John and Peter took part in a marathon race. They each ran 42 km.

(a)
John ran at a constant speed of x kilometres per hour.

Write down an expression, in terms of x, for the number of hours he took.

Ans: ___ h
(b)
Peter ran at a constant speed which was 1/2 km/h more than John's speed.

Write down an expression, in terms of x, for the number of hours he took.

Ans: ___ h
(c)
The difference between their times was 15 minutes.
1. (i)
  Write down an equation in x to represent this information, and show that it reduces to $2x^2 + x - 168 = 0$ .
  
  Answer:
2. (ii)
  Solve the equation $2x^2 + x - 168 = 0$ , giving your answers correct to 3 decimal places.
  
  Ans: x = ___ or ___
3. (iii)
  Calculate the time that Peter took to complete the race, giving your answer in hours and minutes.
  
  Ans: ___ h ___ min

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Worked solution

1(a)

1. Form the time taken using distance divided by speed.

Distance run $=42$ km

Speed $=x$ km/h

$\text{Time}=\frac{\text{distance}}{\text{speed}}=\frac{42}{x}$

Answer

$\dfrac{42}{x}$ h

1(b)

1. Express Peter's speed and use distance divided by speed to form his time.

Peter's speed $= x+\frac12$ km/h.

Time taken $= \dfrac{\text{distance}}{\text{speed}} = \dfrac{42}{x+\frac12}$ h.

Answer

$\dfrac{42}{x+\frac12}$ h

1(c)(i)

1. Write an equation for the difference between John's and Peter's times.

John's time $=\dfrac{42}{x}$ h.

Peter's time $=\dfrac{42}{x+\frac12}$ h.

Since Peter's speed is higher, Peter took less time. Also, $15$ minutes $=\dfrac{15}{60}=\dfrac14$ h.

$\frac{42}{x}-\frac{42}{x+\frac12}=\frac14$

2. Clear the denominators.

$\frac{42}{x}-\frac{42}{x+\frac12}=\frac14$

Multiply throughout by $4x\left(x+\frac12\right)$ :

$168\left(x+\frac12\right)-168x=x\left(x+\frac12\right)$

3. Expand and simplify to obtain the required quadratic equation.

$168x+84-168x=x^2+\frac12x$

$84=x^2+\frac12x$

Multiply by $2$ :

$168=2x^2+x$

$2x^2+x-168=0$

Answer

$\frac{42}{x}-\frac{42}{x+\frac12}=\frac14$
$2x^2+x-168=0$

1(c)(ii)

1. Identify the coefficients and substitute them into the quadratic formula.

For $2x^2+x-168=0$ ,
$a=2,\quad b=1,\quad c=-168$
Using $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$
$x=\frac{-1\pm\sqrt{1^2-4(2)(-168)}}{2(2)}$

2. Simplify the discriminant and obtain the exact root form.

$x=\frac{-1\pm\sqrt{1+1344}}{4}$
$x=\frac{-1\pm\sqrt{1345}}{4}$

3. Calculate both roots and round to 3 decimal places.

$x=\frac{-1+\sqrt{1345}}{4}=9.9158\ldots$
$x=\frac{-1-\sqrt{1345}}{4}=-10.4158\ldots$
Correct to 3 decimal places,
$x=9.916\quad \text{or}\quad x=-10.416$

Answer

$x=9.916$ or $x=-10.416$

1(c)(iii)

1. Use the valid positive value of $x$ to find Peter's speed.

From part (c)(ii), John's speed is
$x=9.916\text{ km/h}$
Peter's speed is
$x+\frac12=9.916+0.5=10.416\text{ km/h}$

2. Calculate Peter's time and convert it to hours and minutes.

Peter's time
$=\frac{42}{10.416}=4.032258\ldots\text{ h}$
Convert the decimal part to minutes:
$0.032258\ldots\times 60=1.93548\ldots\text{ min}$
So Peter took approximately
$4\text{ h }2\text{ min}$

Answer

$4$ h $2$ min

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John and Peter took part in a marathon race. They each ran 42 km.

Worked solution

1(a)

1(b)

1(c)(i)

1(c)(ii)

1(c)(iii)

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