
3 types of percent computation problems Finding
the part Finding the percent  Finding
the whole  Adding percents
 Further Applications  Calculator
short cuts  Quick quiz 
We
have already looked at a common application of percent in the section,
Key Ideas: Percentage
increases and decreases. These simple examples fall into the categories
of problems below, but were introduced earlier because they are problems
with which most of us are familiar.
3
types of percent computation problems
There
are 3 basic types of percent computation problems which can be represented
using the following expression,
r%
of b is p
where
r is the percent, b is the whole or base quantity and p is the part
of b.
2% 
of
200 
is
4 
r% 
b 
p 
percent 
whole 
part 
In
each type of percent problem we want to find one of the following:
 the percent (r)
 the whole (b)
 the part (p)
Using
the dualscale number line we can easily see which piece of information
is missing in each type of problem. Please see Meaning
and Models, Percent models and Key
Ideas for more details
Because
of the relationship of percent to common fractions and decimals there
are a number of approaches we can use as a starting point to solve
percent problems:
Benchmark
 Recognise common percents, such as 50% and then perform the
computation if necessary. 
Fractions
 Express the percent as a fraction, such as 54/100, and then
perform the computation using fraction algorithms if necessary. 
Decimal
 Express the percent as a decimal, such as 0.54, and then perform
the computation using decimal algorithms if necessary. 
Unitary
 Find 1% of the quantity and then perform computation if necessary. 
Calculator
 Use
short cut methods provided on some calculators 
We
often use the unitary or benchmark approach when working out percent
problems in our heads. We will look at how we use these methods to
solve each problem type below.
Finding
the part
Example
1: What is 5% of 80 kilometres?
What
is r% of b? We need to find p, the part.
There
are many different ways in which we could solve this problem. If we
used either of the unitary or benchmark methods described below we
would usually do them in our heads.
Estimating
the answer
Before
solving a problem it is a good idea to make an estimate of our answer
so that we can check it for reasonableness when we have completed
the problem. In this case we could note that 5% of 100 is 5 so we
would expect our answer to be less than 5.
Working
Out 
Thinking 
0.8
is 1% of 80
0.8
x 5 = 4
4
is 5% of 80

Unitary
approach
80 is 100% of 80, so 0.8 is 1% of 80.
Now
we have 1% of 80 we can multiply by 5 to get 5%.

8
is 10% of 80
8
÷ 2 = 4
4
is 5% of 80

Benchmark
approach
I
know that 8 is 10% of 80, so to find 5% I divide 8 by 2.

4
is 5% of 80
or,
p
= 4
4
is 5% of 80

Fraction
approach 1
We
could express 5% as a fraction out of 100 and simplify it into
the equivalent fraction 1/20. We know that 1/20th of 80 is 4.
Fraction
approach 2
We
could also say 5 out of 100 equals p out of 80 since we want
to find out what part of 80 equals 5%. We then do the same to
both sides of the equation to find p.

5%
of 80 = 0.05 x 80
4
is 5% of 80

Decimal
approach
We could express 5% as a decimal, 0.05, and then multiply by
80.

Using
my calculator,
Enter
80
Press x
Enter 5
Press %
Answer:
4
4
is 5% of 80

Calculator
approach
We
could use the shortcut method provided on some calculators which
have a '%' button.
NOTE:
the way you enter the numbers and symbols will depend on your
calculator  you will need to experiment. Note also that you
are not necessarily keying in a correct mathematical sentence,
but rather using a shortcut method provided by your calculator.

Example
2: Movie  simple example of finding the percentage of hard
numbers using a calculator.
What
is 12.3% of $1294?

We
can summarise what we have been doing when using the decimal or fraction
approach into a formula.
p
= (r/100) x b,
where
p is the part, r is the percent and b is the whole.

In
order to find the part we need to multiply the whole amount
by the percent.
Example: What is 15% of $300?
p= (15/100) x 300
p= 45
15% of $300 is $45

Finding the percent
Example
3: What percent is $4 of $80?
What
percent is p of b? We need to find r, the percent.
Estimating
our answer
We
will first estimate our answer. 10% of $80 is $8 (10 x 8 = 80) so
we know our answer will be less than 10%.
Working
Out 
Thinking 
4
is 5% of 80
or,
4
is 5% of 80

Fraction
approach
1
Our final answer will be as a fraction with a denominator of
100. How do we get there?
We
can simplify the fraction to a form which we can more easily
convert into hundredths and then express as a percent.
Fraction
approach
2
We
could also say 4 out of 80 equals r out of 100. We then do the
same to both sides of the equation to find r.

4
is 5% of 80

Decimal
approach
We
can express 4/80 as a decimal using our calculator and then
express as a percent.

0.8
is 1% of 80
4
is ?% of 80
4
÷ 0.8 = 5
4
is 5% of 80.

Unitary
approach
80
is 100% of 80, so 0.8 is 1% of 80.
We
know that 0.8 is 1% of 80 and we want to know what percent 4
is of 80. We could find out how many 0.8 's there are in 4.
Four divided by 0.8 is 5.
NOTE:
This approach is not as suited to this problem because we still
have to do a lot of working out to get our answer. We can see
that all other approaches get to the answer quicker.

8
is 10% of 80
4
is 5% of 80

Benchmark
approach
I
know that 8 is 10% of 80. 4 is half of 8, so 4 must be 5% of
80.

Using
my calculator,
Enter
4
Press /
Enter 80
Press x
Enter 100
4
/ 80 x 100 = 5%
4
is 5% of 80
Shortcut:
Enter
4
Press /
Enter 80
Press % gives 5 directly

Calculator
approach
4 is how many percent of 80?
In
the first case we are not using the '%' button but are multiplying
4/80 by 100 to give us the percent.
We
could use the shortcut method provided on some calculators which
have a '%' button.

Example
4: Movie  finding the percent using the fraction approach.
48 people out of 360 people studied drink Brand A softdrink regularly.
What percent is this?
Movie
 verifying our answer to example 4 using the calculator.
Finding the whole
Example
5: 4 is 5% of what number?
p
is r% of what number? We need to find b, the whole.
Estimating
our answer
We
will first estimate our answer. 4 is 1% of 400, so 4 will be 5% of
a much smaller number. We know that our answer will be a lot less
than 400.
Setting
Out 
Thinking 
4
is 10% of 40
4
is 5% of (2 x 40)
4
is 5% of 80

Benchmark
approach
4
is 5% of what number. We know 4 is 10% of 40 (4 x 10 = 40),
so for 4 must be 5% of two times 40.

4
x100 = 5 x b
5b
= 400
So,
4 is 5% of 80

Fraction
approach
We can express the problem in terms of fractions. 4 divided by
some number, b, is equivalent to 5/100 (5%). We can then do the
same to both sides of the equation to find b.

5%
= 0.05
b=80
4
is 5% of 80

Decimal
approach
We
can express 5% as a decimal fraction, 0.05. 4 divided by b is
equivalent to 0.05 (5%). We can then do the same to both sides
of the equation to find b.

4
is 1% of 400
4
is 5% of (400 ÷ 5)
4
is 5% of 80

Unitary
approach
4 is 5% of what number? We know 4 is 1% of 400, so 4 must be
5% of a number which is 5 times smaller.

4
is 5% of b
Enter 4
Press / (divide)
Enter 5
Press % gives 80 directly

Calculator
approach
There is a shortcut for this problem using
my calculator.

Example
6: Movie  finding the whole amount.
Con noticed that the GST he paid on his TV was $36. Currently, the
GST is 10% of the preGST price. How much did the TV cost excluding
GST? How much did it cost including GST?
Example
7: Movies of a worded problem, finding the percent, percentage and
the whole.
Movie
A: Finding the percent.
Sarah shot 7 points out of her team's total of 29 points at the basketball
match last Saturday, what percent of points did she shoot?
Movie
B: Finding the part.
Mary
shot 17% of points, how many points did she shoot?
Movie
C: Finding the whole  estimating our answer.
If,
in another game, Mary and Sarah shot 4 and 6 goals respectively, which
amounted to 27% of their team's total score, what was their team's
total score?
Movie
D: Finding the whole  calculating our answer
Adding
percents
Example
8: If 30% of Grade 10 students come to school by bus and 40% of Grade
10 students come to school by train, what percent of students come
to school by public transport? (NOTE: Assume only one form of transport
is used by each student)
Thinking

Here
I am talking about one group of students  Grade 10 students.
I know 30% of all students come to school by bus and 40%
of all students come to school by train. Both these forms
of transport are public transport, so 40% + 30% = 70% of all
students come to school by public transport. 

Example
9: If Classes A and B each contain the same number of students and
30% of students in Class A are boys and 40% of students in Class B
are boys, what percent of Classes A and B combined are boys?
Thinking

Here
there are 2 equal groups of students, Class A and Class
B. I know that 30% of students in Class A are boys and 40% of
students in Class B. If both classes had 100 students, there
would be 30 + 40 boys altogether out of the total of 200 (=
100 + 100) students.
70
out of 200 is the same as 35 out of 100 (because 70/200 = 35/100
are equivalent fractions).
So
we can say that while 30% of students in Class A are boys and
40% of students in Class B are boys, 35% of Classes A and
B combined are boys.



In
the first example, example 8, we are referring to one group of students
so we can add the percents of students using the bus and train to
get to school to get the percent of students using public transport.
In the second example, example 9, we are talking about two equal
groups of students.
NOTE:
A common mistake is to add percents for groups which aren't equal.
This is not correct. When we want to find the combined percent of
more than one group, where the groups are of different size, we need
to take into account the proportions of the groups.
Example
10: If 30% of students in Class A are boys and 40% of students in
Class B are boys, and Class A has 30 students in total and Class B
has 20 students, what percent of Classes A and B combined are boys?
Thinking

Here
Class A and Class B do not have the same number of students.
I have to convert the percentages to numbers, add the numbers
and then convert back to a percent. In this case Class A has
9 boys and Class B has 8 boys. Altogether, there are 17 boys
out of a total of 50 pupils. This is 34% of the combined classes.



Further Applications
Making
comparisons
Although
most people understand the concept of percent, they still get confused
when working with more complicated percent problems, particularly
those which ask us to compare the percents of different quantities.
The following examples work through these types of problems.
Example
11: Comparing percents.
I have a 2 litre carton of milk of Brand A which is 95% fat free and
1 litre carton of Brand B which is 98% fat free, if I pour one glass
of each, which glass of milk contains more fat?
Thinking

Although
the 2 litre carton of milk contains more milk than the 1 litre
carton the fat free percent is 95%. This means that in the 2
litre carton there are 95 parts out of 100 parts that do not
contain fat (or 5 parts out of every 100 which do contain fat).
The
1 litre carton is 98% fat free. This means that in this carton
98 parts out of 100 parts do not contain fat (or 2 parts out
of every 100 do contain fat).
If
we are comparing 2 glasses of milk, the glass containing Brand
A milk will have 95 parts out of 100 fat free and the glass
containing Brand B milk will have 98 parts out of 100 fat free.
Therefore the glass of Brand A milk contains more fat.

Example
12: Finding the percent of combined quantities.
Lee
has 2 different sized packets of coloured jelly beans: one is 100g
and the other 250g. Each packet has 50g of black jelly beans. Lee
makes up a bowl with both packets of jelly beans. What is the percent
of black jelly beans in the bowl?
Thinking

Seeing
as the packets have been mixed in the bowl we are interested
in the percent of both packets together

we know that both packets have 50g of black jelly beans, so
combined there is 100g of black jelly beans
 we know that the total weight of the packets is
100g + 250g = 350g
 so we have 350 g of jelly beans in the bowl, 100 g of which
are black jelly beans
To
find the percent of black jelly beans in the bowl we need to
ask, what percent is 100g out of 350g?
100/350
= r/100
r/100 = 100/350
r = 100/350 x 100
r= 28.57 % (rounded)
Because
we have different proportions of black jelly beans in different
sized packets, when combined the total number of black jelly
beans is about 29%.
This
explanation might make this problem sound fairly straightforward,
but a common error is to find the percent of black jelly beans
in each packet and then add them. If we think about this it
does not make sense. If we start off with (50/100) 50% of black
jelly beans in a small packet and (50/250) 20% of black jelly
beans in a large packet and then we add our answers, we get
the percent of black jelly beans in the bowl as 70%. This means
we have a higher percent of black jelly beans in the bowl than
we had in either packet to start with! We cannot have more black
jelly beans in the bowl than we started with.
NOTE:
In problems like this where we are talking about combining amounts
and finding percents we need to look at the total quantities;
in this case the total number of black jelly beans and the total
number of jelly beans.

Calculator
short cuts
Simple
electronic calculators with a percent sign can be very handy for adding
percents. These calculators usually offer a shortcut method.
Example
13: Using a calculator to find 110% of 57
Working
Out 
Thinking 
Using
my calculator,
Enter 57
Press x
Enter 110
Press
%
Answer: 62.7

To
find 110% of 57 I can key in
57
x 110 %
Note
that the % key tells the calculator to work out an answer: it
is analogous to the = key.

The
more elementary the calculator the more likely a shortcut method is
offered, but it will vary from calculator to calculator. Scientific
calculators may not have this shortcut. You will notice that they
usually do not have a '%' sign and you have to work out the problem
by constructing a proper mathematical sentence.
NOTE:
The method will depend on the calculator  you will need to experiment!
Example
14: Using a calculator to add 5% of 30 to 7% of 45
Working
Out 
Thinking 
Using
my calculator,
Enter 30
Press x
Enter 5
Press
%
to calculate
Press M+ to add to memory
Press CE to clear screen
Enter 45
Press x
Enter 7
Press
% to calculate
Press M+ to add to memory
Press MR to read answer
Answer:
4.65

5%
of 30 I can key in as 30 x 5 %.
I can then use the memory function to store this answer
7% of 45 I can key in as 45 x 7 %.
I can then add this to the memory and recall the final memory
to see the answer.
(NOTE:
If you are not sure how to use the memory function of a calculator,
see the section Whole
Numbers, Multiplication, Using a calculator.)

Example
15: Using a calculator to increase $120 by 7.2%
Working
Out 
Thinking 
Enter
120
Press +
Enter 7.2
Press %
Answer:
$128.64

This
is a special short cut for adding a percent of a number. Check
that it works on your calculator.

Example
16: Using a calculator to take a 15% discount from $850.
Working
Out 
Thinking 
Using
my calculator,
Enter
850
Press 
Enter 15
Press %
Answer:
$722.50

Again
this is a special shortcut. Check that it works on your calculator
before you rely on it.

Quick
quiz
1. 
A
long distance bus which travels from Melbourne to Sydney has 45
passenger seats. 
(a) 
If
80% of the seats have been booked, how many seats is this? 
(b) 
If
only 19 seats have been booked, what percent of seats is this? 


2. 
There
are 13 girls in Sari's class which make up 45% of the whole
class. How many students are in Sari's class?



3. 
In
an exam, Georgie answered 35 out of 40 questions correctly.

(a) 
What percent of the questions did she answer correctly? 
(b) 
If the pass mark is 85%, did Georgie pass? 
(c) 
If
Georgie resat the test and now scored 97%, how many questions
did she answer correctly? 


4. 
If
a puppy put on 300 grams over the last week and this is 15% of
its total body weight, what does the puppy weigh now? 


5. 
Ryan
bought 15 concert tickets to sell to his friends.

(a)

So
far he has sold 5 tickets to his friends, what percent has he
sold? 
(b) 
Ryan marked up the price of each ticket by 10% to pay for his
own ticket. If he paid $65 for the tickets originally, how much
is he selling them for now? 
(c) 
Ryan has not had much success at selling the tickets and now has
to discount them to get rid of them. If he discounts the price
of $65 by 20%, how much do the tickets cost now? 
(d) 
Eventually Ryan managed to sell 73% of the 15 tickets, how many
did he sell? 


6. 
Vinh
purchased a pair of jeans and a Tshirt that were on sale at a
25% discount. The original price of the jeans was $79 and the
Tshirt was $35. How much did Vinh pay for each item? 


7. 
In
a city office of 35 workers, 14 catch public transport to work,
8 ride their bikes, 10 drive their cars and 3 walk. 
(a) 
What percent of workers ride their bikes to work? 
(b) 
What percent catch public transport? 
(c) 
What
percent drive or walk? 
(d) 
What percent use any method other than car to get to work? 
(e) 
If these figures were used as the basis of a transport survey
how many people out of 100 could be said to use each method of
transport to get to work? 


8. 
One
brand of pumpkin soup weighs 440 gs and contains 35% fat. Another
brand weighs 1 kg and contains 20% fat. If I have a bowl of each,
which bowl of soup contains more fat?

To
view the quiz answers, click here.
