Question 53

Challenge question #53 (19/06/2017)

Junior Question

The petrol tank in Jim’s car is $\tfrac{2}{3}$ full. He drove to his friend’s house and used $2$ litres of petrol. The petrol tank is now $\tfrac{5}{8}$ full.

How many litres does the petrol tank hold when full?

Answer

The petrol tank holds $48$ litres when full.

Worked Solution

Jim’s trip to his friend’s house used $\tfrac{2}{3} - \tfrac{5}{8} = \tfrac{16}{24} - \tfrac{15}{24} = \tfrac{1}{24}$ of a tank of petrol.

\therefore\quad 2

litres

=

\tfrac{1}{24}

of the tank

\therefore\quad 48

litres

=

the whole tank

The petrol tank holds $48$ litres when full.

Senior Question

5

23

In the grid alongside, the blank squares must be filled in such that the number in each blank square is the average of the numbers in its adjacent squares.

What number must be filled in the shaded blank square?

Answer

The number is $17$ .

Worked Solution

For any three adjacent squares numbered $a$ , $b$ , $c$ , we have

b=

\dfrac{a+c}{2}

\therefore\quad 2b=

a+c

\therefore\quad b-a=

c-b

\therefore

any three adjacent squares form an arithmetic sequence.

So, all of the numbers in the grid must form an arithmetic squence.

Suppose the sequence has common difference $d$ .

The sequence has

u_{1}=5\ \

and

\ \ u_{7}=

23

\therefore\quad u_{1}+6d=

23

\therefore\quad 6d=

18

\therefore\quad d=

3

The number in the shaded square is

u_{5}=

u_{1}+4d

=

5+4(3)

=

17

Question 53

Haese Mathematics Mailing List