A motorboat whose speed is 24
kilometres per hour in still water took one more hour to go 32 kilometres upstream
than it did to return downstream to the same spot. Find the speed of the stream.
Firstly, we will let the speed of
the stream be 𝑆 kilometres per hour. As the speed of the motorboat in
still water is 24 kilometres per hour, the speed of the motorboat upstream will be
24 minus 𝑆 and the speed of the motorboat downstream will be 24 plus 𝑆. The distance travelled in each
direction was 32 kilometres.
As it took one hour longer to go
upstream than downstream, we can say that the time it took to go upstream minus the
time it took to come downstream is equal to one hour. Time is equal to the distance
divided by the speed. We can therefore write the equation
32 divided by 24 minus 𝑆 minus 32 divided by 24 plus 𝑆 is equal to one, as we can
divide the distance by the speed of the boat going upstream and the speed of the
boat going downstream.
In order to calculate the speed of
the stream, we now need to solve this equation. In order to solve any problem with
algebraic fractions, we need to ensure that the denominators are the same. Whatever you do to the top of a
fraction you must do to the bottom. Therefore, we’ll multiply the
numerator and denominator of the first fraction by 24 plus 𝑆. And we can multiply the numerator
and denominator of the second fraction by 24 minus 𝑆.
Now that the fractions have a
common denominator, 24 minus 𝑆 multiplied by 24 plus 𝑆, we can rewrite it as a
single fraction. We do this by grouping the
numerators together. This gives us 32 multiplied by 24
plus 𝑆 minus 32 multiplied by 24 minus 𝑆 divided by 24 minus 𝑆 multiplied by 24
plus 𝑆. This is all equal to one.
Expanding the brackets on the
numerator gives us 768 plus 32𝑆 minus 768 plus 32𝑆. Expanding the two brackets on the
bottom gives us 576 minus 𝑆 squared. This is because we have the
difference of two squares. 24 multiplied by 24 is 576. Multiplying the outside terms gives
us 24𝑆. Multiplying the inside terms gives
us negative 24𝑆. And multiplying the last terms
gives us negative 𝑆 squared.
The middle terms cancel, leaving us
with 576 minus 𝑆 squared. Simplifying the numerator gives us
64𝑆. So we are left with 64𝑆 divided by
576 minus 𝑆 squared is equal to one. Multiplying both sides of this
equation by 576 minus 𝑆 squared gives us 64𝑆 is equal to 576 minus 𝑆 squared. Adding 𝑆 squared and subtracting
576 from both sides of this equation gives us a quadratic equation: 𝑆 squared plus
64𝑆 minus 576 equals zero.
We can solve this by factorising,
where the first term in both brackets will be 𝑆, as 𝑆 multiplied by 𝑆 is 𝑆
squared. The two numbers in the brackets
will have a product of negative 576 and a sum of 64. In this case, they’ll be positive
72 and negative eight, as positive 72 multiplied by negative eight is negative 576
and 72 minus eight is 64. This gives us two possible values
for 𝑆: 𝑆 equals negative 72 or 𝑆 equals eight. As the speed of a stream must be a
positive answer, we can say that the speed is equal to eight kilometres per