Mathematics Brain Teasers.

[Ozymandias]

6 hours ago, Harte said:

The disc centers form a 3-4-5 triangle. The right angle in that triangle sweeps out a fourth of the 1 meter circle area. The other two angles are not as clean (apprx. 53.13 degrees and it's complement.)

The area of the triangle is 6. The sum of the areas areas of the sectors of the discs is approximately 5.59. The difference is 0.41 m² .

That's with rounding. I didn't feel like putting in the time to express it in terms of pi. Left my good calculator in my classroom.

That's a nice problem and I'm certain there are several other ways to solve it.

Harte

That was quick, Harte. I bet you enjoyed that! Whatever about other solution methods your method is the way I had envisaged solving it.

Your result, 0.41m^2, is correct but without your calculator it is not very accurate. A more precise answer is 0.4643m^2 to the nearest cm^2.

Looking forward to your problem. 

[Harte]

4 hours ago, Ozymandias said:

That was quick, Harte. I bet you enjoyed that! Whatever about other solution methods your method is the way I had envisaged solving it.

Your result, 0.41m^2, is correct but without your calculator it is not very accurate. A more precise answer is 0.4643m^2 to the nearest cm^2.

Looking forward to your problem. 

The difference is from rounding. My good calculator lets me carry pi through the work. I had to use the one on the laptop, which gives only a numerical value for pi.

Had to jot down the values as I found them, and I'm too lazy to carry more than a couple of decimal places when I do that.

Regarding another problem, all I have are actual math problems from my lessons. If that's acceptable, I'll need to paste in a figure.

Here's an example. It might serve:

Harte

[Ozymandias]

11 hours ago, Harte said:

The difference is from rounding. My good calculator lets me carry pi through the work. I had to use the one on the laptop, which gives only a numerical value for pi.

Had to jot down the values as I found them, and I'm too lazy to carry more than a couple of decimal places when I do that.

Regarding another problem, all I have are actual math problems from my lessons. If that's acceptable, I'll need to paste in a figure.

Here's an example. It might serve:

Harte

The triangles PFD and MED are congruent with |PF| = 6 and |ME| = 3, i.e. in the ratio 2 : 1.

That means the side |FD| = 2 x |ED| and |PD| = 2 x |MD|  and |PM| = |MD| = 9

Applying Pythgoras' Theorem to triangle MED:      |ED|² + |ME|² = |MD|²     ->      |ED|²  +  3²  =   9²                       

                                                                                |ED|²  =  9²  -  3²   =  81  -  9  =  72

                                                                                |ED|   =   sq.rt(72)  =  8.4853  (to 4 Decimal Places)

 

[Harte]

Correct.

 

[Ozymandias]

14 hours ago, Harte said:

Correct.

I'll set another but this time with a change of tack in the hope of encouraging our economists to have a go:

I invest 50K at a guaranteed annual interest rate of 4.5%. How long to the nearest whole month must I leave it invested in order for it to double its nominal value? 

[danydandan]

50 minutes ago, Ozymandias said:

I'll set another but this time with a change of tack in the hope of encouraging our economists to have a go:

I invest 50K at a guaranteed annual interest rate of 4.5%. How long to the nearest whole month must I leave it invested in order for it to double its nominal value? 

Is this a compound interest thing?

So it's 50000*(1+0.045)^n=100,000.

In my head it's around 16 years/ so 192 months give or take.

But if I do it like a loan, I'm getting 270 months.

If I use I=P*r*t.

So I don't really know. 

[Ozymandias]

1 hour ago, danydandan said:

Is this a compound interest thing?

Yes,Dan.

1 hour ago, danydandan said:

.... 50000*(1+0.045)^n=100,000 ....

This is it. Just solve this for n. 

[danydandan]

8 hours ago, Ozymandias said:

Yes,Dan.

This is it. Just solve this for n. 

Eh I think I worked it out. But I had to use logarithms to work it out. 

n=ln(100000/50000)/ln(1+0.045)

n=15.7 years or 189 months. 

So I cheated and used OpenOffice to calculate it. Would have taken ages to work out in my head.

 

Edited April 13, 2019 by danydandan

[Ozymandias]

2 hours ago, danydandan said:

Eh I think I worked it out. But I had to use logarithms to work it out. 

n=ln(100000/50000)/ln(1+0.045)

n=15.7 years or 189 months. 

So I cheated and used OpenOffice to calculate it. Would have taken ages to work out in my head.

That's it, Dan - 15.747... years or 189 months. You used natural logs (base e) but the more usual base 10 logs give the same answer.

Your turn.

[danydandan]

On 13/4/2019 at 10:04 AM, Ozymandias said:

That's it, Dan - 15.747... years or 189 months. You used natural logs (base e) but the more usual base 10 logs give the same answer.

Your turn.

Ok I'll set a simple one.

Eh............ Never considered 10logs.

My question:

Using two sevens (7) and two threes (3)...... Show me an equation that the result equals twenty four (24).

[Ozymandias]

20 hours ago, danydandan said:

Ok I'll set a simple one.

Eh............ Never considered 10logs.

My question:

Using two sevens (7) and two threes (3)...... Show me an equation that the result equals twenty four (24).

I've figured this one out but am holding back my answer to see if others might like to post a solution (although I doubt it).

[danydandan]

I'll give a hint. 

7,3,7,3.

[Ozymandias]

On 15/04/2019 at 12:26 PM, Ozymandias said:

I've figured this one out but am holding back my answer to see if others might like to post a solution (although I doubt it).

24 = 3(7) + 3     which is one 7 shy

24 = 3(7) + 3(1)

24 = 3(7) + 3(7/7)

24 = 7[3 + (3/7)]

I'm done now. No further contributions from me. There's absolutely no interest.

[danydandan]

1 hour ago, Ozymandias said:

24 = 3(7) + 3     which is one 7 shy

24 = 3(7) + 3(1)

24 = 3(7) + 3(7/7)

24 = 7[3 + (3/7)]

I'm done now. No further contributions from me. There's absolutely no interest.

Yeah that's the shot. 

Yeah I'll post periodically when I get a eureka moment for a teaser.

Some savage weather here at the moment Ozy, isn't there?

[Ozymandias]

5 hours ago, danydandan said:

Yeah that's the shot. 

Yeah I'll post periodically when I get a eureka moment for a teaser.

Some savage weather here at the moment Ozy, isn't there?

I'm sure there is in Dublin  but I've been in West Cork for the last week - it's been mixed. Some lovely sunny days between being overcast or low cloud mist.  But we'll take it all as we find it.

[Ozymandias]

Anyone interested in reviving this thread?

Problem:  [xe^(pi)i + x] / x^2 = 0. Solve for x.

 

[Ozymandias]

On 9/2/2020 at 4:27 PM, Ozymandias said:

Anyone interested in reviving this thread?

Problem:  [xe^(pi)i + x] / x^2 = 0. Solve for x.

 

bump

[ant0n]

On 9/2/2020 at 5:27 PM, Ozymandias said:

Anyone interested in reviving this thread?

Problem:  [xe^(pi)i + x] / x^2 = 0. Solve for x.

 

Let's consider x is a real number.

e^(i.pi) = cos(pi) + i.sin(pi) = -1

[xe^(pi)i + x] / x^2 = 0: x must be different from zero as one cannot divise by zero here.

[xe^(pi)i + x] / x^2 = 0 -> 0/x² = 0 -> multiplied by x² (with x different from 0) -> 0 = 0.x² = 0, which is true for any real number x different from zero.

 

Hence [xe^(pi)i + x] / x^2 = 0 is true for any real number x different from zero.

 

[Ozymandias]

2 hours ago, ant0n said:

Let's consider x is a real number.

e^(i.pi) = cos(pi) + i.sin(pi) = -1

[xe^(pi)i + x] / x^2 = 0: x must be different from zero as one cannot divise by zero here.

[xe^(pi)i + x] / x^2 = 0 -> 0/x² = 0 -> multiplied by x² (with x different from 0) -> 0 = 0.x² = 0, which is true for any real number x different from zero.

 

Hence [xe^(pi)i + x] / x^2 = 0 is true for any real number x different from zero.

Well done, Anton. Your turn to set a problem.

[ant0n]

Alright

EDIT: there are NO parentheses. There are only simple operators to guess (in total, there are several multiplications, divisions and additions but only one substraction).

 

Edited September 6, 2020 by ant0n

[Ozymandias]

27 minutes ago, ant0n said:

Alright

 

I am assuming there are no parentheses in the solution. It is simply a matter of replacing each '?' with a simple operator.

[ant0n]

3 minutes ago, Ozymandias said:

I am assuming there are no parentheses in the solution. It is simply a matter of replacing each '?' with a simple operator.

Yes, exactly. Thanks for questioning that. I've just added an edit to the main topic:

 

EDIT: there are NO parentheses. There are only simple operators to guess (in total, there are several multiplications, divisions and additions but only one substraction).

[ant0n]

33 minutes ago, ant0n said:

Alright

EDIT: there are NO parentheses. There are only simple operators to guess (in total, there are several multiplications, divisions and additions but only one substraction).

 

 

[Ozymandias]

I could not find a subtraction to suit the expression. My solution :

2019 = 10 x 9 x 8 x 7 / 6 / 5 x 4 x 3 + 2 + 1

[ant0n]

34 minutes ago, Ozymandias said:

I could not find a subtraction to suit the expression. My solution :

2019 = 10 x 9 x 8 x 7 / 6 / 5 x 4 x 3 + 2 + 1

 

"2019 - 0 = 2019" is a substraction.

 

Well done! Your turn.

Edited September 6, 2020 by ant0n