94 Unit 3 Problem solving: basic skills
Commentary
This is another problem where an
intermediate calculation is necessary. In order
to calculate Petra’s bill, we need to know the
monthly charge and the rate per unit. We
know the difference between the two quarters’
bills - this difference is only due to the
reduced consumption, so 1400 fewer units
saved $112. This means units cost 8¢ each.
The three-monthly fixed charge, therefore, is:
$250 − 2000 × 8¢ = $250 − $160
= $90 (or $30 per month)
If Petra reduced her January to March
consumption by 25%, this would then be 1500
units, so her bill would be:
$90 quarterly charge plus 1500 × 8¢
= $90 + $120 = $210
In fact, the quarterly charge does not have to
be calculated, only the unit rate. The entire
process of solving this problem could be
speeded up by simply recognising that the
relevant three-month bill would be reduced by
500 × 8¢ or $40.Commentary
From the data given, it is easy to find out
when they both arrive at their destinations,
but finding when they cross is not so
straightforward. The problem can be made
much simpler by using an intermediate step.
First calculate where Amy is when David
leaves. She has been travelling for 2 hours, so
she has covered 240 km – that is, she is
160 km from David’s house. The problem is
now quite easy. At 10 a.m. they are 160 km
apart and rushing towards each other at a
joint speed of 240 km/h. Therefore, they will
meet 40 minutes later (160 km/240 km per
hour is^23 hour or 40 minutes). The time they
pass each other is 10.40 a.m.
If we had been asked to find the place where
they pass, the passing time could have been
used as a second intermediate value. David
travels 120 km in an hour. 40 minutes
represents^23 of this, or 80 km, so they cross
80 km from David’s house.
In this case the numbers were very easy but
the same method of solution could have been
used whatever the distances, times and speeds.
The method of solution, which was not
immediately obvious, became easier by using
the intermediate step.Amy and her brother David live 400 km apart.
They are going to have a week’s holiday by
exchanging houses. On the day they are
starting their holiday, Amy leaves home at
8 a.m. and David at 10 a.m. They both drive
at 120 km/h on a motorway that travels
directly between their homes.
At what time do they pass each other on
the road?Petra’s electricity supply company charges
her a fixed monthly sum plus a rate per unit
for electricity used. In the most expensive
quarter last year (January to March), she
used 2000 units and her bill was $250.
In the least expensive quarter (July to
September), she used 600 units and her bill
was $138.
She is now adding extra insulation to her
home which is expected to reduce her overall
electricity consumption by 25%. What can
she expect her January to March bill to be
next year (if there are no increases in overall
tariffs)?Activity Activity