Parallelogram 32 • Level 2 • 11 Apr 2024Chocolate algebra

Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

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

Tackle each Parallelogram in one go. Don’t get distracted.
When you finish, remember to hit the SUBMIT button.
Finish by Sunday night if your whole class is doing parallelograms.
Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

1.

6 marks

1.1 Box P has p chocolates and box Q has q chocolates, where p and q are both odd and p>q.

What is the smallest number of chocolates which would have to be moved from box P to box Q so that box Q has more chocolates than box P?

q−p+22
p−q+22
q+p−22
p−q−22
q+p+22
(Not answered)

Because p and q are both odd, p+q is even. If the chocolates were shared equally between the boxes there would be 12p+q chocolates in each box.

So the least number of chocolates there must be in box Q if it is to have more chocolates than box P is 12p+q+1.

Because box Q starts with q chocolates in it, to end up with 12p+q+1 in box Q the number of chocolates we need to transfer from box P to box Q is

p+q2+1−q=p−q2+1=p−q+22

2.

5 marks

2.1 The diagram shows two arrows drawn on separate 4 cm × 4 cm grids. One arrow points North and the other points West.

When the two arrows are drawn on the same 4 cm × 4 cm grid (still pointing North and West) they overlap. What is the area of overlap?

4 cm²
4 12 cm²
5 cm²
5 12 cm²
6 cm²
(Not answered)

By drawing one arrow on top of the other, as shown, we see that the region of overlap covers the whole of 4 of the 1 cm × 1 cm squares into which the grid is divided, and 4 halves of these squares. So the area of the overlapping region is 4 + 4 (12) = 6 cm².

3.

5 marks

3.1 Which of the following numbers is divisible by 7?

111
1111
11111
111111
1111111
(Not answered)

The problem may be solved by dividing each of the alternatives in turn by 7, but the prime factorisation of 1001, i.e. 1001=7×11×13, leads to the conclusion that 111 111, which is 111×1001, is a multiple of 7.

4.

5 marks

4.1 What is the remainder when the square of 49 is divided by the square root of 49?

0
2
3
4
7
(Not answered)

Because 49=7×7, it follows both that 49=7 and that 7 is a factor of 49.

Therefore 7 is also a factor of 492.

Hence the remainder when the square of 49 is divided by the square root of 49 is 0.

5.

6 marks

5.1 After playing 500 games, my success rate at Spider Solitaire is 49%. Assuming that I win every game from now on, how many extra games do I need to play in order that my success rate increases to 50%?

1
2
5
10
50
(Not answered)

Since I have won 49% of my first 500 games, so far I have won 49100×500=49×5=245 games. So I have lost 500−245=255 games. I need now to win enough games so that I have won as many as I have lost.

So, assuming I win every game from now on, I need to win 255−245=10 more games.

Another approach is to say that I am going play and win W more games. At that point, the number of games that I win will be:

49100×500+W500+W=50%or0.5

Therefore 245+W500+W=0.5

245+W=250+0.5W
0.5W=5
W=10

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.