Parallelogram 17 • Level 4 • 28 Dec 2023Chrismaths - Part 4

Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

a portmanteau word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.

It’s the end of Christmaths, so here is your last instalment of your Christmaths maths challenge paper!

Simon.

PS: I want to say thank you to the UK Mathematics Trust, who own the copyright to these questions.

1.

3 marks

How many minutes will elapse between 20:12 today and 21:02 tomorrow?

A) 50
B) 770
C) 1250
D) 1490
E) 2450
(Not answered)

From 20:12 today until 20.12 tomorrow is 24 hours, that is 24 × 60 = 1440 minutes. There are 50 minutes from 20:12 tomorrow to 21:02 tomorrow. This gives a total of 1440 + 50 = 1490 minutes.

2.

3 marks

An ‘abundant’ number is a positive integer N, such that the sum of the factors of N (excluding N itself) is greater than N.

What is the smallest abundant number?

A) 5
B) 6
C) 10
D) 12
E) 15
(Not answered)

In the IMC, it is only necessary to check the factors of the numbers given as the options.

However, to be sure that the smallest of these which is abundant, is the overall smallest abundant number, we would need to check the factors of all the positive integers in turn, until we find an abundant number.

The following table gives the sum of the factors of N (excluding N itself), for 1≤N≤12.

From this table we see that 12 is the smallest abundant number.

3.

3 marks

Which of the following has a value that is closest to 0?

A) 12+13×14
B) 12+13÷14
C) 12×13÷14
D) 12−13÷14
E) 12−13×14
(Not answered)

When working out the values of these expressions it is important to remember the convention (sometimes known as BODMAS or BIDMAS) that tells us that Divisions and Multiplications are carried out before Additions and Subtractions.

Some work can be saved by noting that the expressions A and B have values greater than 12, whereas the value of expression E lies between 0 and 12. So it must be C, D or E that has the value closest to 0.

Now, noting that 13÷14=13×41=43,
we obtain that the value of C is 12×13÷14=12×43=23;
that of D is 12−13÷14=12−43=−56;
and that of E is 12−13×14=12−112=512.

From these calculations we see that E gives the value closest to 0.

4.

4 marks

A rectangle with area 125 cm² has sides in the ratio 4:5. What is the perimeter of the rectangle?

A) 18 cm
B) 22.5 cm
C) 36 cm
D) 45 cm
E) 54 cm
(Not answered)

Since the side lengths of the rectangle are in the ratio 4:5, they are 4a cm and 5a cm, for some positive number a.

This means that the rectangle has area 4a×5a=20a2cm2.

Hence 20a2=125. So a2 = 12520=254, and hence a=52.

Hence the rectangle has perimeter 24a+5a=18a=18×52=45 cm.

5.

5 marks

The diagram shows a ceramic design by the Catalan architect Antoni Gaudi.

It is formed by drawing eight lines connecting points which divide the edges of the outer regular octagon into three equal parts, as shown.

What fraction of the octagon is shaded?

A) 15
B) 29
C) 14
D) 310
E) 516
(Not answered)

We consider the triangular segment of the octagon formed by joining two adjacent vertices, P and Q to the centre, O. For convenience, we show this segment, drawn on a larger scale, on the right, where we have added the lines RW,ST,TW and UV.

These lines are parallel to the edges of the triangle POQ, as shown and together with the lines RU and SV they divide the triangle OPQ into 9 congruent triangles, of which 2 are shaded.

Thus 29 of the segment is shaded. The same holds for all the other congruent segments of the octagon.

So 29 of the whole octagon is shaded.

Before you hit the SUBMIT button, here are some quick reminders:

You will receive your score immediately, and collect your reward points.
You might earn a new badge... if not, then maybe next week.
Make sure you go through the solution sheet – it is massively important.
A score of less than 50% is ok – it means you can learn lots from your mistakes.
The next Parallelogram is next week, at 3pm on Thursday.
Finally, if you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelograms.

Cheerio, Simon.