Saturday, 24 December 2016

Quibans 51: Monster Carrots

From the Cambridge News:
Frenzied Friday: Supermarkets to see surge in last minute food shoppers before Christmas
Supermarkets across the country are bracing themselves for rush of last minute Christmas food shoppers today (December 23).
Already dubbed 'Frenzied Friday', many stores are expected to have their busiest day of the year as millions of customers stock up for some Christmas feasting.
Tesco's has predicted that over 10 million customers will be visiting its stores throughout the day, around 15,000 per minute.
Tesco expects to selling 40 million Brussels sprouts across today and yesterday as well as 3.5 million carrots - enough to stretch around the world 11 times.

The big thing here is having an idea about the size of numbers. An initial question (before showing the other questions):

Question zero) Do the numbers in the question seem reasonable ones?


Q1) “10 million customers […], around 15,000 per minute”. What does this tell us?

Q2) The distance from the north pole to the equator is 10,000 km. How big would a carrot need to be if the news story is accurate?


Q0) The number of customers seems reasonable, but the number of carrots seems to be much too small to stretch around the world 11 times.

Q1) 10 million divided by 15,000 equals just over 11 hours. So either Tesco stores are open for 11 hours a day, or they have sensibly decided that more people shop during the day rather than in the middle of the night.

Q2) The circumference of the earth is 40 million metres. 11 times round the world would be 440 million metres. Dividing that by 3.5 million carrots gives an average carrot-length of 126 metres. Monster carrots! 12.6cm seems more reasonable for the length of a carrot so dividing by 1000 would be useful.

There are three easy ways to do this: make an error and assume the circumference of the world is actually 40,000 metres (rather than km); sell 3.5 billion carrots, or say that the carrots will go 0.011 times round the world!

If you decide that the Brussels sprouts should be included (the sentence is potentially ambiguous, even if you ignore the grammatical typo) then that doesn’t help (unless you have Monster Sprouts…).


Friday, 16 December 2016

Quibans 50: A million cyclists

From the Cambridge News:
One million cyclists in Cambridge have passed Parker's Piece bike counter this year
14 Dec 2016
One million cyclists have ridden past the cycle counter in Parker's Piece since January, breaking the all-time record.
Roxanne de-Beaux, of the Cambridge Cycling Campaign, said last year the total had fallen frustratingly short, coming in 50,000 under the million mark.
Cllr Noel Kavanagh, cycling champion at Cambridgeshire County Council, said the number was brilliant news, but not a total surprise.
"I am not surprised," he said. "I cycle through it several times and have noticed the total going up. Most days, it is two or three thousand. I think this is really good.”
The counter was put in place in 2014, in part to commemorate the historic start of the 2014 Tour de France in the city.

Here are some of the comments under the article:

Commenter 1:
Interesting to note that a person who uses their car exclusively probably pays less per mile in VED, than a car owning cyclist... in other words, a car owning cyclist pays more "road tax" per mile than average Joe Car.
Commenter 2:
"a car owning cyclist pays more" - interesting, I never thought of it like that.
Commenter 1:
I have a car and a bicycle. I do ~2000 miles in my car (I cycle most places within 5 miles) and pay ~£220 VED, thus pay ~##p per mile.
The average car does around 8000 per year, with an average VED of £166, so ~##p per mile. (source for averages... Google).
So, thus as mainly a cyclist in town, I pay a lot more road tax per mile than most car drivers. Given, there are a lot of caveats to this.

The comments under a Cambridge News article that involve bikes often turn into a cyclist vs driver argument. This little exchange between two commenters involves some nice Core Maths skills!

Possible questions:

1) What does the headline mean?

2) How many bikes pass this point per day? What are the difficulties with carrying out this sort of calculation?

3) What will the counter read at the end of the year?

4) What is the percentage increase from last year?

5) Is the councillor correct with his statements?

6) VED is “vehicle excise duty” – commonly referred to as “car tax” or “road tax”. The symbol ~ means “approximately”. What are the numbers I have blanked out in the comments?


1) It doesn’t mean that there were a million cyclists. Just that a bike has been read by the sensor a million times. One person cycling past it twice each day would contribute 365 x 2 of these, for example.

2) There are 31-13 = 18 days to go until the end of the year. This was a leap year, so there have been 366-18 = 348 days. 1 million divided by 348 = 2874 per day.

I have no idea whether more people cycle during the week (getting to work), or at the weekend (because they have time to do so). Maybe best to say it is between 2500 and 3000 per day?

We also don’t know when the figure of a million was reached (we only have the date of the article to go on.)

3) What will the counter read at the end of the year? Again, we don’t know whether more people cycle close to Christmas (for leisure reasons, coming back from Christmas parties, etc), or fewer (they drive because they won’t fit the presents they buy in their basket, they are away visiting relatives, etc. If we go for 2500 per day between now and the end of the year then that would be an additional 45000.

4) What is the percentage increase from last year? 1,045,000 this year compared to 950,000 last year. Dividing these gives 1.1, which means a 10% increase.

5) Is the councillor correct with his statements? “have noticed the total going up” – yes- that is correct (!). “Two or three thousand” is right too.

6) ~11p compared to ~2p.


Saturday, 10 December 2016

Quibans 49: Bike thefts

From the Cambridge News:

Cambridge branded bike theft capital and rail station ranked third worst cycle theft hotspot

Criminals swiped 2,173 cycles in 12 months and streets within the city account for 13 of the top 100 bike theft hotspots listed in police recorded crime data, more than any other local authority area.

The top hotspots for theft are Station Road at the railway hub with 63 thefts, and Parkside, next to the city police station and Parker’s Piece, with 62, following new analysis of crime data from every police force in England and Wales analysed by online insurer Protect Your Bubble.

The bike theft problem in Cambridge is underlined when the thefts are calculated per 1,000 residents. The overall rate for England and Wales is 1.4 thefts per 1,000 people. In Cambridge it’s 16.6.

Government figures show that across England 9.5 per cent of adults cycle at least once a week. But Cambridge is way ahead of other cities with 52 per cent.

Can students create their own questions?
Here are mine:
1) What percentage of bikes stolen in Cambridge were taken from Parkside (near the police station)?
2) How many residents does Cambridge have?
3) Roughly how many people in Cambridge cycle at least once a week?

1) 62/2173 = 2.85%
2) (2173/16.6) * 1000 = 130,904 (This is very close to the official figure of 130,907 and rounding is likely to be the issue.)
3) Hmm. The article says that 52% of _adults_ cycle at least once a week. If we assume that the same is true of children then we would get about 65,000. If more children cycle then it will be a little higher.


Thursday, 10 November 2016

Quibans 48: Trump and the pollsters

From The Guardian:

This was a good starting point for a Quibans.

First of all, if there was a “15% chance” and that happened, it doesn’t necessarily mean the prediction was wrong – just that something less likely did occur.

I didn’t understand that comparison that was given here.

  1. What does this mean:  15% chance is roughly the same as rolling a six if you have two dice.
  2. Can you come up with a better comparison that uses coins, dice, cards, etc.

  1. It might mean

  • Getting a sum of 6 when you roll two dice.
  • Getting exactly one six if you roll two dice.
  • Getting one or more sixes if you roll two dice
A probability table might be useful here:

This shows the total of the two dice.  There is therefore a probability of 5/36 of getting a total of 6, which is about 13.9%

This table shows (in yellow) the ones where exactly one die shows 6.  10/36 = 27.8%

If we include a double-six too then that rises to 30.6%

The first of these (a total of 6) is nearest to 15%.  If it had said “getting 8 when you roll two dice” the probability would have been the same but the ambiguity would have been removed.

A better comparison
The probability of getting a 6 on a six-sided die is 1/6, which is 16.7%.  This is so much clear but is only a tiny bit less accurate than the suggestion in the article, so this is arguably better.
The probability of getting an Ace or a King from a pack of cards is 8/52 = 15.4%, which is very close and is also easy to understand.

[Other issues related to this also appear on my blog.  Find it here.]


Friday, 4 November 2016

Quibans 47: Shortest flights

From the Daily Telegraph:

The world's 10 shortest flights

A list of the world's 10 shortest regular commercial flights contains an international service for the first time, thanks to a 13-mile, eight-minute route between Switzerland and Germany that launched today.
In more excitement for fans of ludicrously brief forays into the sky, this week the world’s largest commercial aircraft - the Airbus A380 - was handed a regular outing on its shortest ever flight - a mere 235-mile jaunt between Doha and Dubai. Emirates, the airline behind the route, regularly uses the behemoth "superjumbos" on 8,000-mile odysseys, but instead thought it could be put to good use nipping between the two Middle Eastern cities. The journey could conceivably be made 40 times before the need to refuel.
But what of the rest of the world’s tiniest hops?
The competition to become a member of this club is not as fierce as it is at the other end of the scale, where airlines battle to out-fly each other to the longest non-stop schlep, but is nevertheless one of intrigue.
Topping the table is the wonderfully unconventional two-minute Loganair service between Westray and Papa Westray in Scotland’s Orkney Islands, which are separated by just 1.7 miles of land and sea. It can be completed in as little as 47 seconds, depending on the direction of the wind, and welcomed its millionth passenger this week. 
Featuring more obscure airlines, such as Pacific Wings and Guam’s finest carrier, Freedom Air, read on to find out about 10 flights that cover a total of just 91.5 miles - less than the distance between London and Bristol.

What a great story!  What can we work out?

  1. What is the average speed for each journey? (We could do this for the new route and also for the Orkney flight.)
  2. How far can the A380 fly without needing to refuel?
  3. What sort of correlation is there between the length of the flight in miles and the length in minutes?

Here is the data from the image, shown in a spreadsheet:

Cayman Brac
14 miles
10 mins
13 miles
20 mins
12 miles
5 mins
11 miles
15 mins
10.8 miles
10 mins
10.4 miles
6 mins
8.8 miles
10 mins
7 miles
15 mins
Caye Chapel
2.4 miles
2 mins
1.7 miles
2 mins

13 miles in 8 minutes means 1.625 miles in 1 minute and 97.5 mph.
1.7 miles in 47 seconds means 0.036170213 miles in 1 second and (multiply by 3600)
235 x 40 = 9400 miles.  But this might not be accurate, because it will presumably depend on wind-direction and wind-speed, there will be 40 take-offs and 40 landings rather than just one of each, which might use a different amount of fuel.  Presumably there are also rules about having to have a certain amount of fuel in reserve in case of bad-weather diversions, etc.
The pmcc is 0.58, which shows a weak positive correlation.  We could also draw a scattergraph.

Sunday, 30 October 2016

Quibans 46: Commuting in London

From the London Evening Standard:

One in three Londoners lose one day every week from commuting
A third of Londoners spend the equivalent of a whole day’s work every week commuting to and from the office, a study revealed today.
The findings were revealed in the annual London Commuter Index which looks at attitudes towards daily commutes.
It found that 30 per cent of workers spent an average of eight hours and 45 minutes a week on their commute, and 31 per cent were commuting for an average of six hours and 15 minutes.
Cost of travel was also shown to put workers under financial strain, with more than half spending an average of £1,500 on annual season tickets — the equivalent of paying the average London rent for about two months.
Three quarters of Londoners said reducing this expense would improve their journeys.
To beat rush-hour blues more than 64 per cent of commuters said they listened to music — and more than one in 10 practised mindfulness during their journeys.
About 61 per cent of people thought overcrowding was the worst part of commuting, while 28 per cent said it was delays. A survey of Tube passengers found a third liked the Victoria line best, while 36 per cent said the Central line was worst and would be improved by air conditioning.
The survey was carried out by season ticket loan company CommuterClub which quizzed 554 customers.


1)  How many people made each statement? 

2)  How long is each journey if the total in the week is 8 hours 45 mins?

3)  If everyone not mentioned has an average commute of 3 hours, what is the average commute for everyone in London?

4)  Which part of the article doesn’t seem sensible? 

5)  How might they have collected the data?
What can we do with the information about the cost of the tickets?


1)  How many people made each statement? 
% given
lower bound
upper bound
lowest integer
highest integer
[To work out the final two columns I used the Excel formula “Ceiling.Math” and “Floor.Math”]

2)  How long is each journey if the total in the week is 8 hours 45 mins:  This is 525 minutes.  Divide it by 10 (5 days of travelling there and back) to get 52 and a half mins per journey.

3)  Average:  5 hours 44 mins

4)  Which part of the article doesn’t seem sensible:   How will having a cheaper ticket make the journey more bearable?  Do people get grumpy at delays and overcrowding and then get even crosser because they are paying lots for the privilege?  Or would they be able to afford more takeaway coffee if the tickets were cheaper?

5)  How might they have collected the data:  Asking people at different stations.  Emailing those who use their website.  Are there any problems with this?

Quibans 45: Army fitness

From the Daily Telegraph:
Army fitness slips, but female soldiers close the gap on menThe proportion of Army soldiers failing fitness tests has nearly doubled in three years amid concerns that personnel are losing focus because they are not being sent to war.
Female troops are also narrowing the gap with their male counterparts after years of falling behind, The Telegraph can disclose.
In 2013, 7,120 - or 9.6 per cent - of the ### personnel who were made to complete personal fitness assessments failed at least one, according to figures released by the Ministry of Defence (MoD).
In comparison, between January and mid-September this year ### troops - around 17.7 per cent - out of a possible 63,910 tested failed one or more.
Three years ago, just over 12 per cent of females tested failed at least once, while 9.4 per cent of men did. This year, 18.8 per cent of women and 17.6 per cent of men failed one or more assessments.  
One recently retired Army officer suggested that soldiers were finding it more difficult to focus on keeping fit because they knew they were not going on operations.
He said: "Operations sharpen the mind and focus your efforts on keeping fit - the prospect of going to Afghanistan was certainly an incentive to get fit. After all you don't want to let your mates down by being unfit and not prepared. Being shot at is hard enough when you're fit, let alone when you're out of shape."

1)  There are some blanked out numbers to calculate.
2)  In what way are the female soldiers “closing the gap”?
3)  From these figures there clearly aren’t the same number of men and women who were tested.  What is the ratio of men to women?

If you have used any of the others that focus on rounded values, you could ask students at this stage what the upper and lower bound is for each blanked out number.

1)  Here is the unedited excerpt:
In 2013, 7,120 - or 9.6 per cent - of the 74,010 personnel who were made to complete personal fitness assessments failed at least one, according to figures released by the Ministry of Defence (MoD).
In comparison, between January and mid-September this year 11,300 troops - around 17.7 per cent - out of a possible 63,910 tested failed one or more.
Here are the upper and lower bounds for each calculation (note that the numbers of troops might be rounded to the nearest 10 or might be exact – we can’t tell).

2)  The gap between the performance of the men and the performance of the women is smaller.  Given that both have got worse that doesn’t seem like a result to be excited about!

3)  This is interesting.  I haven’t thought about this sort of thing before.  In 2013, 9.4% of the men and 12% of the women are 9.6% of the total.

If the number of men is m and the number of women is w then we have

9.4m + 12w = 9.6(m+w)
2.4w = 0.2m
12w = m
So there are 12 times as many men as women.

In 2016, 17.6% of the men and 18.8% of the women are 17.7% of the total.
17.6m + 18.8w = 17.7(m+w)
1.1w = 0.1m
11w = m
There are 11 times as many women as men. 

This might mean that there are now more women in the army than there used to be, or that more women were tested second time around, or that the percentages given were not exact …

Quibans 85: Crime and Police figures

From the Cambridge News: Violent crime in Cambridge has nearly doubled in a decade as police numbers drop 9 APR 2018 Bottom of For...