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Can you solve these slightly harder maths problems?

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Last updated: January 14, 2019

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First submitted | January 14, 2019 |

Times taken | 263 |

Average score | 40.0% |

Rating | 4.71 |

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1.
What is the value of ^{2/5} + ^{2/50} + ^{2/500}

0.111
**✓**

0.222
**✓**

0.333
**✓**

0.444
**✓**

0.555
**✓**

2.
How many perfect squares have a 7 as their units digit?

0
**✓**

1
**✓**

2
**✓**

3
**✓**

4
**✓**

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?

What is the difference between the largest angle and the smallest angle?

30º
**✓**

40º
**✓**

50º
**✓**

60º
**✓**

70º
**✓**

4.
For what value of *x* is 64^{x} equal to 512^{5}?

6
**✓**

7.5
**✓**

8
**✓**

16
**✓**

40
**✓**

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?

How much water does the tank hold when full?

180 litres
**✓**

360 litres
**✓**

540 litres
**✓**

720 litres
**✓**

900 litres
**✓**

6.
What is the only positive integer that has exactly 8 divisors including 6 and 15?

21
**✓**

30
**✓**

45
**✓**

60
**✓**

90
**✓**

7.
How many numbers can be written as a sum of two different positive integers, each at most 100?

100
**✓**

196
**✓**

197
**✓**

198
**✓**

199
**✓**

8.
How many weeks are there in 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 minutes?

1
**✓**

2
**✓**

3
**✓**

4
**✓**

5
**✓**

9.
Suppose that 8^{n} = 27. What is the value of 4^{n}?

3
**✓**

4
**✓**

9
**✓**

13.5
**✓**

no such *n* exists
**✓**

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?

14
**✓**

28
**✓**

42
**✓**

56
**✓**

70
**✓**

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 2^{2} x 3^{3} x 5^{5} x 7^{7} is divided by 8?

2
**✓**

3
**✓**

4
**✓**

5
**✓**

6
**✓**

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?

What was her average speed over the whole ride?

13 km/h
**✓**

14 km/h
**✓**

15 km/h
**✓**

16 km/h
**✓**

17 km/h
**✓**

14.
How many of the following four equations has a graph that does __not__ pass through the origin?

*y* = *x*^{4} - *x*

*y* = *x*^{4} + 1

*y* = *x*^{4} + *x*

*y* = *x*^{4} + *x*^{2}

0
**✓**

1
**✓**

2
**✓**

3
**✓**

4
**✓**

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?

The mean of these eleven integers is 7. What is the median?

6
**✓**

7
**✓**

7.5
**✓**

8
**✓**

9
**✓**

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"Use it or lose it"in action! Haha, will try harder next time :o)PS

I'm sure you'll know already but the

× HTML entity should render the multiplication sign okay i.e. × vs xThese here should all work as they're from c. 2001, maybe the minus one is of use too?! Thanks! :o)

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 :)

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!