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Distance, speed and time: comparing routes

The plane and space, grade 10 (3 units) Lesson 6 of 15Learning

Distance, speed and time: comparing routes

Distance, speed and time

Speed is the distance covered in one unit of time. A car that drove 150 km in two hours covered 150÷2=75\displaystyle {150\div2}={75} km each hour: its speed is 75 km/h (kilometres per hour).

  • s\displaystyle s – the distance, t\displaystyle t – the time, v\displaystyle v – the speed.
v\displaystyle v =\displaystyle = st\displaystyle \frac{s}{t}
  • From the same relation: the distance is speed times time, and the time is distance divided by speed.
s\displaystyle s =\displaystyle = v⋅t\displaystyle v\cdot t
t\displaystyle t =\displaystyle = sv\displaystyle \frac{s}{v}
Time in hours and in minutes

When the speed is in km/h, the time t=sv\displaystyle {t}={\frac{s}{v}} comes out in hours. To change to minutes, multiply by 60.

  • Half an hour is 30 minutes, a quarter of an hour 15 minutes, a third of an hour 20 minutes.
  • Example. 12 km at 48 km/h: t=1248=0.25\displaystyle {t}={\frac{12}{48}}={0.25} hours, that is 0.25⋅60=15\displaystyle {0.25\cdot60}={15} minutes.
  • And back. 40 minutes are 4060=23\displaystyle {\frac{40}{60}}={\frac{2}{3}} of an hour.
Exercise 1

Calculate.

a
The speed of a bus that drove 180 km in 2.5 hours, in km/h

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b
    The distance a lorry covers in 4 hours at 65 km/h, in km

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      c
      The time of a 210 km drive at 70 km/h, in hours

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        d
        The time of a 5 km walk at 6 km/h, in minutes

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          Example: comparing two routes

          A family drives to the beach. There are two ways:
          By the motorway: 60 km, at an average speed of 90 km/h.
          Through the towns: 45 km, at an average speed of 54 km/h.
          The example below works out how long each way takes.

          • The motorway: t=6090=23\displaystyle {t}={\frac{60}{90}}={\frac{2}{3}} hours, 23⋅60=40\displaystyle {\frac{2}{3}\cdot60}={40} minutes.
          • Through the towns: t=4554=56\displaystyle {t}={\frac{45}{54}}={\frac{5}{6}} hours, 56⋅60=50\displaystyle {\frac{5}{6}\cdot60}={50} minutes.
          • The longer way is 10 minutes faster. The shortest route is not always the fastest.
          Exercise 2

          A cyclist can choose between two routes:
          A flat route: 24 km, at 16 km/h.
          A route over the hill: 18 km, at 9 km/h.

          a
          The time on the flat route, in minutes

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            b
            The time on the route over the hill, in minutes

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              c

              Which route is faster, and by how many minutes?

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                Exercise 3

                A field shaped like a rectangle is 400 m by 300 m.
                You can walk from corner to corner along two sides, on a path, at 5 km/h.
                You can also cross the field diagonally, through mud, at 3 km/h.

                a
                The time along the sides, in minutes (700 m is 0.7 km)

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                  b
                  The length of the diagonal, in metres

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                    c
                    The time across the diagonal, in minutes

                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                      d

                      Which way is worth choosing? Explain.