Arithmetic+ 03
Problem A
I have purchased a share of some company XYZ for price P0=$100.
After a month the price was lower by ρ=10%.
Another month later the price went up by the same percentage ρ=10% and I sold this share.
Have I lost or gain money, or got even?
Solution A
After the first month the price dropped from P0=100 by ρ=10%, that is by P0·0.1 and was
P1 = P0 − P0·ρ =
= 100 − 100·0.1 = 90
After the second month the price went up from P1=90 by ρ=10%, that is by P1·0.1 and was
P2 = P1 + P1·ρ =
= 90 + 90·0.1 = 99
Therefore, since I bought a share for P0=$100 and sold it for P2=$99, my net result is
ΔP = P2 − P0 = −$1
which is a loss.
Notice that percentage of my loss was equal to percentage of my gain, but in dollars it was larger because the same percentage was applied to a larger amount on the way down, than to a smaller amount on the way up.
Problem B
I have purchased a share of some company XYZ for price P0=$100.
After a month the price was higher by ρ=10%.
Another month later the price went down by the same percentage ρ=10% and I sold this share.
Have I lost or gain money, or got even?
Solution B
After the first month the price went up from P0=100 by ρ=10%, that is by P0·0.1 and was
P1 = P0 + P0·ρ =
= 100 + 100·0.1 = 110
After the second month the price went down from P1=110 by ρ=10%, that is by P1·0.1 and was
P2 = P1 − P1·ρ =
= 110 − 110·0.1 = 99
Therefore, since I bought a share for P0=$100 and sold it for P2=$99, my net result is
ΔP = P2 − P0 = −$1
which is a loss.
Notice that percentage of my loss was equal to percentage of my gain, but in dollars it was larger because the same percentage was applied to a larger amount on the way down, than to a smaller amount on the way up.
IMPORTANT CONCLUSION:
If market price of a stock regularly goes up and down by the same percentage, the price per share goes down, investors lose money.
To be profitable, the price should go up more frequently or/and by a greater percentage than down.
Problem C
(a) Within the framework of Problems A, what the percentage of gain would neutralize the loss after price moving down by ρ=10%?
(b) Within the framework of Problems B, what the percentage of loss would neutralize the gain after price moving up by ρ=10%?
Solution C
(a) If we lost ρ=10%, the price became
P1 = P0 − P0·ρ =
= 100 − 100·0.1 = 90
To return back to $100 from $90, the growth in price σ should satisfy the equation
100 = 90 + 90·σ
from which
σ = 1/9 ≅ 0.1111
Therefore, if we add approximately 11.11% to P1=90, the final price will be
P2 = 90 + 90·0.1111 =
= 99.999 ≅ 100
(b) If we gain ρ=10%, the price became
P1 = P0 + P0·ρ =
= 100 + 100·0.1 = 110
To return back to $100 from $110, the diminishing in price σ should satisfy the equation
100 = 110 − 110·σ
from which
σ = 10/11 ≅ 0.0909
Therefore, if we subtract approximately 9.09% from P1=110, the final price will be
P2 = 110 − 110·0.0909 =
= 99.999 ≅ 100
Answer C:
(a) To neutralize a loss of 10% we need a larger (in %) gain of approximately 11.11%.
(b) To neutralize a gain of 10% we need a smaller (in %) loss of approximately 9.09%.
Problem D (suggested by Max)
Concentration of salt dissolved in water is measured in percentage of the mass of salt to the total mass of solution (salt + water).
Initially, a glass contained
Question 1: how much salt and how much water was in a glass?
This same glass was standing on a table for a week and part of water has evaporated, so the concentration of salt after a week became 2%.
Question 2: how much salt and how much water was in a glass after this process of evaporation?
Question 3: how much water has evaporated in a week?
Solution D
If 100 gram of solution contained 1% of salt, the mass of salt was
msalt = 100·0.01 = 1 (gram)
Hence, the mass of water was
mwater = 100 − 1 = 99 (gram)
Answer 1 (initially):
msalt = 1 (gram)
mwater = 99 (gram)
After some water has evaporated, the glass contained the same amount salt, that is
Since we know that it constitutes 2% of a solution, the total mass of solution must be 50 gram because
1/0.02 = 50 (gram)
To make 50 gram of solution with 1 gram of salt, we need
Answer 2 (after a week):
msalt = 1 (gram)
mwater = 49 (gram)
Answer 3:
That means,
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