1) Two angles of an isosceles triangle are \((x-20)\)° and \((x+25)\)°. Find the two possible values of \(x\).

As the triangle is isosceles, its angles will add to \(180\)° and two of its three sides must be equal. So either

\(2(x-20)+(x+25)=180\)

or

\(2(x+25)+(x-20)=180\)

For the first case

\begin{align}2(x-20)+(x+25) & =180\\ 2x-40+x+25 & =180\\ 3x-15 & =180\\ 3x & =165\\ x & =55\end{align} And for the second case \begin{align}2(x+25)+(x-20) & =180\\ 2x+50+x-20 & =180\\ 3x+30 & =180\\ 3x & =150\\ x & =50\end{align}

So our two possible values for \(x\) are \(55^{\circ}\) and \(50^{\circ}\)

2) A rectangle has a length of \((5x-2)cm\), a width of \((4-2x)cm\) and a perimeter of \(25cm\) . Find \(x\).

First form an equation using the above information, then solve.

\begin{align} 2(5x-2)+2(4-2x) & =25\\ 10x-4x-4+8 & = 25\\ 6x+4 & =25\\ 6x & = 21\\ x & =3.5 \end{align}

Therefore \(x\) is \(3.5cm\).

3) A ship is carrying 115 passengers. It stops at a port and 28 passengers disembark, while \(x\) embark. At the next port, \(\frac{1}{3}\) of the passengers disembark and 23 embark. There are now 58 passengers on the ship. Find \(x\), the number of passengers who embarked at the first port.

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

\begin{align}(115-28+x)-\frac{(115-28+x)}{3}+23 & =58\\ 87+x - \frac{(115-28+x)}{3} & =58\\ 87+x-(29+\frac{x}{3}) & = 58\\ 56 +x-\frac{x}{3} & = 58\\ \frac{2}{3}x & =2\\ x & =3 \end{align} Therefore 3 passengers embarked at the first port

4) A man wins a cash prize in a lottery and gives it all to his sons, his nephews and his brother. He gives each of his four sons three times as much as he gives each of his three nephews. He gives his brother $18,000 which is a quarter of the total. How much did he give each nephew and each son?

The total win must be \(4(18,000)=72,000\). We know the brother received 18,000, so the amount distributed to the others is

\(72,000-18,000=54,000\)

So we can now form the equation, then solve. Let each nephew's share be \(x\)

\begin{align}3x+3x+3x+3x+x+x+x & =54,000\\ 15x & = 54,000\\ x & =3,600\\ \end{align}

By substitution, \(3(3,600) =10,800\)

Therefore each son receives $10,800 and each nephew receives $3,600

5) A woman is 29 years older than her daughter. Eight years ago, the woman was twice as old as her daughter. What are their ages now?

Form the equation, then solve. Let the daughter's age be \(x\).

\begin{align}(x+29)-8 & = 2(x-8)\\ x+21 & = 2x-16\\ x & = 37\\ \end{align}

And by substitution 37+29 = 66

Therefore the daughter is 37 years old and the mother is 66 years old.

6) Solve for \(x\).

\((2x-1)(4x+3)=(8x-2)(x+1)\)

\begin{align}(2x-1)(4x+3) & =(8x-2)(x+1)\\ 8x^2+6x-4x-3 & =8x^2+8x-2x-2\\ 2x-3 & =6x-2\\ -1 & = 4x\\ x & = -\frac{1}{4}\\ \end{align}

7) The hypotenuse of a right angled triangle measures \((x+4)cm\). The other two sides measure \(9cm\) and \((x-1)cm\). Find the length of each side.

Using Pythagoras' Theorem, form the equation then solve.

\begin{align}9^2+(x-1)^2 & = (x+4)^2\\ 81+x^2-2x+1 & = x^2+8x+16 \\ 10x & =66\\ x & = 6.6\\ \end{align}

By substitution, \(6.6+4=10.6\),  and  \(6.6-1=5.6\)

Therefore the respective lengths are: \(10.6cm\) (hypotenuse),  \(9cm\),  and \(5.6cm\)