You may also like

problem icon

Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

problem icon

Air Routes

Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.

problem icon


The probability that a passenger books a flight and does not turn up is 0.05. For an aeroplane with 400 seats how many tickets can be sold so that only 1% of flights are over-booked?

Debt Race

Age 16 to 18 Challenge Level:

To solve this problem we must first set up a linear difference equation and then solve the equation to find n when the debt remaining = 0.

$A_n$ =amount owed in the nth year of repayment
I = Interest rate
L= Lump sum payed annually

In general $A_n= (1+I)A_{n-1}- L$

$A_n - (1+I)A_{n-1} = - L$

Set the right hand side = 0 and solve for the complementary function.

$z - (1+ I) = 0$
$z = 1+I$

$A_{CF} = C(1+I)^{n}$

Now solve for the particular integral:

Let $A_n = K$ then $A_{n-1}$ also = K

Substituting into the general form we find:

$K -(1+I)K =-L \to K = \frac{L}{I}$

General solution:

$A_n = A_{CF} + A_{PI}= C(1+I)^n + \frac{L}{I}$

Now use the boundary condition that at n = 0 , $A_n = 100,000$

We find that the constant $c = (10^5 - \frac{L}{I})$

Therefore $A_n = (10^5 - \frac{L}{I}) (I +1)^n +\frac{L}{I}$

If we now set $A_n$ = 0 and solve for n we find:

$n =\frac{log(\frac{-LI^{-1}}{10^5 - LI^{-1}}) }{log(1+I)}$

If we now substitute the values of I and L for each person given in the intitial question we find:

Person A: n=9.73years
Person B: n= 9.60 years
Person C: n= 9.59 years
Person D: n= 9.69 years
Person E: n= 9.92 years