 # Mathematics Multiple Choice #2

Can you solve these slightly harder maths problems?
Quiz by Stewart
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Last updated: January 14, 2019
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 First submitted January 14, 2019 Times taken 160 Average score 40.0% Rating 4.67 Report this quiz Report
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1. What is the value of
2/5
+
2/50
+
2/500
2. How many perfect squares have a 7 as their units digit?
3. The angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6.
What is the difference between the largest angle and the smallest angle?
4. For what value of x is 64x equal to 5125?
5. A water tank is
5/6
full. When 30 litres of water are removed from the tank, the tank is
4/5
full.
How much water does the tank hold when full?
6. What is the only positive integer that has exactly 8 divisors including 6 and 15?
7. How many numbers can be written as a sum of two different positive integers, each at most 100?
8. How many weeks are there in 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 minutes?
9. Suppose that 8n = 27. What is the value of 4n?
10. The positive integers a, b and c are all different. None of them is a perfect square, but all the products ab, ac and bc are squares. What is the least value that a + b + c can be?
11. What is a half of a third, plus a third of a quarter, plus a quarter of a fifth?
12. What is the remainder when 22 x 33 x 55 x 77 is divided by 8?
13. On a training ride, Laura averages speeds of 12 km/h for 5 minutes, then 15 km/h for 10 minutes and finally 18 km/h for 15 minutes.
What was her average speed over the whole ride?
14. How many of the following four equations has a graph that does not pass through the origin?
y = x4 - x
y = x4 + 1
y = x4 + x
y = x4 + x2
15. The following eleven integers are written in ascending order:
1, x, x, x, y, y, y, y, 8, 9, 12

The mean of these eleven integers is 7. What is the median?
+1
Level 49
Jan 14, 2019
Oh dear... only got 8/15! "Use it or lose it" in action! Haha, will try harder next time :o)

PS

I'm sure you'll know already but the &times; HTML entity should render the multiplication sign okay i.e. × vs x

+2
Level 68
Jan 15, 2019
I did not know that! I was so worried trying to find a simple enough html for a latex x that I didn’t realise I could distguish them another way!
+1
Level 49
Jan 15, 2019
I worked with media-neutral XML files for a bit and had to know lots of these! If the files were later used in e.g. a TTS engine it would correctly "read/speak" symbols etc.

These here should all work as they're from c. 2001, maybe the minus one is of use too?! Thanks! :o)

+1
Level 59
Jan 14, 2023
Hardcore stuff; love it.
+1
Level 57
Jan 20, 2023
Is there a trick for the a + b + c question? Spotted most of the tricks (aside from the remainder question, assume it has something to do with prime factors given the way it's written) but couldn't see how to do this one aside from brute force and I ran out of time.
+1
Level 68
Jan 20, 2023
Yes there is!

So you know that the minimum a, b or c can be is 2, since 1 is square. IF a = 2, then that means b = 2 * a square, and c = 2 * a square. This gives us 2, 2 * 4 and 2 * 9 = 2, 8, 18. Added together gives us 28.

However, the real trick is spotting that all of a, b and c must be multiples of squares, and all the same multiple of different squares. Otherwise the condition ab, bc and ca being squares won't work.

I'll let you work out how to see those bits, if not feel free to comment again and I'll explain :)

+1
Level 68
Jan 20, 2023
Regarding the remainders, this one is actually really simple, that nobody realised it :)

The multiplication is clearly a multiple of 4, since we have 2^2 at the start. This means that, when dividing by 8, we can only get either remainder 4 or 0. Since none of the other prime factors are divisible by 2, it means it is not divisible by 8, so it must be remainder 4!