Problems and Solutions

Jai Ganesh · 2005-07-19 16:51:33

Problem #6

A salesman's commission is 5% on all sales up to $10,000 and 4% on all sales exceeding this. He remits $31,100 to his parent company after deducting his commission. His sales was worth $_________.

mathsyperson · 2005-07-19 19:06:43

MathsIsFun · 2005-07-19 19:16:37

S = Sales
C = Commission
R = Remittance to Company after sales

For S <= $10,000, C = 0.05S
For S > $10,000, C = 500 + 0.04(S-10,000)

R = S - C

Now, we know that R = 31,100

So ... sorry, gotta go!

Jai Ganesh · 2005-07-19 19:25:30

31,100 = S - [500+0.04(s-10,000)]
31,100 = S -[500 + 0.04s - 400]
31,100 = S - 100 - 0.04s
31,200 = 0.96 S
S = 31,200 / 0.96
S = 32,500

Jai Ganesh · 2005-07-19 21:58:49

#7
The ratio of annual incomes of two persons is 9:7 and the ratio of their annual expenditure is 4:3. Each of them saves $2000 yearly. Find their annual incomes.

mathsyperson · 2005-07-20 03:10:35

Wow, four simultaneous equations...

MathsIsFun · 2005-07-20 09:41:07

BY THE WAY, since my little {hide}{/hide} tag is getting used a bit, it will soon become too late to rename it to {popup} or {boing} or whatever.

So, is "hide" the best name, then (even though it is just hidden under a little button)

im really bored · 2005-07-20 11:14:20

For MathIsFuns lake problem I got 6,931,471.8055994530941723212145818 liters of water that needs to be added.

Zach · 2005-07-20 12:37:45

There's a Hide tag?!

mathsyperson · 2005-07-20 18:18:16

im really bored wrote:

For MathIsFuns lake problem I got 6,931,471.8055994530941723212145818 liters of water that needs to be added.

Was that with calculus or a more accurate version of my method?

im really bored · 2005-07-20 19:21:56

Nope no calculas or your method, I just got a tip from a friend that its related to the natural log of 2, which is 0.69314718055994530941723212145818.

Jai Ganesh · 2005-07-20 19:27:16

mathsyperson wrote:

Wow, four simultaneous equations...

There are only two:-
9x - 4y = 2,000
7x - 3y = 2,000

Solving for 'x', the annual income of the two can be calculated. They would be 9x and 7x.

Last edited by Jai Ganesh (2005-07-20 19:27:58)

Jai Ganesh · 2005-07-21 16:35:07

Which is greater?
Cos (Sin A) or Sin (Cos A)?

mathsyperson · 2005-07-21 17:57:18

Degrees, radians or grads?

I can tell just by thinking about it the answer for degrees and grads, but if you're working with radians, there might be a few exceptions, though I doubt it.

MathsIsFun · 2005-07-21 18:14:40

Oooh ... (reaches for Excel) ...

cos(sin(a)) fluctuates between 0.734664609.. and 1
sin(cos(a)) fluctuates between 0.841470985.. and -0.841470985..

They cross over first at about 28.9° (0.5031796.. radians), then back again at 331.1° (5.779671333.. radians)

Now, are 0.734664609.., 0.841470985.. or 0.5031796.. magical numbers?

mathsyperson · 2005-07-21 18:51:40

Plotting cos(sin a) against sin(cos a) in Excel gives a smiley mouth. The important thing, though, is to see whether one is always bigger than the other, and this graph cannot do that.

I plotted radian value against cos(sin a)-sin(cos a) and I got a beautiful swirly pattern. This pattern always stayed above 0, so cos(sin a) is always bigger than sin(cos a).

Jai Ganesh · 2005-07-21 20:52:37

Cos(Sin A) is always greater than Sin (Cos A),
as rightly pointed out by Mathsy!
We shall test this for three values in the first quadrant, in degrees.
First, 0.01°.
Cos (Sin A) > Sin(CosA).
Next, for 45°
Again, Cos (Sin A) > Sin(CosA).
Finally, for 89.9°
Yet again, we get the same result.
I shall try to search for a proof, or prove it myself!

MathsIsFun · 2005-07-21 21:02:16

Ooops, my bad in some calculations.

Yes, I see same result now.

I will make up for my mistake by posting this nice graph that mathsy talked about:

Jai Ganesh · 2005-07-21 22:07:33

Thank you, Admin, for the neat graph.

Jai Ganesh · 2005-07-22 21:55:17

I just remembered a question asked to me when I participated in a quiz when I was 14 :

A three digit number is written on a window glass,
the difference between the number and as it can be seen from the other side is 693,
what's the number?

mathsyperson · 2005-07-22 23:04:50

Well, it has to be made up of 0's 1's and 8's (and maybe 2's and 5's that swap when reversed, depending on the writing style)

I've got

, but there might be others.

Jai Ganesh · 2005-07-29 19:54:39

Two candles A and B, of equal height but different circumferences, burn for 4 hours and 3 hours respectively. If the two are lit at the same time, after what time would one candle be half the height of the other?

mathsyperson · 2005-07-29 22:02:39

This is the closest answer I could find. I get about 3 minutes more than what the picture says.

Jai Ganesh · 2005-07-31 18:54:04

As always, you are corect, Mathsy!
I shall show how I did it.
Assume both candles are 'h' inches in height.
The height of the first candle reduces by h/4 every hour and that of the second reduces by h/3.
Lets assume in time 't' hours, the first is twice the height of the second.
Therefore,
h - t(h/4) = 2 [h - t(h/3)]
h - ht/4 = 2h - 2ht/3
h = 2ht/3 - ht/4
h = 5ht/12
Cancelling h on both sides,
1 = 5t/12
or t = 12/5
The unit we had taken was hours,
therefore, in 12/5 hours, that is 2 2/5 hours, one will be double the height of the other.
Simplified, it is 2 hours and 24 minutes.

Jai Ganesh · 2005-08-02 16:10:54

Problem #n
The distance between two cities is 840 kilometres.
If I start driving from one city to the other at a speed
5 kilometres/hour more than my normal speed,
I save 3 hours. What is my normal speed?

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