Tennis balls

Lene and Hans each have some tennis balls.

If Lene gives Hans 8 of her balls they would have an equal number.

If Hans gives Lene 8 of his balls, Lene would have 3 times as many balls as Hans.

How many balls does Lene have?

Write your answer in the box:

   tennis balls

The bike ride

Anne, Bente and Cecilie wanted to try three different courses for bike rides. They all averaged different rates of speed. Here are some facts about the courses.

• All of the racers began and ended at C.
• The length of CE is the length of AB.
• EF = CB
• ACB is an isosceles triangle; angles CAB and ABC are congruent.
• DAC is a right triangle.
• Angles CBF and CEF are right angles.
• CD is 5/4 AC.
• AB is AD.
• BF = 2 km
• EF = 8 km

All girls started at the same time.

Anne biked the CDAC route at a rate of 30 km per hour.
Bente biked the route around the isosceles triangle at a rate of 20 km per hour.
Cecilie biked the CEFBC route at a rate of 24 km per hour.

In what order did they finish, and what was the finishing time (in minutes) for the one who was first back at C?

Write A for Anne, B for Bente and C for Cecilie in the right boxes:

What is the side length?

The figure shows a regular hexagon with side length 10 cm. An equilateral triangle has corners on the mid points of three of the sides of the hexagon, as shown in the figure.

What is the length of the triangle sides?

Cross out your answer:

12,5 cm

15 cm

16 cm

17,5 cm

20 cm

Lunchtime

Half of the students who eat lunch in the cafeteria didn't buy anything at the cafeteria. Out of the remaining half who bought something,  bought fruit. Half of the students who bought fruit bought apples and one-fourth bought pears. The remaining 15 students bought bananas.

How many students had lunch in the cafeteria?

Write your answer in the box::

Tangents

8 different tangents are drawn om a circle. These tangents will divide the plane into some closed areas and some open areas.

How many open areas are there?

Cross out your answer:

a)    2

b)    4

c)    8

d)    16

e)    24

Body guards

Wonderful Violet and Fancy Fia, the world famous models, together had more than 10 but fewer than 30 bodyguards. One day, one of the bodyguards, Knuckles, decided to leave Wonderful Violet and join Fancy Fia. Now both the females had the same number of bodyguards.

Eventually, Brage rejoined Wonderful Violet. Also, Mandig decided to leave Fancy Fia and join Wonderful Violet. Now, both the females had a prime number of bodyguards.

How many bodyguards did each have now?

Write your answers in the boxes:

Wonderful Violet has  body guards.

What time is it?

At a certain time I looked at a 24-hour digital clock and noticed some things about the time:

1) The number in the hours section plus the number in the minutes section makes 60.
2) One of the digits in the minutes section is the square root of the other digit in the minutes section.
3) The sum of the digits in the minutes section is the number in the hours section reversed.

What was the time?

Write your answer in the boxes:

Problem 1

Tennis balls

Lene has 40 balls (and Hans has 24 balls). Then they both have 32 balls if Lene gives 8 of hers to Hans. If Hans gives 8 of his balls to Lene, he has 16 and she has 48 (=3∙16).

If Lene has L balls and Hans has H, we know that:

H+8=L-8 and

L+8 = 3 (H-8)

From the first equation we have: H – 8 = L – 24 that substituted in the second equation gives

L + 8 = 3 (L – 24) = 3L – 72

or

2L = 80

L = 40

Problem 2

The bike ride

The order is:  

1.      Anne – 48 minutes 
2.      Cecilie – 50 minutes 
3.      Bente – 60 minutes

BF = 2km and EF = 8 km. Then CB = 8km and CE = 4 km. AB = 2 ∙ CE = 4 km.   AD = 3/2 AB = 6 km. AC = CB = 8 km. CD = 10 km (find it either by measuring or using Pythagoras on triangle DAC).

Anne´s route is 10 km + 6 km + 8 km = 24 km.  
The time she uses is 24 km : 30 km/h = 4/5 ∙ 60 min = 48 min 
Bente´s route is 8 km + 4 km + 8 km = 20 km. 
The time she uses is 20 km : 20 km/h = 1 h = 60 min 
Cecilie´s route is 2 km + 8 km + 2 km + 8 km = 20 km 
The time she uses is 20 km : 24 km/h = 5/6 ∙ 60 min = 50 min

Problem 3

Triangles

The possibility is 25%.

The bundles consist of the following combinations of sticks:

1cm – 1cm – 10cm
1cm – 2 cm – 9cm
1cm – 3cm – 8cm
1cm – 4cm – 7cm
1cm – 5cm – 6cm
2cm – 2cm – 8cm
2cm – 3cm – 7cm
2cm – 4cm – 6cm
2cm – 5cm – 5cm
3cm – 3cm – 6cm
3cm – 4cm – 5cm
4cm – 4cm – 4cm

3 out of 12 bundles contain sticks that can be formed as a triangle. That means that the possibility to pull out such a bundle is 3/12=1/4=25% (the sum of the two shortest sticks must be greater than the longest stick).

Problem 4

What is the side length?

15 cm 

You find the solution by drawing perpendicular lines from two corners of the hexagon to a side of the triangle. That gives two right angle triangles on each side of a rectangle.  The two triangles can be put together to form an equilateral triangle with sides (10 : 2) cm = 5 cm. Then the sides of the equilateral triangle we seek are cm + 5 cm = 15 cm.

Problem 5 

Lunchtime

160 students had lunch in the cafeteria.

The 15 students who bought bananas, were ¼ of ¾, which means 3/16, of half of the students. The total number of students was

2 ∙ ((15 ∙ 16) : 3) = 160 

that had lunch in the cafeteria.

Problem 6

Tangents

There is 16 open areas.

Problem 7

Boy guards

Wonderful Violet has 11 bodyguards and Fancy Fia has 7 bodyguards.

In the beginning Wonderful Violet had two more bodyguards than Fancy Fia. Since they had less than 30 bodyguards all together, Fancy Fia had less than 14 bodyguards.

Now Wonderful Violet has 4 bodyguards more than Fancy Fia, since Fia has 1 less and Violet one more than in the beginning.

We must look for two prime numbers with difference 4. The prime numers are among these numbers:

2, 3, 5, 7, 11 og 13

Then only 7 and 11 are possible.

13 and 17 also have difference 4, but the sum is 30, and that is against the assumptions.

Problem 8

What time is it?

From clue 2 we know the time has to be : ??:00, ??:11, ??:24, ??:42 or ??:39.

For clue 1 to work the times would have to be : 60:00, 49:11, 36:24, 18:42 or 21:39. Of these only 18:42 and 21:39 could show up on a 24 hour clock.

4+2 does not equal 81 so 18:42 doesn't work for clue 3. 3+9 does equal twelve, so the time was 21:39.