x^2-18xy\)
\(9x(x-2y)\) | \(=9x\times x-9x\times 2y\) |
\(=9x^2-18xy\) |
Use the distributive law to expand the expression \(b(3-b)\). (2 marks)
\(3b-b^2\)
\(b(3-b)\) | \(=b\times 3-b\times b\) |
\(=3b-b^2\) |
Use the distributive law to expand the expression \(x(x+2)\). (2 marks)
\(x^2+2x\)
\(x(x+2)\) | \(=x\times x+x\times 2\) |
\(=x^2+2x\) |
Use the distributive law to expand the expression \(-(a-8)\). (2 marks)
\(-a+8\)
\(-(a-8)\) | \(=-1\times a-(-1)\times 8\) |
\(=-a+8\) |
Use the distributive law to expand the expression \(-10(y+4)\). (2 marks)
\(-10y-40\)
\(-10(y+4)\) | \(=-10\times y+(-10)\times 4\) |
\(=-10y-40\) |
Use the distributive law to expand the expression \(4(m-3)\). (2 marks)
\(4m-12\)
\(4(m-3)\) | \(=4\times m-4\times 3\) |
\(=4m-12\) |
Use the distributive law to expand the expression \(3(x+1)\). (2 marks)
\(3x+3\)
\(3(x+1)\) | \(=3\times x+3\times 1\) |
\(=3x+3\) |
Use the algebraic expression \(4-x\) to complete the table. (3 marks)
\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(4-x\) |
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\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(4-x\) | \(3\) | \(2\) | \(1\) | \(0\) | \(-1\) | \(-6\) |
\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(4-x\) | \(3\) | \(2\) | \(1\) | \(0\) | \(-1\) | \(-6\) |
Use the algebraic expression \(-x\) to complete the table. (3 marks)
\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(-x\) |
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\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(-x\) | \(-1\) | \(-2\) | \(-3\) | \(-4\) | \(-5\) | \(-10\) |
\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(-x\) | \(-1\) | \(-2\) | \(-3\) | \(-4\) | \(-5\) | \(-10\) |
Use the algebraic expression \(5x\) to complete the table. (3 marks)
\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(5x\) |
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\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(5x\) | \(5\) | \(10\) | \(15\) | \(20\) | \(25\) | \(50\) |
\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(5x\) | \(5\) | \(10\) | \(15\) | \(20\) | \(25\) | \(50\) |
Use the algebraic expression \(x^2-3x\) to complete the table. (3 marks)
\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(x^2-3x\) |
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\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(x^2-3x\) | \(-2\) | \(-2\) | \(0\) | \(4\) | \(10\) | \(70\) |
\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(x^2-3x\) | \(-2\) | \(-2\) | \(0\) | \(4\) | \(10\) | \(70\) |
Use the algebraic expression \(x^2+1\) to complete the table. (3 marks)
\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(x^2+1\) |
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\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(x^2+1\) | \(2\) | \(5\) | \(10\) | \(17\) | \(26\) | \(101\) |
\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(x^2+1\) | \(2\) | \(5\) | \(10\) | \(17\) | \(26\) | \(101\) |
Use the algebraic expression \(2x+1\) to complete the table. (3 marks)
\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(2x+1\) |
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\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(2x+1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(21\) |
\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(2x+1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(21\) |
Use the algebraic expression \(x-5\) to complete the table. (3 marks)
\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(x-5\) |
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\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(x-5\) | \(-4\) | \(-3\) | \(-2\) | \(-1\) | \(0\) | \(5\) |
\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(10\) |
\(x-5\) | \(-4\) | \(-3\) | \(-2\) | \(-1\) | \(0\) | \(5\) |
Use the algebraic expression \(x+3\) to complete the table. (3 marks)
\(x\) | 1 | 2 | 3 | 4 | 5 | 10 |
\(x+3\) |
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\(x\) | 1 | 2 | 3 | 4 | 5 | 10 |
\(x+3\) | 4 | 5 | 6 | 7 | 8 | 13 |
\(x\) | 1 | 2 | 3 | 4 | 5 | 10 |
\(x+3\) | 4 | 5 | 6 | 7 | 8 | 13 |
Simplify \(\dfrac{3x}{2}\times \dfrac{8}{5x}\ ÷\ \dfrac{4}{x}\) giving your answer as an algebraic fraction in simplest form. (2 marks)
\(\dfrac{3x}{5}\)
\(\dfrac{3x}{2}\times \dfrac{8}{5x}\ ÷\ \dfrac{4}{x}\) | \(=\dfrac{3x}{2}\times \dfrac{8}{5x}\times \dfrac{x}{4}\) |
\(=\dfrac{3x}{5}\) |
Simplify the following quotients, giving your answer as an algebraic fraction in simplest form.
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a. \(\dfrac{5x}{4y}\)
b. \(\dfrac{9ab}{7c}\)
c. \(\dfrac{4rs}{45}\)
a. | \(\dfrac{5}{y}\ ÷\ \dfrac{4}{x}\) | \(=\dfrac{5}{y}\times \dfrac{x}{4}\) |
\(=\dfrac{5x}{4y}\) |
b. | \(\dfrac{9a}{7}\ ÷\ \dfrac{c}{b}\) | \(=\dfrac{9a}{7}\times\dfrac{b}{c}\) |
\(=\dfrac{9ab}{7c}\) |
c. | \(\dfrac{r}{3}\ ÷\ \dfrac{5}{4s}\) | \(=\dfrac{r}{3}\times \dfrac{4s}{5}\) |
\(=\dfrac{4rs}{15}\) |
Simplify the following quotients, giving your answer as an algebraic fraction in simplest form.
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a. \(\dfrac{5}{2}\)
b. \(\dfrac{8x}{y}\)
a. | \(\dfrac{5m}{8}\ ÷\ \dfrac{m}{4}\) | \(=\dfrac{5m}{8}\times \dfrac{4}{m}\) |
\(=\dfrac{20m}{8m}\) | ||
\(=\dfrac{5}{2}\) |
b. | \(\dfrac{14x}{3}\ ÷\ \dfrac{7y}{6}\) | \(=\dfrac{14x}{3}\times \dfrac{6}{7y}\) |
\(=\dfrac{7x\times 2\times 3\times 2}{7y\times 3}\) | ||
\(=\dfrac{4x}{y}\) |
Simplify the following products, giving your answer as an algebraic fraction in simplest form.
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a. \(\dfrac{6bc}{5}\)
b. \(\dfrac{10xy}{3}\)
a. | \(\dfrac{2b}{3}\times \dfrac{9c}{5}\) | \(=\dfrac{2b\times 9c}{3\times 5}\) |
\(=\dfrac{18bc\ ÷\ 3}{15\ ÷\ 3}\) | ||
\(=\dfrac{6bc}{5}\) |
b. | \(\dfrac{8x}{5}\times \dfrac{25y}{12}\) | \(=\dfrac{8x\times 25y}{5\times 12}\) |
\(=\dfrac{200xy\ ÷\ 20}{60\ ÷\ 20}\) | ||
\(=\dfrac{10xy}{3}\) |
Simplify the following products, giving your answer as an algebraic fraction in simplest form.
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a. \(\dfrac{a}{10}\)
b. \(\dfrac{6xy}{35}\)
c. \(\dfrac{4rs}{9}\)
a. | \(\dfrac{a}{2}\times \dfrac{1}{5}\) | \(=\dfrac{a\times 1}{2\times 5}=\dfrac{a}{10}\) |
b. | \(\dfrac{3x}{5}\times \dfrac{2y}{7}\) | \(=\dfrac{3x\times 2y}{5\times 7}=\dfrac{6xy}{35}\) |
c. | \(\dfrac{5r}{9}\times \dfrac{4s}{5}\) | \(=\dfrac{5r\times 4s}{9\times 5}=\dfrac{4rs}{9}\) |
For the expression \(\dfrac{a}{2}+\dfrac{2a}{3}-\dfrac{a}{4}\), simplify and write an equivalent algebraic fraction. (3 marks)
\(\dfrac{11a}{12}\)
\(\dfrac{a}{2}+\dfrac{2a}{3}-\dfrac{a}{4}\) | \(=\dfrac{a}{2}\times \dfrac{6}{6}+\dfrac{2a}{3}\times \dfrac{4}{4}-\dfrac{a}{4}\times \dfrac{3}{3}\) |
\(=\dfrac{6a}{12}+\dfrac{8a}{12}-\dfrac{3a}{12}\) | |
\(=\dfrac{11a}{12}\) |
Ben recieved \($x\) for his birthday.
He spent \(\dfrac{1}{2}\) of his money on shoes and \(\dfrac{1}{3}\) on Gold Class movie tickets.
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a. \(\dfrac{$x}{2}\)
b. \(\dfrac{$x}{3}\)
c. \(\dfrac{$5x}{6}\)
a. | \(\text{Shoes}\) | \(=\dfrac{1}{2}\times x\) |
\(=\dfrac{$x}{2}\) |
b. | \(\text{Movie tickets}\) | \(=\dfrac{1}{3}\times x\) |
\(=\dfrac{$x}{3}\) |
c. | \(\text{Total Spent}\) | \(=\dfrac{x}{2}+\dfrac{x}{3}\) |
\(=\dfrac{x}{2}\times \dfrac{3}{3}+\dfrac{x}{3}\times \dfrac{2}{2}\) | ||
\(=\dfrac{3x}{6}+\dfrac{2x}{6}\) | ||
\(=\dfrac{$5x}{6}\) |
Yesterday Gareth walked \(x\) kilometres in Kosciuszko National park.
He started at Thredbo village and walked up Merritts Nature Track to the Kosciuszko express chairlift. When he arrived at the chairlift he had complete \(\dfrac{1}{5}\) of the total distance.
Gareth then joined a walking group and walked a further \(\dfrac{1}{3}\) of the total distance before beginning his descent back to the village.
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a. \(\dfrac{x}{5}\ \text{kilometres}\)
b. \(\dfrac{x}{3}\ \text{kilometres}\)
c. \(\dfrac{8x}{15}\ \text{kilometres}\)
a. | \(\text{First leg}\) | \(=\dfrac{1}{5}\times x\) |
\(=\dfrac{x}{5}\ \text{kilometres}\) |
b. | \(\text{Second leg}\) | \(=\dfrac{1}{3}\times x\) |
\(=\dfrac{x}{3}\ \text{kilometres}\) |
c. | \(\text{Distance travelled}\) | \(=\dfrac{x}{5}+\dfrac{x}{3}\) |
\(=\dfrac{x}{5}\times \dfrac{3}{3}+\dfrac{x}{3}\times \dfrac{5}{5}\) | ||
\(=\dfrac{3x}{15}+\dfrac{5x}{15}\) | ||
\(=\dfrac{8x}{15}\ \text{kilometres}\) |
For the sum \(a+\dfrac{a}{4}\), simplify and write an equivalent algebraic fraction. (2 marks)
\(\dfrac{5a}{4}\)
\(a+\dfrac{a}{4}\) | \(=\dfrac{a}{1}\times \dfrac{4}{4}+\dfrac{a}{4}\) |
\(=\dfrac{4a}{4}+\dfrac{a}{4}\) | |
\(=\dfrac{5a}{4}\) |
For the difference \(\dfrac{a}{3}-\dfrac{b}{4}\), simplify and write an equivalent algebraic fraction. (3 marks)
\(\dfrac{4a-3b}{12}\)
\(\dfrac{a}{3}-\dfrac{b}{4}\) | \(=\dfrac{a}{3}\times \dfrac{4}{4}-\dfrac{b}{4}\times \dfrac{3}{3}\) |
\(=\dfrac{4a}{12}-\dfrac{3b}{12}\) | |
\(=\dfrac{4a-3b}{12}\) |
For the difference \(\dfrac{5r}{8}-\dfrac{3r}{8}\), simplify and write an equivalent algebraic fraction. (2 marks)
\(\dfrac{r}{4}\)
\(\dfrac{5r}{8}-\dfrac{3r}{8}\) | \(=\dfrac{2r}{8}\) |
\(=\dfrac{r}{4}\) |