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