More Practise Calculations
on the
fx-991ES and fx-115ES
Calculators
Dr Allen Brown
Cambridge
Paperbacks
Cambridge Paperbacks
www.CambridgePaperbacks.com
First published by Cambridge Paperbacks 2019
© Allen Brown 2019
All rights reserved. No part of this publication may be reproduced or
transmitted in any form or by any means, electronic or mechanical, including
photocopy, recording, or any information storage and retrieval system without
permission in writing from the author.
Disclaimer
Although the author and publisher have made every effort to ensure that the
information in this book was correct during preparation and printing, the
author and publisher hereby disclaim any liability to any party for any errors
or omissions.
Read this First
The assumption is being made that you have already
worked through the ebook Practise Calculations on
the fx-991ES PLUS or fx-115ES Calculators. This ebook
will enhance your skill set by expanding your working
knowledge of the fx calculator.
As with the previous ebook, this ebook is all about
practise on your fx calculator and as you progress
through it, you are strongly advised to work through
the examples on your own calculator. Re-create all the
keystrokes yourself to verify the correct answers. You
will not gain very much by just looking at the pages in
a passive manner. You need active learning through
doing.
There are several books on GCSE and A Level Maths
published by Cambridge Paperbacks and if you wish to
strengthen your working knowledge of maths using
your calculator you will find these books extremely
useful.
Dr Allen Brown
Cambridgeshire
Contents
1 Calculations with Indices ................................................... 4
Example 1.1: ...................................................................... 5
Example 1.2: ...................................................................... 5
Example 1.3: ...................................................................... 6
Example 1.4: ...................................................................... 6
Example 1.5: ...................................................................... 7
Example 1.6: ...................................................................... 7
Example 1.7: ...................................................................... 8
x
x
2 Calculations with 10 and e .............................................. 9
Example 2.1: ...................................................................... 9
Example 2.2: ...................................................................... 9
Example 2.3: .................................................................... 10
Example 2.4: .................................................................... 10
Example 2.5: .................................................................... 11
Example 2.6: .................................................................... 12
Example 2.7: .................................................................... 13
Example 2.8: .................................................................... 14
Example 2.9: .................................................................... 15
3 Calculations with Logarithms .......................................... 16
Example 3.1: .................................................................... 16
1
Example 3.2: .................................................................... 17
Example 3.3: .................................................................... 17
Example 3.4: .................................................................... 18
Example 3.5: .................................................................... 19
Example 3.6: .................................................................... 20
Example 3.7: .................................................................... 21
Example 3.8: .................................................................... 21
4 Calculations with big and small numbers ........................ 23
Example 4.1: .................................................................... 24
Example 4.2: .................................................................... 24
Example 4.3: .................................................................... 25
Example 4.4: .................................................................... 26
Example 4.5: .................................................................... 26
Example 4.6: .................................................................... 27
Example 4.7: .................................................................... 28
Example 4.8: .................................................................... 28
5 Trigonometry Calculations .............................................. 30
Example 5.1: .................................................................... 30
Example 5.2: .................................................................... 31
Example 5.3: .................................................................... 31
Example 5.4: .................................................................... 32
2
Example 5.5: .................................................................... 33
Example 5.6: .................................................................... 34
Example 5.7: .................................................................... 35
Example 5.8: .................................................................... 35
Example 5.9: .................................................................... 36
6 Calculations with Physics Equations ............................... 37
Example 6.1: .................................................................... 37
Example 6.2: .................................................................... 38
Example 6.3: .................................................................... 40
Example 6.4: .................................................................... 41
Example 6.5: .................................................................... 42
3
1 Calculations with Indices
You are already familiar with the rules of algebraic
indices and just to remind you,
= ( + )
−
= = ( − )
( ) =
1
√ = 2
1
3
√ =
3
2
3
2
√ =
3
√ =
When you examine your fx-991ES calculator you will
see the following,
4
All the power calculations on page 4 can be performed
using these key functions and the following examples
will demonstrate how they are used.
Example 1.1:
Perform the following calculation,
9.87 2
5.9 2.6 + 6.8 1.87
The fx-991ES keystrokes for this calculation are,
9.87da5.9^2.6$
+6.8^1.87=
Example 1.2:
Perform the following calculation,
2
4.46 × 4.46 3 4.46 (2+3)
=
7.3 1.7 × 7.3 2.88 7.3 (1.7+2.88)
The keystrokes for this calculation are,
5
Ca4.46^5R7.3^(
1.7+2.88)=
Example 1.3:
Perform the following calculation,
√2.973
| |
2
2.58 − 3.09 3.8
Cq(Abs)as2.973R2.
58dp3.09^3.8=
Example 1.4:
Perform the following calculation,
3 4
√22.43 − √19.81
6
CqS22.43$pqF
4$19.81=
Example 1.5:
Perform the following calculation,
2 3 3 1.8
(3 ) + (4 )
7 8
CqA3$2R7$q(x )+
3
qA4$3R8$^1.8=
Example 1.6:
Perform the following calculation,
3 4
√5.58 + √5.58 + √5.58
7
Since the number 5.58 appears three times, to save
on keystrokes make 5.58 the M.
C5.58=n
sM$+qSM$+qF4
$M=
Example 1.7:
Perform the following calculation,
3
3
3
√87.46 − 63.91
CqS87.46q(x )p63
3
.91q(x )=
3
8
x
x
2 Calculations with 10 and e
You will find these
functions on your fx-
991ES calculator as
shown on the left.
Example 2.1:
Perform the following calculation,
10 2.43 + 10 1.86 − 10 0.79
CqG2.43$+qG1
.86$pqG0.79=
Example 2.2:
Perform the following calculation,
10 1.96 + 10 1.73
10 1.4 − 10 0.7
9
CaqG1.96$+qG
1.73RqG1.4$pq
G0.7=
Example 2.3:
Perform the following calculation,
10 1.87
√4.812
CaqG1.87Rs4.8
12=
Example 2.4:
Perform the following calculation,
10
10 + 10
10 − 10
where A = 2.43, B = 1.54, C = 1.65 and D = 1.97
CaqGQ(A)$+qGQ(B)
RqGQ(C)$pqGQ(D)r
A? 2.43=
B? 1.54=
C? 1.65=
D? 1.97=
Example 2.5:
Perform the following calculation,
0.117
2 + 0.687
CaqH0.117R2+q
H0.687=
11
Example 2.6:
Perform the following calculation,
0.8752 − −0.8752
0.8752 + −0.8752
Since 0.8752 occurs four times in this expression we
shall make it an M with the keystrokes,
C0.8752=
Now enter the following keystrokes,
aqHM$pqHzMRq
HM$+qHzM=
You can reduce the number of keystrokes to perform
this calculation by making e 0.8752 the M,
CqH0.8752=
12
aMpMuRMpMu=
The number of keystrokes is reduced from 29 to 21.
The number of keystrokes can be reduced even
further as this expression is the hyperbolic tanh
function,
Cc3 0.8752)=
Example 2.7:
Perform the following calculation,
3
+ 2 +
when x = 0.559. Although it can be calculated
directed, it can be expressed as,
(1 + (1 + ))
The keystrokes for this two-stage calculation are,
13
CHQ(X)$Q(:)
M(1+M(1+M))r
X? 0.559==
Example 2.8:
Perform the following calculation,
+ 1.3
2.1
where A = 3.57, x = 0.94, B = 2.55 and C = 4.07.
The keystrokes for this calculation are,
CaQ(A)qHQ(X)$+Q(B)q
H1.3J(X)RQ(C)qH2.
1J(X) r
A? 3.57=
X? 0.94=
B? 2.55=
14
C? 4.07=
Example 2.9:
Find the value of x which satisfies the expression,
2.3 − 1.63 = 18.57
We shall use the SOLVE feature on your fx calculator,
the keystrokes are,
CqH2.3Q(X)$pqH1
.63J(X)$Q(=)18.57
q(SOLVE)
Solve for X 1.0= (this is a guess value).
The answer is 1.473007496 to 9 decimal places.
15
3 Calculations with Logarithms
In the following examples we shall be using the keys
on your calculator,
As you are aware, g is a logarithm of base 10 and
h is the natural logarithm of base e. The i key
allows a logarithm of any base to be calculated.
Example 3.1:
Perform the following calculation,
log(0.588)
ln(3.221)
The keystrokes are,
Cg0.588)ah3.22
1)=
16
You will note this number is negative since a logarithm
of a number in the range 0 < < 1 is always
negative.
Example 3.2:
Perform the following calculation,
ln(30.886)
103.4 +
ln(1.114)
C103.4+h30.886
)ah1.114)=
Example 3.3:
Perform the following calculation,
log(1.582) + log(2.813) − log(0.677)
17
Cg1.582)+g2.81
3)pg0.677)=
Example 3.4:
Perform the following calculation,
log(1 + )
log(1 + )
For y = 2.406 and x = 3.815
Cg1+Q(Y))ag(1+
Q(X))r
Y? 2.406=
X? 3.815=
18
Example 3.5:
Find the value of x in the following expression,
log(2.11 + ) = 2.94
Use the SOLVE feature on your calculator with a guess
value of 1, enter the following keystrokes,
Cg2.11+Q(X))Q(=)26
.94q(SOLVE)
Solve for X 1=
x = 868.85, a graph of the expression is shown below
and you can see where the function intercepts the x-
axis.
19
You can confirm the result by rearranging the
expression to give,
= 10 2.94 − 2.11
The keystrokes for this calculation are,
CqG2.94$p2.11
=
which corresponds to the result from using the SOLVE
feature on your fx calculator.
Example 3.6:
Perform the calculation,
log (2.788) + log (4.857) − log (3.019)
4
2
3
Enter the following keystrokes,
Ci2$2.788$+i3$
4.857$pi4$3.01
9=
20
Example 3.7:
Perform the calculation,
35.74 + log (82.44)
6
| |
1.81 − log (156.95)
7
Cq(Abs)a35.74+i6$
82.44R1.81pi7$
156.95=
Example 3.8:
Perform the calculation,
+ log 2.3 ( )
+ log 2.8 ( )
21
where A = 2.46, B = 9.44 and x = 18.52.
You will notice in this expression logarithms whose
basis are not integers, this example shows how to
evaluate these on your fx calculator. The keystrokes
are,
CaQ(A)+i2.3$Q(X)R
Q(B)+i2.8$J(X) r
A? 2.46=
X? 18.52=
B? 9.44=
Should you be wondering about log2.3(x), consider the
following,
= log 2.3 (4.6)
log 2.3 (2.3) = log 2.3 (4.6)
log 2.3 (2.3) = log 2.3 (4.6)
2.3 = 4.6
y is the power you have to raise 2.3 to give the value
4.6, which is 1.8322
22
4 Calculations with big and small
numbers
In this chapter we shall be using the K key to
perform very big calculations and very small
calculations. Consider the following numbers; for the
first one, start on the right-hand side and count how
many decimal places you have until you reach the 1.
This is the index value on the 10.
10
14,506,000,000 = 1.4506 × 10
For a very small number, start on the left-hand side,
count how many decimal places until you pass the 6.
This is the index value on the 10 except it has a
negative – in front of it,
-8
0.00000006827 = 6.827 × 10
In the following examples set your fx to show the
result in scientific mode, q(SETUP)
Selection option 7
23
Select the number of decimal values after the decimal
point 4. The following examples will illustrate how
K key is used.
Example 4.1:
Perform the calculation,
6
7
4.567 × 10 + 2.668 × 10
C4.567K6+2.668
K7=
Example 4.2:
Perform the calculation,
9.39 × 10 8
5.022 × 10 3
24
The keystrokes are,
C9.39K8a5.022K
3=
Example 4.3:
Perform the calculation,
+
5
7
6
where A = 5.82×10 , B = 1.971×10 and C = 6.433×10 .
The keystrokes are,
CaQ(A)+Q(B)RQ(C)r
A? 5.82K5=
B? 1.97K6=
C? 6.433K7=n
25
-2
The result of the calculation is 3.87 × 10 .
Example 4.4:
Perform the calculation,
8.297 × 10 5
3 7
√7.701 × 10
The keystrokes for this calculation are,
C8.297K5aqS7.
701K7=
Example 4.5:
When you take a time
lapse photo of the
centre of the Milky
Way you will get a
result similar to the
image on the left. The
26
number of stars (just like our sun) in the Milky Way is
estimated to be
(250 ± 150) billion
What is the estimate of the maximum and minimum
values? The keystrokes for your calculator are,
C250K9+150K9Q(:)
250K9p150K9
= =
Maximum Minimum
This is a serious number of stars in our galaxy. It all
goes to show how silly sci-fi movies are when a race
of beings on one planet plan to take over the whole
galaxy.
Example 4.6:
Perform the calculation,
−6
6.28 × 10 −7 + 2.286 × 10
27
C6.28Kz7+2.286
Kz6=
Example 4.7:
Perform the calculation,
1.326 × 10 −9
8.832 × 10 −4 − 5.132 × 10 −5
C1.326Kz9a8.83
2Kz4p5.132Kz5=
Example 4.8:
Perform the calculation,
1.4 −
2
28
-18
-14
Where A = 7.226 × 10 , B = 6.581 × 10 and C =
-13
5.53 × 10 .
The keystrokes are,
Caq(A)^1.4$pQ(B)R
Q(C)dr
A? 7.226Kz14=
B? 6.581Kz18=
C? 5.53Kz13=
By now you will have gained a good grasp of using very
large and very small numbers on your fx calculator.
29
5 Trigonometry Calculations
It is assumed you have a basic knowledge of
trigonometric functions; in this chapter you will see
several examples where your fx calculator is used to
perform trig operations. Looking at your calculator
you will see the trigonometry keys as shown in the
image on the left. Set
your calculator to
q(SETUP)66.
Example 5.1:
Perform the calculation,
sin(26.5) + sin(2 × 26.5) + sin(3 × 26.5)
Since 26.5 appears three times in this expression, let
it become M. 26.5=
jM)+j2M)+j3M)=
30
Example 5.2:
Perform the calculation,
cos ( ) + 2 sin ( )
− −
o
o
o
where A = 60.3 , B = 32.7 and C = 53.1
This is a two-stage calculation; the keystrokes are,
CQ(A)aQ(B)pQ(C)$Q(:)
kM)+2jM)r
A? 60.3=
B? 32.7=
C? 53.1
=n =
Example 5.3:
Perform the calculation,
21.49[1 + 1.36 tan(73.1)]
31
C21.49(1+1.36l
73.1))=
Example 5.4:
Perform the calculation,
2
1 − sin (22.4)
1 + sin (22.4)
2
Since the sine occurs twice, this becomes a two-stage
calculation; the keystrokes are,
Cj22.4)d=
a1pMR1+M
32
Example 5.5:
Perform the calculation,
3
1 − cos ( )
3
1 + cos ( )
Where A = 48.28 and also when A = 76.5.
Again a two-stage calculation, but this time with a
variable value A. The keystrokes are,
CkQ(A))Q(:)
a1pMq(x )R1+Mq(x )r
3
3
A? 48.28
= =
r
A? 76.5
= =
33
Example 5.6:
Perform the calculation,
cos( ) −cos( )
sin( ) + sin( )
Where A = 2.14 radians and B = 1.65 radians.
This calculation requires the angle measurement in
radians; when you enter q(DEG►)
By choosing 2 you have a radian value. Enter the
following keystrokes,
CakQ(A)q(DEG►)2)pkQ
(B)q(DEG►)2)RjJ(A)q(DEG►)2
)+jJ(B)q(DEG►)2)r
A? 2.14=
B? 1.65=
34
Although you can reduce the number of keystrokes by
putting your fx calculator in its radian mode first, the
example demonstrates the use of the (DEG►)
function.
Example 5.7:
Perform the calculation,
4
√|1.15 − tan(68.3) |
th
The keystrokes for this calculation involve a 4 root,
CqF4$q(Abs)1.15pl
68.3)=
Example 5.8:
Perform the calculation,
2.55
sin −1 ( )
4.81
35
The keystrokes are,
Cq<2.55a8.81$
)=
In this example you can consider
a triangle as shown on the left.
What you have just calculated is
the ? angle.
Example 5.9:
Perform the calculation,
6.77
√1 + tan −1 ( )
8.29
Cs1+q?6.77a8.
29$)=
36
6 Calculations with Physics Equations
Here are some examples, although they may not be
part of a GCSE Physics syllabus, they do however
provide good practice for you to increase your
proficiency in the use of your fx-991ES calculator.
Example 6.1:
The time it takes for a swinging
pendulum to complete one cycle
(released from its starting
position and returning to its
starting position) is given by,
= 2 √
where g is the gravitational constant and X is its
length. Calculate the value of T when X = 0.12 m. The
value of g is stored in your calculator (look at the
reverse of the calculator cover and you will see its
code 35). The keystrokes are,
C2qLsQ(X)aq(CONST)35r
37
X? 0.12=
The cycle time is close to 0.7 seconds.
Example 6.2:
Although not within the
GCSE or A Level physics
syllabus, to calculate the
collisional cross section of
two molecules is very
straight forward. If the
radius of the purple
molecule is rA and the radius of the red molecule is rB,
the collisional cross section is given by,
2
= ( + )
A collision occurs when the distance between the
centre of the molecule’s is ≤ the sum of their radii.
Looking at the diagram, the thin blue circle shows the
collisional area of the molecules. When two molecules
38
of the same type collide, the collisional cross section
becomes,
2
= (2 )
When the centre of a second molecule is within this
area a collision occurs. Consider the reaction for
hydrogen,
+ →
2
Two hydrogen atoms collide to form a hydrogen
molecule. If the radius of a hydrogen atom is
-11
5.3×10 m, what is the collisional cross section?
Using the equation, the keystrokes for your fx
calculator are,
CqL(2O5.3Kz11)
d=
-20
The collision cross section for hydrogen is 3.53×10
2
m , which is a very small area.
39
Example 6.3:
The distance travelled s by a particle starting with a
velocity B and an acceleration A in a time C is given by,
1
2
= +
2
If the initial velocity is B = 10.43 m/s and its
2
acceleration is A = 3.56 m/s how far will it travel in
C = 32.6 seconds?
The keystrokes for this calculation are,
CQ(B)Q(C)+1a2$Q(A)J(C)
dr
B? 10.43=
C? 32.6=
A? 3.56=b
The particle would have travelled 2.232 km in first
32.6 seconds. As expected, if it continues to
accelerate its velocity will continue to increase.
40
Example 6.4:
The mass M is about to fall over the
edge of height A (2.83 m). Its initial
potential energy is MgA. After falling a
distance A all it potential energy will
become kinetic energy. What will be
the velocity as it hits the ground?
We know that,
1
2
=
2
Rearranging we get,
= √2
g is the gravitational constant (code 35), the
keystrokes are,
Cs2q(CONST)35Q(A)r
A? 2.83=
41
The velocity of the mass on impact is 7.45 m/s. Now
convert this value to miles per hour. First convert it to
km/hour; the keystrokes are,
Mq(CONV)20=
Now convert from km to miles,
Mq(CONV)08=
Velocity just before impact is 16.66 miles/hour.
Example 6.5:
An enclosure contains argon gas. The
atoms are moving around in all
directions as shown in the diagram
on the left. The mass of an argon
42
-26
atom is m = 6.636 × 10 kg. The average velocity is
given by,
8 ( + 273.15)
̅ = √
where k is Boltzmann’s constant (code 25) and the
temperature is X in degrees C. What is the mean
o
velocity at X = 43 C; the keystrokes are,
Cs8q(CONST)25(Q(X)+27
3.15)aqLO6.636
Kz26r
X? 43=
o
The average velocity of argon atoms at 43 C is 409.23
km/s.
This concludes this ebook on practice to improve your
skill set in using your fx-991ES or fx-115ES calculator.
43
There are more interesting ebooks on maths by the
same author.
44
Have a look at the Cambridge Paperbacks Catalogue
for the complete range of books on GCSE and A Level
Maths.
http://online.anyflip.com/pchj/vykd/
45