Homework #1
Examples Pertaining to the Additional Exercises Appearing Below
Example 1: An investment rose by 15% after a while, then rose by another 5%,
and finally rose by another 2%. What was the overall percentage change?
Solution: When we state that the investment rose by 15% at first, we mean
that, at some point, we checked our brokerage account and found that the val-
ue of the investment at that point was 15% higher than its value had been
when checked earlier. We aren’t saying anything about the details of the be-
havior of the value during the time in between (except that the value was
higher by 15% when we did check the account). Those details about the inter-
vening time, including its length, are irrelevant here.
Now, notice that, if we let A denote the earlier value of the invest-
ment, then after the 15% rise, the new value is
A + 0.15A = 1A + 0.15A = (1 + 0.15)A = 1.15A .
Notice that, from the point of view of doing arithmetic, the fourth of these
four expressions is simpler than the first since in the fourth expression we
need perform only one arithmetical operation (namely, multiply by 1.15),
whereas in the first expression we need to perform two arithmetical opera-
tions (namely multiply by 0.15 and then add A). Yes, we did have to perform
an arithmetical operation just to get the 1.15, namely we had to add 1 and
0.15; but that’s so easy, we could do it without any mental strain whatsoev-
er. The savings in work – one operation instead of two – might not seem like
much, but in the given example, where three changes occur, we will find it
much easier to do three arithmetical operations instead of six.
Anyway, what we have just seen is that, for a quantity A to rise by
15%, we simply need to multiply it by a magnification factor 1.15. (Don’t
forget the initial 1.) In the present example, the quantity 1.15A resulting
from that first change then enjoyed an additional change, namely a rise by
5%, so we will need to multiply by another magnification factor, namely 1.05.
And the quantity (1.05)(1.15)A resulting from that second change rose by an
additional 2%, so we will need to multiply by yet another magnification fac-
tor, namely 1.02. The final value we end up with is therefore
(1.02)(1.05)(1.15)A = 1.23165A = (1 + 0.23165)A =
A + (0.23165)A = A + (23.165%)A
(where we have written the successive magnification factors in order from
right to left, as they occur in the course of time). We see that the value
of our investment is up overall by 23.165%.
Notice how easy our calculation was since we worked in terms of magni-
fication factors: we had to perform only three multiplications, instead of
performing six arithmetical operations! Also notice that the overall rise of
23.165% is different from the 22% that we get by merely adding the three per-
centage rises, namely 15%, 5%, and 2%.
Example 2a: The value of an investment increases by 3% at first, and then in-
creases by another 2%. What is the overall relative change (or overall per-
centage change)?
Solution: To increase by 3% is to be multiplied by 1.03; to increase by 2% is
to be multiplied by 1.02. Since (1.03) (1.02) = 1.0506, the value has in-
creased by 5.06%.
Example 2b: What if the 2% rise came before the 3% rise? What difference
would that make to the overall relative change?
Solution: No change. After all, (1.03) (1.02) = (1.02)(1.03).
Example 3a: The value of an investment increases by 3% at first, and then de-
creases by 4%. What is the overall relative change (or overall percentage
change)?
Solution: To increase by 3% is to be multiplied by 1.03; to decrease by 4% is
to be multiplied by 0.96. Since (1.03) (0.96) = 0.9888 = 1 – 0.0112, the val-
ue has decreased by 1.12%.
Example 3b: What if the 4% drop came before the 3% rise? What difference
would that make to the overall relative change?
Solution: No change. After all, (1.03) (0.96) = (0.96) (1.03).
Example 4: The value of an investment increases by 100% at first, and then
increases by another 100%. What is the overall relative change (or overall
percentage change)?
Solution: To rise by 100% is to double. The value of the investment doubled
and then doubled again. Since 2 2 = 4, the value has quadrupled and therefore
has increased by 300%.
Note that in none of the four examples above do we get the exact answer
