1) The sum of three consecutive even numbers is 234. Find the numbers.

Every even number is divisible by 2, so let the first of our three even numbers be \(2x\). This means the next even number will be \(2x+2\), and the next after that will be \(2x+4\).

Now we form an equation using the above information, then solve.

\begin{align} 2x+(2x+2)+(2x+4) & =234\\ 6x+6 & = 234\\ 6x & =228\\ x & = 38 \end{align}

We know our first even number is \(2x\), so by substitution our first even number must be

2(38)=76

Therefore our three consecutive even numbers are 76,78 and 80.

2) The sum of three consecutive odd numbers is 285. Find the numbers.

Let the first of our three odd numbers be \(2x+1\). (Note: as \(2x\) is even, \(2x+1\) must be odd). This means the next odd number will be \(2x+3\), and the next after that will be \(2x+5\).

Now we form an equation using the above information, then solve.

\begin{align} (2x+1)+(2x+3)+(2x+5) & =285\\ 6x+9 & = 285\\ 6x & =276\\ x & = 46 \end{align}

We know our first even number is \(2x+1\), so by substitution our first even number must be

2(46)+1=93

Therefore our three consecutive odd numbers are 93,95 and 97.

3) The perimeter of a rectangle is 52cm. The rectangle is \(3x+1\) long and \(2x-3\) wide. Find \(x\).

First form an equation with the given information, then solve.

\begin{align}2(3x+1)+2(2x-3) & =52\\ 6x+2+4x-6 & =52\\ 10x-4 & =52\\ 10x & =56\\ x & =5.6 \end{align} Therefore \(x=5.6 \;cm\)

4) The perimeter of a triangle is 74cm. The lengths of the sides are \(2x+1,\;2x-2,\) and\( \;4x-3\). Find \(x\).

First form an equation with the given information, then solve.

\begin{align}(2x+1)+(2x-2)+(4x-3) & =74\\ 8x-4 & =74\\ 8x & =78\\ x & =9.75\\ \end{align} Therefore \(x=9.75 \;cm\)

5) The angles of a triangle are in the ratio \(2:5:11\). Find the angles.

The interior angles of a triangle add to \(180^{\circ}\). We combine this fact with the given ratio information to form an equation, then solve.

\begin{align}2x+5x+11x & =180\\ 18x & =180\\ x & =10\\ \end{align}

So by substitution: \( 2x =20, \; 5x=50,\;11x=110\)

Therefore the three angles are \(20^{\circ},\;50^{\circ},\;110^{\circ}\)

6) The sum of three numbers is 54. The first number is half the third number, and the second number is 6 more than the first. Find the three numbers.

First form an equation with the given information, then solve. Let the third number be \(x\).

\begin{align}(\frac{1}{2}x)+(\frac{1}{2}x+6)+(x) & =54\\ 2x+6 & =54\\ 2x & =48\\ x & = 24\\ \end{align}

So by substitution: \( \; x=24,\;\frac{1}{2}x=12,\; \frac{1}{2}x+6=18\)

Therefore the three numbers are \(24,\;12,\;18\)

7) The desk is 13 cm lower than the table. The stool is 24 cm lower than the desk. If the combined height of the stool, desk and table is \(1.9m\), how tall is each item?

First form an equation with the given information, then solve. Let the height of the table be \(x\).

\begin{align}(x)+ (x-13)+(x-13-24) & = 190\\ 3x-50 & = 190 \\ 3x & =240\\ x & = 80\\ \end{align}

So by substitution: \( \; x=80,\;x-13=67,\; x-38=42\)

Therefore the respective heights are: table= \(80 cm\), desk=\(67 cm\),  stool=\(42 cm\)

x x 6 m 4 m Courtyard Path

8 a) Find an expression for the area of the path.

8 b) Find an expression for the outside perimeter of the path.

8 c) Find the value of \(x\) when the outside perimeter of the path is \(30 m\).

a) Each 'corner' of the path has an area of \((x \; \times \; x) = x^2\).

If we subtract these four 'corners' from the path, we are left with four rectangles. Two of the rectangles each have the dimensions \((6\; \times \;x)\). The other two rectangles each have the dimensions \((4 \;\times \; x)\)

So our expression for the area of the path is \begin{align}Area & = x^2+2(6x)+2(4x)\\ & = x^2 +20x\\ \end{align}

b) \begin{align}Perimeter & = 2(x+x+4)+2(x+x+6)\\ & = (4x+8)+(4x+12)\\ & = 8x+20 \end{align}

c) Substitute 30 into our expression for perimeter from part (b) \begin{align} 30 & = 8x+20\\ 10 & = 8x\\ x & = \frac{10}{8}\\ \end{align} Therefore \(x=1.25 m\)