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AS IS OUR annual tradition for MATHS WEEK, we’ve been setting our readers daily puzzles. Give them a go!
Interesting Rates
Yesterday, we discussed time and the measurement of time.
Once there was a good system for units of time and more accurate ways of measuring time, the rate of scientific advancement accelerated.
In many circumstances it was desirable to be able to measure the rates things changed with respect to time. An example would be speed which is the rate at which distance changes with time.
For instance, if you travelled 60km in one hour your speed would be 60kph.
The rate of change of speed with respect to time is called acceleration.
In the 17th century a new mathematics emerged which we call calculus.
Differential calculus measures the rate at which things change relative to other things, such as distance and time.
There are many applications in multiple fields where rates of change are important and calculus is invaluable.
We are not going to try any calculus today, but we will look at some rate-of-change puzzles. For example:
A man-and-a-half can dig a hole-and-a-half in a day-and-a-half. How many holes can two men dig in three days?
These are a bit mind-bending. Ignore the practicalities of a half-a-man digging a hole, or the precise definition of a half-a-hole, and try to work out how much one man can dig in a day-and-a-half.
A man-and-a-half can dig a hole-in-a-half in a day-and-a-half. Therefore, this man can dig at a rate of one hole a day (number of holes divided by time taken).
If one man can dig one hole in a day-and-a-half, then he can dig two holes in twice the time – 3 days.
Therefore 2 men can dig twice as many holes. Therefore 4 holes in 3 days.
These puzzles get quite a bit more difficult when we have people working at different rates.
Try some yourself:
1. A farmer hires three people to dig a drain 300m long. One can dig 15 metres in a day, one can dig 20 metres in a day and one can dig 25 metres a day. If they work together, how long will it take them to finish the job?
2. A farmer wants a square field walled in. She hires three masons who set to work on a wall each. Ann finishes her wall in 4 days, Brian finishes his wall in 6 days and Ciaran finishes his wall in 12 days. She gets them to work together on the last wall. If they work at the same rates how long will it take them to finish the job?
3. A farmer found…
that his cow and goat would eat a bag of meal in 45 minutes,
that the cow and the turkey would eat it in 60 minutes,
but that it would take the goat and the turkey 90 minutes to finish the bag.
How long would it take the cow, goat and turkey together to eat the bag of meal?
***
The answers: Thursday’s puzzle
1. 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
2. 5 o’clock
3. 8h 12m 5s
24 hours is 10 hours decimal
1 hour decimal is 24/10 = 2.4 hours
7.30pm is 7.5 hours which in decimal will be 7.5/2.4 = 3.125 hours
As 12 Noon is 5 O’Clock decimal:
7.30pm would be 5 + 3.125 = 8.125 hours which is 8 hours 12 minutes 50 seconds in the decimal system.
Come back tomorrow at 7.30pm for the answers to today’s questions and a brand new challenge.
These puzzles were prepared exclusively for The Journal by Eoin Gill, co-founder and co-ordinator of Maths Week Ireland.
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