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Equations, SM-Bank 035

Two consecutive integers have a sum of 147.

  1. Given the first number is \(x\), write an algebraic equation to represent this information.  (2 marks)

  2. Solve your equation algebraically to find the two numbers.  (2 marks)

Show Answers Only

a.    \(2x+1=147\)

b.    \(73 , 74\)

Show Worked Solution

a.    \(\text{Given the 1st number is}\ x\text{, the 2nd number is }x+1.\)

\(\therefore\ \text{Equation is:}\)

\(x+(x+1)\) \(=147\)
\(2x+1\) \(=147\)

 

b.    \(2x+1\) \(=147\)
  \(2x\) \(=146\)
  \(x\) \(=73\)

 
\(\therefore\ \text{Numbers are }73\ \text{and }74.\)

Equations, SM-Bank 034

Two consecutive integers have a sum of 25.

  1. Given the first number is \(x\), write an algebraic equation to represent this information.  (2 marks)

  2. Solve your equation algebraically to find the two numbers.  (2 marks)

Show Answers Only

a.    \(2x+1=25\)

a.    \(12 , 13\)

Show Worked Solution

a.    \(\text{Given the 1st number is}\ x\text{, the 2nd number is }x+1.\)

\(\therefore\ \text{Equation is:}\)

\(x+(x+1)\) \(=25\)
\(2x+1\) \(=25\)
b.    \(2x+1\) \(=25\)
  \(2x\) \(=24\)
  \(x\) \(=12\)

 
\(\therefore\ \text{Numbers are }12\ \text{and }13.\)

Special Properties, SMB-010 MC

Which of these are always equal in length?

  1. the diagonals of a rhombus
  2. the diagonals of a parallelogram
  3. the opposite sides of a parallelogram
  4. the opposite sides of a trapezium
Show Answers Only

`C`

Show Worked Solution

`text{Consider each option:}`

`A:\ \text{rhombus diagonals are perpendicular but not always equal}`

`B:\ \text{parallelogram diagonals not always equal (see below)}`

`C:\ \text{always true (see above)}`

`D:\ \text{at least 1 pair of opposite sides of a trapezium are not equal}`
\(\Rightarrow C\)

Special Properties, SMB-009 MC

`PQRS` is a parallelogram.

Which of these must be a property of `PQRS`?

  1. Line `PS` is perpendicular to line `PQ`.
  2. Line `PQ` is parallel to line `PS`.
  3. Diagonals `PR` and `SQ` are perpendicular.
  4. Line `PS` is parallel to line `QR`.
Show Answers Only

`D`

Show Worked Solution

`text{By elimination:}`

`A\ \text{and}\ B\ \text{clearly incorrect.}`

`C\ \text{true if all sides are equal (rhombus) but not true for all parallelograms.}`

`text(Line)\ PS\ text(must be parallel to line)\ QR.`

`=>D`

Special Properties, SMB-004 MC

Which one of the following triangles is impossible to draw?

  1. a right angled triangle with two acute angles
  2. an isosceles triangle with one right angle
  3. a scalene triangle with three acute angles
  4. a right angled triangle with one obtuse angle
Show Answers Only

`D`

Show Worked Solution

`text(A right angle = 90°.)`

`text{Since an obtuse angle is greater than 90°, it is impossible for}`

`text(a triangle, with an angle sum less than 180°, to have both.)`

`=>D`

Special Properties, SMB-002 MC

A triangle has two acute angles.

What type of angle couldn't the third angle be?

  1. an acute angle
  2. an obtuse angle
  3. a right-angle
  4. a reflex angle
Show Answers Only

`D`

Show Worked Solution

`text(A triangle’s angles add up to 180°, and a reflex angle is)`

`text(greater than 180°.)`

`:.\ text(The third angle cannot be reflex.)`

`=>D`

Equations, SM-Bank 032

Verify that \(x=0.3\) is a solution of the equation \(8x+0.5=2.9\).  (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution

\(\text{When}\ x=7\)

\(LHS\) \(=8x+0.5\)
  \(=8\times 0.3+0.5\)
  \(=2.4+0.5=2.9\)
  \(=RHS\)

  
\(\therefore\ x=0.3\ \text{ is a solution}\)

Equations, SM-Bank 031

Verify that \(x=7\) is a solution of the equation \(\dfrac{x-3}{2}=2\).  (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution

\(\text{When}\ x=7\)

\(LHS\) \(=\dfrac{x-3}{2}\)
  \(=\dfrac{7-3}{2}\)
  \(=\dfrac{4}{2}=2\)
  \(=RHS\)

  
\(\therefore\ x=7\ \text{ is a solution}\)

Equations, SM-Bank 030

Verify that \(x=\dfrac{3}{4}\) is a solution of the equation \(4x-5=-2\).  (2 marks)

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\(\text{See worked solution}\)

Show Worked Solution

\(\text{When}\ x=\dfrac{3}{4}\)

\(LHS\) \(=4x-5\)
  \(=4\times \dfrac{3}{4}-5\)
  \(=3-5=-2\)
  \(=RHS\)

  
\(\therefore\ x=\dfrac{3}{4}\ \text{ is a solution}\)

Equations, SM-Bank 029

Verify that \(x=1.5\) is a solution of the equation \(\dfrac{2x}{3}+3=4\).  (2 marks)

Show Answers Only

\(\text{See worked solution}\)

Show Worked Solution

\(\text{When}\ x=1.5\)

\(LHS\) \(=\dfrac{2x}{3}+3\)
  \(=\dfrac{2\times 1.5}{3}+3\)
  \(=1+3=4\)
  \(=RHS\)

  
\(\therefore\ x=1.5\ \text{ is a solution}\)

Equations, SM-Bank 028

Verify that \(x=2\) is a solution of the equation \(2x+3=7\).  (2 marks)

Show Answers Only

\(\text{See worked solution}\)

Show Worked Solution

\(\text{When}\ x=2\)

\(LHS\) \(=2x+3\)
  \(=2\times 2 +3\)
  \(=7\)
  \(=RHS\)

  
\(\therefore\ x=2\ \text{ is a solution}\)

Equations, SM-Bank 025

\(3\) is subtracted from a quarter of \(x\) and the result is \(-\dfrac{1}{2}\).

Write an equation and solve it algebraically to find the value of \(x\).  (3 marks)

Show Answers Only

\(x=10\)

Show Worked Solution
\(\dfrac{x}{4}-3\) \(=-\dfrac{1}{2}\)
\(\dfrac{x}{4}\) \(=-\dfrac{1}{2}+3\)
\(\dfrac{x}{4}\) \(=2\dfrac{1}{2}\)
\(x\) \(=4\times 2\dfrac{1}{2}\)
\(x\) \(=10\)

 

Equations, SM-Bank 020

Solve  \(8-\dfrac{x}{9}=-1\)  (2 marks)

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\(x=81\)

Show Worked Solution
\(8-\dfrac{x}{9}\) \(=-1\)
\(\dfrac{-x}{9}\) \(=-1-8\)
\(\dfrac{-x}{9}\) \(=-9\)
\(-x\) \(=-9\times 9\)
\(-x\) \(=-81\)
\(x\) \(=81\)

 

Equations, SM-Bank 015

A square with side length  \(\large x\)  has an area of \(81\ \text{cm}^2\).

Write an equation and solve it to find the side length.  (2 marks)

Show Answers Only

\(x^2=81\  \ ,\  \  x=9\ \text{cm}\)

Show Worked Solution
\(x^2\) \(=81\)
\(x\) \(=\sqrt{81}\)
\(x\) \(=9\)

 
\(\therefore\ \text{The side length is }9\ \text{cm}\)

Equations, SM-Bank 014

A number  \(\large p\)  is halved and the result is 125.6.

Write an equation and solve it to find the number.  (2 marks)

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\(\dfrac{p}{2}=125.6\  \ ,\  \  p=251.2\)

Show Worked Solution
\(\dfrac{p}{2}\) \(=125.6\)
\(p\) \(=2\times 125.6\)
\(p\) \(=251.2\)

 
\(\therefore\ \text{The number is }251.2\)

Equations, SM-Bank 013

A number  \(\large w\)  is tripled and the result is 141.

Write an equation and solve it to find the number.  (2 marks)

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\(3w=141\  \ ,\  \  w=47\)

Show Worked Solution
\(3w\) \(=141\)
\(w\) \(=\dfrac{141}{3}\)
\(w\) \(=47\)

 
\(\therefore\ \text{The number is }47\)

Equations, SM-Bank 012

Verity and three of her friends won  \(x\) dollars in a lottery. When the money was divided evenly between the four friends they each received \($124\ 500\).

Write an equation and solve it to find out the total amount of their lottery winnings.  (2 marks)

Show Answers Only

\(\dfrac{x}{4}=124\ 500\  \ ,\  \  x=$498\ 000\)

Show Worked Solution
\(\dfrac{x}{4}\) \(=124\ 500\)
\(x\) \(=4\times 124\ 500\)
\(x\) \(=498\ 000\)

 
\(\therefore\ \text{The amount of their total winnings was}\ $498\ 000\)

Equations, SM-Bank 011

Josh is currently \(x\) years old. In \(15\) years he will be 42.

Write an equation and solve it to find Josh's current age.  (2 marks)

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\(x+15=42\  \ ,\  \  x=27\ \text{years old}\)

Show Worked Solution
\(x+15\) \(=42\)
\(x\) \(=42-15\)
\(x\) \(=27\)

 
\(\therefore\ \text{Josh’s current age is}\ 27\ \text{years}\)

Equations, SM-Bank 010

Preston is paid \(x\) dollars per basket of grapes he picks. Yesterday he picked \(8\) baskets and he earned $306.

Write an equation and solve it to find Preston's rate of pay per basket.  (2 marks)

Show Answers Only

\(8x=306\  \ ,\  \  x=$38.25\)

Show Worked Solution
\(8x\) \(=306\)
\(x\) \(=\dfrac{306}{8}\)
\(x\) \(=38.25\)

 
\(\therefore\ \text{Preston’s rate of pay per basket is}\ $38.25\)

Equations, SM-Bank 009

Missy is paid \(x\) dollars per hour. Last week she worked \(20\) hours and she earned $350.

Write an equation and solve it to find Missy's hourly rate of pay.  (2 marks)

Show Answers Only

\(20x=350\  \ ,\  \  x=$17.50\)

Show Worked Solution
\(20x\) \(=350\)
\(x\) \(=\dfrac{350}{20}\)
\(x\) \(=17.50\)

 
\(\therefore\ \text{Missy’s hourly rate of pay is}\ $17.50\)

Equations, SM-Bank 008

Solve the following one-step equations.

  1. \(s-3=-4\)  (1 mark)

  2. \(c+4=-2\)  (1 mark)

  3. \(-3x=36\)  (1 mark)

  4. \(\dfrac{b}{6}=-4\)  (1 mark)

Show Answers Only

a.    \(s=-1\)

b.    \(c=-6\)

c.    \(x=-12\)

d.    \(b=-24\)

Show Worked Solution
a.    \(s-3\) \(=-4\)
  \(s\) \(=-4+3\)
  \(s\) \(=-1\)

 

b.    \(c+4\) \(=-2\)
  \(c\) \(=-2-4\)
  \(c\) \(=-6\)

 

c.    \(-3x\) \(=36\)
  \(x\) \(=\dfrac{36}{-3}\)
  \(x\) \(=-12\)

 

d.    \(\dfrac{b}{6}\) \(=-4\)
  \(b\) \(=-4\times 6\)
  \(b\) \(=-24\)

Equations, SM-Bank 007

Solve the following one-step equations.

  1. \(x+4=10\)  (1 mark)

  2. \(m-7=2\)  (1 mark)

  3. \(2u=28\)  (1 mark)

  4. \(\dfrac{r}{2}=5\)  (1 mark)

Show Answers Only

a.    \(x=6\)

b.    \(m=9\)

c.    \(u=14\)

d.    \(r=10\)

Show Worked Solution
a.    \(x+4\) \(=10\)
  \(x\) \(=10-4\)
  \(x\) \(=6\)

 

b.    \(m-7\) \(=2\)
  \(m\) \(=2+7\)
  \(m\) \(=9\)

 

c.    \(2u\) \(=28\)
  \(u\) \(=\dfrac{28}{2}\)
  \(u\) \(=14\)

 

d.    \(\dfrac{r}{2}\) \(=5\)
  \(r\) \(=5\times 2\)
  \(r\) \(=10\)

Equations, SM-Bank 006

Jace is paid \(x\) dollars per hour. Last week he worked \(15\) hours and he earned $300.

Write an equation describing his wages for the week.  (2 marks)

Show Answers Only

\(15x=300\)

Show Worked Solution

\(15x=300\)

Equations, SM-Bank 004

Write an equation for:

  1. the difference between \(x\) and \(3\) is \(4\). (1 mark)

  2. the product of \(m\) and \(7\) is \(21\). (1 mark)

  3. the sum of \(p\) and \(3\) is doubled and the answer is 8. (1 mark)

  4. half of \(z\) is added to 6 and the result is 12. (1 mark)

Show Answers Only

a.    \(x-3=4\)

b.    \(7m=21\)

c.    \(2(p+3)=8\)

d.    \(\dfrac{z}{2}+6=12\)

Show Worked Solution

a.    \(x-3=4\)

b.    \(7m=21\)

c.    \(2(p+3)=8\)

d.    \(\dfrac{z}{2}+6=12\)

Equations, SM-Bank 002 MC

Which of the following equations is not correct?

  1. \(3+2+7=2\times 6\)
  2. \(x+1=2x-1,\ \text{when }x=2\)
  3. \(18-3\times 4=3\times 2\)
  4. \(19-13=3^2\)
Show Answers Only

\(D\)

Show Worked Solution

\(\text{Checking each option:}\)

\(\text{Option A:}\longrightarrow\) \(3+2+7=2\times 6\ \ \checkmark\)
\(\text{Option B:}\longrightarrow\) \(2+1=2\times 2 -1\ \ \checkmark\)
\(\text{Option C:}\longrightarrow\) \(18-3\times 4=3\times 2\ \ \checkmark\)
\(\text{Option D:}\longrightarrow\) \(19-13\ne 3^2\)

 
\(\Rightarrow D\)

Indices, SM-Bank 101

Simplify  \(2^3\times 5^4\times 2^5\ ÷\  5^2\), giving your answer in simplified index form.  (2 marks)

Show Answers Only

\(2^8\times 5^2\)

Show Worked Solution
\(2^3\times 5^4\times 2^5\ ÷\  5^2\) \(=2^{3+5}\times 5^{4-2}\)
  \(=2^8\times 5^2\)

Indices, SM-Bank 099

Simplify the following, giving your answers in index form.

  1. \((2+3)^0\)  (1 mark)

  2. \(2(4^0)-2\)  (2 marks)

  3. \(3\times 6^0+4\times 2^0\)  (2 marks)

Show Answers Only

a.    \(1\)

b.    \(0\)

b.    \(7\)

Show Worked Solution

a.   \((2+3)^0=5^0=1\)

b.   \(2(4^0)-2=2\times 1-2=0\)

c.   \(3\times 6^0+4\times 2^0=3\times 1+4\times 1=7\)

Indices, SM-Bank 098

Simplify the following, giving your answers in index form.

  1. \(4^0\)  (1 mark)

  2. \(6^0+2^0\)  (2 marks)

  3. \(4+2^0\)  (2 marks)

Show Answers Only

a.    \(1\)

b.    \(2\)

b.    \(5\)

Show Worked Solution

a.   \(4^0=1\)

b.   \(6^0+2^0=1+1=2\)

c.   \(4+2^0=4+1=5\)

Indices, SM-Bank 097

Simplify the following, giving your answers in index form.

  1. \(4^2\ ÷\ 4^2\)  (1 mark)

  2. \(6^3\ ÷\ 6^3\)  (1 mark)

  3. \(12^7\ ÷\ 12^7\)  (1 mark)

Show Answers Only

a.    \(4^0\)

b.    \(6^0\)

b.    \(12^0\)

Show Worked Solution

a.   \(4^2\ ÷\ 4^2=4^{2-2}=4^0\)

b.   \(6^3\ ÷\ 6^3=6^{3-3}=6^0\)

c.   \(12^7\ ÷\ 12^7=12^{7-7}=12^0\)

Indices, SM-Bank 096

Simplify the following, giving your answers in index form.

  1. \((5^2)^2\)  (1 mark)

  2. \((6^3)^4\)  (1 mark)

  3. \((7^4)^5\)  (1 mark)

Show Answers Only

a.    \(5^4\)

b.    \(6^{12}\)

b.    \(7^{20}\)

Show Worked Solution

a.   \((5^2)^2=5^{2\times 2}=5^4\)

b.   \((6^3)^4=6^{3\times 4}=6^{12}\)

c.   \((7^4)^5=7^{4\times 5}=7^{20}\)

Indices, SM-Bank 095

Simplify the following, giving your answers in index form.

  1. \((2^4)^4\)  (1 mark)

  2. \((4^3)^5\)  (1 mark)

  3. \((3^3)^3\)  (1 mark)

Show Answers Only

a.    \(2^{16}\)

b.    \(4^{15}\)

b.    \(3^9\)

Show Worked Solution

a.   \((2^4)^4=2^{4\times 4}=2^{16}\)

b.   \((4^3)^5=4^{3\times 5}=4^{15}\)

c.   \((3^3)^3=3^{3\times 3}=3^9\)

Indices, SM-Bank 094

  1. Write  \((5^2)^4\)  in expanded form.  (2 marks)

  2. Write your answer to (a) in simplified index form.  (1 mark)

Show Answers Only

a.    \((5\times 5)\times (5\times 5)\times (5\times 5)\times (5\times 5)\)

b.    \(5^8\)

Show Worked Solution

a.   \((5^2)^4=(5\times 5)\times (5\times 5)\times (5\times 5)\times (5\times 5)\)
 

b.   \((5\times 5)\times (5\times 5)\times (5\times 5)\times (5\times 5)=5^8\)

Indices, SM-Bank 093

  1. Write  \((2^3)^3\)  in expanded form.  (2 marks)

  2. Write your answer to (a) in simplified index form.  (1 mark)

Show Answers Only

a.    \((2\times 2\times 2)\times (2\times 2\times 2)\times (2\times 2\times 2)\)

b.    \(2^9\)

Show Worked Solution

a.   \((2^3)^3=(2\times 2\times 2)\times (2\times 2\times 2)\times (2\times 2\times 2)\)
 

b.   \((2\times 2\times 2)\times (2\times 2\times 2)\times (2\times 2\times 2)=2^9\)

Indices, SM-Bank 092

Find the missing term in the following number sentence.  (1 mark)

Show Answers Only

\(2^7\)

Show Worked Solution

\(a^m\ ÷\ a^n=a^{m-n}\)

\(2^n\ ÷\ 2^2\) \(=2^5\)
\(\therefore \ n-2\) \(=5\)
\(n\) \(=7\)

 
\(\therefore \ \text{the missing term is }2^7\)

Indices, SM-Bank 091

Find the missing term in the following number sentence.  (1 mark)

Show Answers Only

\(3^{11}\)

Show Worked Solution

\(a^m\ ÷\ a^n=a^{m-n}\)

\(3^n\ ÷\ 3^4\) \(=3^7\)
\(\therefore \ n-4\) \(=7\)
\(n\) \(=11\)

 
\(\therefore \ \text{the missing term is }3^{11}\)

Indices, SM-Bank 090

Simplify the following, giving your answers in index form.

  1. \(\dfrac{2^{16}}{2^9}\)  (1 mark)

  2. \(\dfrac{4^8}{4^3}\)  (1 mark)

  3. \(\dfrac{10^{11}}{10^2}\)  (1 mark)

Show Answers Only

a.    \(2^7\)

b.    \(4^5\)

b.    \(10^9\)

Show Worked Solution

a.   \(\dfrac{2^{16}}{2^9}=2^{16-9}=2^7\)

b.   \(\dfrac{4^8}{4^3}=4^{8-3}=4^5\)

c.   \(\dfrac{10^{11}}{10^2}=10^{11-2}=10^9\)

Indices, SM-Bank 089

Simplify the following, giving your answers in index form.

  1. \(2^8 \ ÷\ 2^4\)  (1 mark)

  2. \(5^{15} \ ÷\ 5^9\)  (1 mark)

  3. \(3^4 \ ÷\ 3\)  (1 mark)

Show Answers Only

a.    \(2^4\)

b.    \(5^6\)

b.    \(3^3\)

Show Worked Solution

a.   \(2^8\ ÷\ 2^4=2^{8-4}=2^4\)

b.   \(5^{15}\ ÷\ 5^9=5^{15-9}=5^6\)

c.   \(3^4\ ÷\ 3^1=3^{4-1}=3^3\)

Indices, SM-Bank 088

  1. Write  \(\dfrac{7^5}{7^2}\)  as a fraction in expanded form.  (2 marks)

  2. Write your answer to (a) in simplified index form.  (1 mark)

Show Answers Only

a.    \(\dfrac{7\times 7\times 7\times 7\times 7}{7\times 7}\)

b.    \(7^3\)

Show Worked Solution

a.   \(\dfrac{7^5}{7^2}=\dfrac{7\times 7\times 7\times 7\times 7}{7\times 7}\)
 

b.   \(\dfrac{7\times 7\times 7\times 7\times 7}{7\times 7}=7^3\)

Indices, SM-Bank 087

  1. Write  \(\dfrac{3^6}{3^4}\)  as a fraction in expanded form.  (2 marks)

  2. Write your answer to (a) in simplified index form.  (1 mark)

Show Answers Only

a.    \(\dfrac{3\times 3\times 3\times 3\times 3\times 3}{3\times 3\times 3\times 3}\)

b.    \(3^2\)

Show Worked Solution

a.   \(\dfrac{3^6}{3^4}=\dfrac{3\times 3\times 3\times 3\times 3\times 3}{3\times 3\times 3\times 3}\)
 

b.   \(\dfrac{3\times 3\times 3\times 3\times 3\times 3}{3\times 3\times 3\times 3}=3^2\)

Indices, SM-Bank 086

Find the missing term in the following number sentence.  (1 mark)

Show Answers Only

\(3^7\)

Show Worked Solution

\(a^m\times a^n=a^{m+n}\)

\(3^n\times 3^4\) \(=3^{11}\)
\(\therefore \ n+4\) \(=11\)
\(n\) \(=7\)

 
\(\therefore \ \text{the missing term is }3^7\)