BELLAHOUSTON
ACADEMY
BIOLOGY DEPARTMENT
PROBLEM SOLVING
SELF HELP PACK
CONTENTS
PAGE (S)
CALCULATING AVERAGES
3- 4
PERCENTAGES
5 - 10
RATIOS
11 - 13
PIE CHARTS
14 - 17
LINE GRAPHS
17 -
2
AVERAGES
The average of a set of numbers is the
number that is most typical. Averages are
often used when it is difficult to be sure of
the true number. A box of matches or a
packet of pins will have the average
contents listed on the side. Some boxes
may contain a few more, some a few less,
but on average each box would have
what is printed on the side.
READ
CALCULATING AN AVERAGE
To find the average of a set of values we add
together all the figures and then divide by
the number of figures we added together.
This can be written like this:Total of all the figures
Number of figures
EXAMPLE
1. Here are the lengths of six sunflower seeds in millimetres.
13, 12, 14, 10, 13, 10
To find the average:
13 + 12 + 14 + 10 + 13 + 10
72
=
6
6
= 12 mm
DONT FORGET TO INCLUDE THE
UNITS YOU ARE WORKING IN !!!
3
Calculate the averages in the following examples:
2. Average Heart rate in beats per minute.
70 , 75 , 65 , 80 , 65 , 70 , 80 , 90
3. Average number of petals on daisies.
27 , 25 , 32, 26, 26 , 28 , 31 , 32 , 35, 29
4. Average length of human pregnancy in weeks.
38 , 40, 39 , 37 , 40 , 36 , 39 , 40 , 33 , 32 , 37, 37
(State your answer to the nearest number of
weeks)
TABLES OF INFORMATION
Often the figures will be in a table. Here are the lengths of broad
bean seeds.
LENGTH (mm)
16
18
19
20
22
NUMBER OF SEEDS
1
8
10
4
1
To find how many seeds there are we need to add up the number
in the column. There are 24 seeds.
To find the total length of all the seeds, we need to multiply each
length by the number of seeds for that length before we add
the figures together.
i.e.
(16 x 1) + (18 x 8) + (19 x 10) + (20 x 4) + (22 x 1)
24
=
16 + 144 + 190 + 80 + 22
=
24
452
24
=
4
19mm
PERCENTAGES
Percentages are often used in everyday life:-
READ
e.g. 20% off all prices
a wage rise of 4%
a pass mark of 65%
It is easy to change a fraction into a
percentage. All you do is multiply the fraction
by :100
1
e.g. if you get 15 out of 20 for a test. You can multiply this by
100 to change it into a percentage
1
15 out of 20 is the same as
15 (“out of” means “divided by”)
20
Multiplying it by 100 will change it into a percentage.
15
100
= 75% (means “out of 100”)
X
20
5
Percentages are often used to allow fair comparisons to be
made: A pupil sat 4 tests and achieved the following results: 15
30
10
20
53
14
15
x
and
22
100
75
= 75%
or
20
30
100
x
100
56
= 56%
or
53
10
100
x
100
71
= 71%
or
14
18
18
100
x
100
81
= 81%
22
or
100
This makes the marks much easier to compare. It can clearly be
seen that the best mark was for test 4 and the worst mark was for
test 2.
6
Change the following fractions into percentages:1.
a)
1
b)
1
2
c)
4
1
d) 1
5
20
2. Change the following test results into percentages:a)
11
b)
25
20
c) 45
50
60
d)
17
20
3. In a class of 25 children, the number who can roll their
tongue is 20. What percentage of the class can roll their
tongues?
4. A field has 18 trees in it. 12 of them are horse chestnut.
What percentage of the trees are horse chestnuts?
5. Calculate the percentage of protein in cereal. The table
shows the composition of a 50g serving of cereal.
Component
Carbohydrate
Protein
Fat
Water
Weight (g)
37
5
0.5
7.5
20% of all prices means 20
or 20 out of 100 of all prices.
100
READ
This cancels down to 1
5
To change a percentage to a fraction , divide by 100.
25%
=
25
100
=
1
5
80%
=
80
=
4
100
5
7
TO FIND THE PERCENTAGE OF A
NUMBER
We will work out 20% of £80.
Same as 20
means “multiplied by”
100
20% of £80
= 20
X 80
=
1600 = £16
Now try 5% of 61 kg.
5% of 61 =
5
100
x
61
=
305
100
= 3.05kg
1. Of the 900 children in a school. 75% are blood group
O, 15% are blood group A, 12% are blood group B and
the rest are blood group AB.
Complete the table below:
BLOOD GROUP
O
A
B
AB
NUMBER OF CHILDREN
2. A marathon runner weighs 55kg at the start of a race.
During the race his weight is reduced by 4%. What
does he weigh at the end of the race?
3. A lizard weighs 500g. Whilst escaping from a predator, it
loses it’s tail which weighs 30g. What percentage of its
weight has it lost?
8
PERCENTAGE INCREASE
A percentage increase is calculated as follows.
READ
Eg. A runner’s heart rate is 75 beats per minute at
rest. After five minutes brisk exercise his pulse rate
rose to 105 beats per minute. Calculate the
percentage increase in his heart rate.
To find the percentage increase in his heart rate we
first find the increase in his heart rate and then find
this as a percentage of the starting heart rate.
The increase in his heart rate = 105 – 75 = 30bpm
The percentage increase in his heart rate
=
30
75
X 100
= 40%
Calculate the percentage increase in each of the
examples below:
1. Percentage increase in dry weight of beans
shoots, starting weight 15g, final weight 21g.
2. Percentage increase in length of a worm,
starting length 6cm, final length 9cm.
3. Percentage increase in milk yield. Starting yield
1110 litre per year, final yield 1125 litres per
year.
9
PERCENTAGE DECREASE
Percentage decrease is calculated in a very similar way:-
READ
The runner’s breathing rate was a maximum of 32 breaths
per minute when he was running. After a few minutes rest,
his breathing slowed to 18 breaths per minute.
Calculate the percentage decrease in his breathing rate.
To find the percentage decrease in his breathing rate we first
find the decrease in his breathing rate and then find this as a
percentage of the starting breathing rate.
Decrease in breathing rate = 32 – 18 = 14 breaths per min
Percentage decrease in breathing rate
= 14
X 100 = 43.75%
32
Calculate the percentage decrease in each of the
examples below :
1. Percentage decrease in body mass of an individual.
Initial body mass is 85kg, final body mass is 60kg.
2. Percentage decrease in volume of air in lungs. Initial
volume is 1.4 litres, final volume is 0.8 litres.
2. Percentage decrease in length of a lizard which has
lost its tail. Initial length is 15 cm, final length is 8 cm.
10
RATIOS
Ratios are a way of comparing several things at
the same time.
READ
They are always written as small whole
numbers
E.g. At a bird table, 6 birds were seen feeding – 2 blackbirds
3 thrushes
1 sparrow.
This can be written as a ratio:2 blackbirds : 3 thrushes : 1 sparrow
(or two blackbirds to three thrushes to 1 sparrow.)
Later that same day, 12 blackbirds were seen at the bird table
with 9 thrushes and 3 sparrows.
12 blackbirds : 9 Thrushes : 3 sparrows
To simplify the ratio we divide each number by the
smallest number (in this case 3) to get the simplest whole
number ratio.
i.e.
4 blackbirds : 3 thrushes : 1 sparrow.
11
This works most times but if one of the figures contains a ½ we
multiply all the numbers by 2 to get rid of the ½.
E.g.
10 blackbirds : 8 Thrushes 4 sparrows
The smallest number is 4. We divide each number by 4.
2½ blackbirds
: 2 thrushes : 1 sparrow
The blackbirds is not a whole number so we multiply by 2.
5 blackbirds
:
4 thrushes
;
2 sparrows
This is the ratio.
Rules
for finding
a ratio
are :Rules
for finding
a ratio
are :1. Divide all the numbers by the
smallest number
2. If a ½ is obtained, multiply all
your answers by 2
3. Other part numbers should be
rounded up or down
4. When a question asks for a ratio
of two figures, the ratio should
always be stated in the same order
as the question.
12
Express each of the following in the simplest whole
number ratios.
1.
12 pupils had the following eye colours – 6 brown 4
blue and two grey.
2.
96 hamsters had brown and white coats
and 12 hamsters had white coats.
3.
In a garden there were 70 red snapdragon flowers, 72
white and 141 pink snapdragons.
3.
A blood sample contains 36,400 red blood cells and 52
white blood cells. Calculate the ratio of red blood cells
to white blood cells.
4.
The table shows the percentage of the food groups
found in rice.
Food Group
Carbohydrate
Fat
Protein
Other
Percentage (%)
80
5
10
5
Calculate the ratio of Protein to Carbohydrate in rice.
13
PIECHARTS
Pie charts are another way that
figures can be compared and are
useful because it gives a visual
representation of the data.
READ
Suppose we want to show the amount of land used in different ways in a
certain country. One way would be to show them as percentages.
USE
TOWN
% OF COUNTRY
30
FARMLAND
25
FOREST
25
DESERT
5
OTHER
20
A better way would be to
draw a pie chart.
A pie chart is a circle
representing the whole
country with sections of
the circle representing the
different uses.
The pie chart is divided
into 20 segments.
The whole circle = 100%
Therefore each of the
segments represents
100
= 5%
20
14
We can now start to complete the pie chart:DESERT
Town uses 30% = 6
segments
Farmland and forest use
25% each = 5 segments
each.
OTHERS
TOWN
The remaining 20% =
other uses = 4 segments
FOREST
FARMLAND
DRAWING A PIE CHART FROM FIGURES
Forty pupils were asked which was their favourite subject.
TO DRAW A PIE CHART FROM FIGURES
12 chose Biology (of course!!!)
8 chose English
4 chose Maths
The rest didn’t know.
We can now insert the information into the pie chart.
40 pupils
= 10 segments
Therefore 4 pupils = 1 segment
Biology =
12 pupils =
English =
8 pupils =
Maths =
4 pupils =
Don’t know = 16 pupils =
15
3 segments
2 segments
1 segment
4 segments
BIOLOGY
30%
DON'T KNOW
40%
ENGLISH
20%
MATHS
10%
1.The table shows the percentage of the
food groups found in rice.
Food Group
Percentage (%)
Carbohydrate
80%
Fat
5%
Protein
10%
Other
5%
16
(a) Use the information in the table to complete and label
the pie chart below.
(b) Calculate The ratio of Fat to Carbohydrate in rice.
__________
:
Fat
_____________
Carbohydrate
2.
A survey was carried out to
determine the blood groups of a
sample of 1000 people in a town.
There were 20 with blood group
AB, 500 with blood group A,
400 with blood group O and 80
with blood group B.
(1) Complete the pie chart of
these results by drawing and
labelling the remaining segments.
17
(2) What percentage of people in the town had blood group O ?
____________%
LINE GRAPHS
A line graph lets us see information at a glance. During Biology
experiments, a lot of information is collected. If we show the results
on a line graph, it makes them easier to interpret.
The following set of results was obtained from an
experiment to see how the heartbeat changes after
hard exercise. A pupil exercised for one minute and
then sits down. She has her heart rate taken every
minute for ten minutes.
READ
Time from Sitting Down
(minutes)
Number of Heart Beats
1
2
3
4
5
6
7
8
9
10
120
90
85
80
75
73
71
69
68
68
We will use these results to plot a line graph.
18
1. We draw our axes.
This is the
Y axis
This is the X axis
We must now choose scales for the axes.
On the X axis, we always plot the
information or factor which is under the
control of the person doing the
experiment. In this case it is the time from
sitting down.
The scale depends on the largest number
we have to plot and the number of
squares on the x axis.
The largest number we have to plot is 10
and there are 11 squares on the x axis
19
120
100
80
60
40
20
1
2
3
4
5
6
7
8
9
On the Y axis, we always plot the
readings taken during the experiment.
In this case, the number of heartbeats.
20
10
Now, we have to label the axes.
The X axis shows the time after sitting down. The unit that time is
measured in for this experiment is minutes.
Heart
beats /
minute
120
100
80
60
40
20
1
2
3
4
5
6
7
8
9
10
Time after sitting down (minutes)
The Y axis shows the heart rate. The unit that heart rate is
measured in is heart beats per minute. This can be written as heart
beats / minute.
We are now ready to plot the information. We deal with pairs of
information – time and the number of heartbeats.
We plot the points one at a time and then join the points to give a
smooth line.
Use a ruler to draw straight lines between them.
21
Sometimes, we are asked to plot more than one set of results on
the same axes. Do one set first. Then plot the second set but use
a different symbol to mark the paper. A small cross or a dot
instead of a square.
140
120
heartbeats / minute
100
80
60
40
20
0
0
.
2
4
6
8
time after sitting down (minutes)
10
In the exam you will be given marks for:1. Labelling the X and Y axes properly.
2. Using suitable scales – should try to use
as much of the graph paper as you can.
You will lose marks if your graph
takes up less than half of the available
graph paper in either direction.
3. Plotting points correctly and joining them
with a straight line.
4. Plot only the results given. If there is no
result for zero in the data don’t join the
plot line to the origin
22
12
Answer the following questions from the graph.
1. What was the girl’s heart rate four minutes after
she stopped exercising?
2. Between which one-minute interval did her heart
rate fall the most ?
3. What is this girl’s heart rate when she is at rest ?
4. How long did it take her heart rate to get back to
normal?
In Biology, some shapes of graphs are very common.
The “uphill” graph.
This graph
happens when
both the factor we
control (X axis )
and the factor we
measure (Y axis)
increase.
Plot this graph.
Length of Plant (cm)
1.0
1.5
2.0
2.5
3.0
3.5
23
day
1
2
3
4
5
6
The “downhill”
graph. Here the
factor we control
(X axis) increases
but the factor we
measure (Y axis)
decreases.
24
Plot this graph .
Weight (g) of germinating seed
2.0
1.8
1.6
1.4
1.2
In this graph the factor
we control (X axis) and
the factor we measure
increase up to a certain
point and then decrease.
25
Day
0
2
4
6
8
Plot this graph.
Amount of alcohol made by
yeast cells (cm3)
2
4
6
8
5
2
READ
Temperature
(0C)
15
20
25
30
35
40
The highest point on the graph
is the point where most is
happening. This is called the
optimum point. Optimum
means best
.
26
In this graph, the factor
we control ( X axis)
increases. The factor we
measure (Y axis
decreases at first then
increases.
1. These are the results of an experiment to investigate
how efficiently a cube of egg white was digested by
an enzyme. Plot the results as a graph.
Volume of egg white (cm3)
8
4
2
4
8
Temperature (0C)
15
25
35
45
55
27
READ
This is a “straight line”
graph. The factor we
control (X axis)
increases but the factor
we measure (Y axis)
stays the same.
28
Usually a “straight line” is found at the end of another graph.
C
This is a very
common graph in
Biology.
B
A – increase in Y
A
B – slower increase
C – no change in Y
Explain what is happening in the following graphs.
The four graphs are all measurements of the
behaviour of a lizard.
(a)
(b)
30
50
45
Period of Sleep (minutes)
Movement (cm)
25
20
15
10
5
40
35
30
25
20
15
10
5
0
1
2
3
4
5
0
1
Temperature (oC)
29
2
3
4
5
6
Temperature (oC)
7
(c)
(d)
6
14
mass of Food Eaten (g)
Number of Fights
5
4
3
2
1
12
10
8
6
4
2
0
1
2
3
4
5
6
7
0
Temperature (oC)
1
2
3
4
5
6
7
Temperature (oC)
BARCHARTS OR BAR GRAPHS
30
8