The Weighted Average
Or
When Can the Wrong Answer Be Correct?
A short essay by Herb Gross
In elementary school some of us were taught such adages as “He who
hesitates is lost”. In almost the same breath we were taught “Look before you
leap.”
This pair of sayings, “He who hesitates is lost” and “Look before you leap”, are
called dueling adages because each one is the contradiction of the other. That
is, if we look before we leap, then we are hesitating – which means we are lost.
And if we hesitate then we are obeying the adage “Look before you leap”, in
which case, according to this adage, we are not lost.
Thus, the logical question seems to be: “How can both adages be correct?”
The answer is relatively simple. Namely, in a given situation, we have to
decide which of the two adages is the correct one to use. Once that decision is
made, and only then, is the other adage the incorrect one to use. Clearly, the
adage this is correct in one case need not be correct in a different case.
The situation involving dueling adages can also be applied to mathematics. For
example, when people see a plus sign between two common fractions, it is
instinctive for many of them to think in terms of adding the numerators and
adding the denominators. Thus we might find someone saying that:
1 1 1+ 1 2
+ =
= .
2 3 2+3 5
• As tempting as it may seem to “add” this way, it violates our rule that we
only add adjectives if they modify the same noun.
• However, the 1 in
1
1
is modifying halves while the 1 in
is modifying
2
3
thirds.
• To see how this applies from a practical point of view, suppose partner A
is reimbursing you for 1 half of your business expenses and that partner
B is reimbursing you for 1 thirds of your business expenses. Clearly
between the two partners you are being reimbursed for more than a half
of your business expenses. However, 2 fifths of your business expenses
is less than half.
Weighted Averages
Page 2
• As another example, suppose we were to believe that
1 1 2
+ = . If we
2 3 5
assume that the three common fractions modify “hours”, we see that 1
half hour is 30 minutes, 1 third (of an) hour is 20 minutes and 2 fifths (of
an) hour is 24 minutes – and it is NOT true that 30 minutes + 20 minutes
5
5
of an hour =
of 60 minutes = 50
6
6
1 1
+
minutes which is the answer we obtain when we add
in the
2 3
= 24 minutes! On the other hand,
“correct” way.
Yet, while it is incorrect to add fractions by adding the numerators and adding
the denominators, the result we get by doing this has practical value in its own
right. Namely, when we combine common fractions this way we get what is
called the weighted average. The weighted average of two common fractions
is always a number that is between the two fractions we started with in value.
2
1
1
being the weighted average of and , notice
5
2
3
2
1
1
that is greater than but less than .
5
3
2
For example, with respect to
More generally, the weighted average of two (unequal) common
fractions is greater than the lesser fraction but less than the greater
fraction. In other words, the weighted average of two fractions “lies
between” the two fractions.
Special Case:
If the two common fractions represent the same number, then the
weighted average is the same as the two fractions.
In order not to confuse the sum of two common fractions with their weighted
average, let’s invent a new symbol for combining two or common fractions in the
way that yields their weighted average.
1 1 1+ 1 2
= or, in more general terms,
2 3 2+3 5
a c a+c
! =
b d b+d
For example, let’s define the symbol ! by ! =
Weighted Averages
Page 3
A correct use of the weighted average of two common fractions is when the
fractions that are being added do not modify the same noun.
 For example,
1
1
2
of 12 + of 18 = 6 + 6 = 12 = of 30, etc.
2
3
5
In terms of a corn bread diagram:
First of all, notice that the sizes of all five pieces below are the same.
(The shaded region =
1
1
of top corn bread. Notice that is modifying
3
3
3 pieces.)
(The shaded region =
1
of bottom corn bread.
2
Notice that
1
is
2
modifying 2 pieces.)
The shaded regions add up to
2
of the total number of pieces
5
because there are 5 equally sized pieces, 2 of which are shaded.
That is:
1 1 2
! =
3 2 5
From the diagram it is not difficult to see that the weighted average is
greater than the smaller common fraction but less than the greater
common fraction. More specifically, it’s easy to see that the shaded
regions add up to less than
1
of the 5 pieces; and by “annexing” an
2
extra piece to the bottom cornbread we see that the shaded regions
add up to more than
1
of the total number of pieces. That is:
3
Weighted Averages
Page 4
is a picture of the “annexed”
cornbreads. Notice that the shaded region is
Therefore the shaded region is more than
1
of the 6 pieces.
3
1
of the original 5 pieces.
3
An Example Using the Diagram Below
Suppose the top row represents Part 1 of a quiz in which the person
answered 1 of 3 questions correctly. The bottom row represents Part
2 of the same quiz in which the person answered 1 of 2 questions
correctly. The composite result is that the person answered 2
questions out of 5 correctly.
That is:
1 answer correct
1 answer correct
2 answers correct
!
=
3 questions asked 2 questions asked 5 questions asked
Notice that
2
1
1
1
(40%) is greater than (33 %) but less than (50%).
5
3
3
2
The following potentially “trick question” will illustrate the use of the weighted
average.
True of False:
A person drives from Town A to Town B at a constant speed of 20
miles per hour. He makes the return trip at a constant speed of 30
miles per hour. Since the average of 20 and 20 is 25, his average
speed for the round trip between Towns A and B is 25 miles per hour.
If your answer was “True”, you are not alone. However, the correct answer is
“False”.
Weighted Averages
Page 5
To make the arithmetic easier, let’s assume that the distance between the two
towns is 60 miles (which is a common multiple of 30 miles and 20 miles). Then
the round trip is 120 miles. At 20 miles per hour, the 60-mile trip from A to B
takes 3 hours and at 30 miles per hour the return trip takes 2 hours. Thus it
takes 5 hours for the person to drive the round trip of120 miles. But, it is the
case that 120 miles ÷ 5 hours = 24 miles per hours – not 25 miles per hour!
The answer lies in the fact that since the distance between A and B is the
same as the distance between B and A, the person spent more time driving at
20 miles per hour than at 30 miles per hour. If the person drove for the same
length of time at each speed then the correct answer would have been 20
miles per hour.
For example, suppose the person drove at 20 miles per hour for 3
hours and at 30 miles for another 3 hours. Then the total distance
traveled would have been 150 miles and the time it took would have
been 6 hours.
150 miles ÷ 6 hours = 25 miles per hour
What Does This Have To Do With The Weighted Average?
To answer this question let’s look at what happens for each mile of the round
trip. In going from A to B at 20 miles per hour, it takes the person 3 minutes to
drive 1 mile. However, on the back driving at 30 miles per hour, it takes him
only 2 minutes to drive the same mile. In summary:
1 mile
1 mile
1 mile + 1 mile
2 miles
24 miles
!
=
=
=
3 minutes 2 minutes 3 minutes + 2 minutes 5 minutes 60 minutes
So, while it is true that the “average” of 20 and 30 is 25, finding the average
was not the correct thing to do in the above problem.
An Application to a Problem Involving Fractions
A rather common problem is to find a fraction whose value is between the
values of two given fractions. The usual method that is taught is to find a
common denominator and then locate a common fraction with the same
denominator whose numerator is between the two given numerators.
Weighted Averages
Page 6
For example, to find a fraction between
1
3
and , we might elect to
3
8
rewrite the fractions as:
1 16
3 18
=
and =
3 48
8 48
17
1
3
in which case
is (half way) between and .
48
3
8
However, this type of computation can become cumbersome when
the denominators are large numbers; and this is where finding the
weighted average comes in very handy.
For example, to find a fraction between
14
32
and
, we need simply
407
109
find the weighted average of the two fractions. More specifically,
14 32
14 + 32
46
!
=
=
407 109 407 + 109 516
" 23 %
$# = 258 '&
Summary:
There is nothing wrong with combining two common fractions by
adding the two numerators separately and the two denominators
separately. What would be wrong is to call this the sum of the tow
fractions.
There are times when the correct answer to a problem requires that
we find the sum of two common fractions and there are other times
when the correct answer to a problem requires that we find the
weighted average of the two common fractions.
Memorizing an algorithm often tends to obscure the above fact. In
other words, for those who say that mathematical computation is
“cut and dried”, the fact is that determining what the correct
computation is in a certain problem is really important. Using a
correct formula in a situation when it doesn’t apply still results in a
wrong answer.