Technique in Math

Calculator Techniques for Solving Progression Problems

This is the first round for series of posts about optimizing the use of calculator in solving math

problems. The calculator techniques I am presenting here has been known to many students who

are about to take the engineering board exam. Using it will save you plenty of time and use that

time in analyzing more complex problems. The following models of CASIO calculator may

work with these methods: fx-570ES, fx-570ES Plus, fx-115ES, fx-115ES Plus, fx-991ES, and

fx-991ES Plus.

This post will focus on progression progression. To illustrate the use of calculator, we will have

sample problems to solve. But before that, note the following calculator keys and the

corresponding operation:

Name Key Operation

Shift

SHIFT

Mode

MODE

Alpha

ALPHA

Stat

SHIFT → 1[STAT]

AC

AC

Name Key Operation

Σ (Sigma)

SHIFT → log

Solve

SHIFT → CALC

Logical equals

ALPHA → CALC

Exponent

x[]

Problem: Arithmetic Progression The 6th term of an arithmetic progression is 12 and the 30th term is 180.

1. What is the common difference of the sequence?

2. Determine the first term?

3. Find the 52nd term.

4. If the nth term is 250, find n.

5. Calculate the sum of the first 60 terms.

6. Compute for the sum between 12th and 37th terms, inclusive.

Traditional Solution For a little background about Arithmetic Progression, the traditional way of solving this problem

is presented here.

Show Click here to show or hide the solution

Calculator Technique for Arithmetic

Progression

Among the many STAT type, why A+BX?

The formula an = am + (n - m)d is linear in n. In calculator, we input n at X column and an at Y

column. Thus our X is linear representing the variable n in the formula.

Bring your calculator to Linear Regression in STAT mode:

MODE → 3:STAT → 2:A+BX and input the coordinates.

X (for n) Y (for an)

6 12

30 180

To find the first term:

AC → 1 SHIFT → 1[STAT] → 7:Reg → 5:y-caret and calculate 1y-caret, be sure to place 1

in front of y-caret.

1y-caret = -23 → answer for the first term

To find the 52nd term, and again AC → 52 SHIFT → 1[STAT] → 7:Reg → 5:y-caret and make

sure you place 52 in front of y-caret.

52y-caret = 334 → answer for the 52nd term

To find n for an = 250, AC → 250 SHIFT → 1[STAT] → 7:Reg → 4:x-caret

250x-caret = 40 → answer for n

To find the common difference, solve for any term adjacent to a given term, say 7th term

because the 6th term is given then do 7y-caret - 12 = 7 for d. For some fun, randomly subtract

any two adjacent terms like 18y-caret - 17y-caret, etc. Try it!

http://www.mathalino.com/blog/romel-verterra/solving-progression-problems-calculator

Sum of Arithmetic Progression by Calculator Sum of the first 60 terms: AC → SHIFT → log[Σ] → ALPHA → )[X] → SHIFT → 1[STAT] → 7:Reg → 5:y-caret →

SHIFT → )[,] → 1 → SHIFT → )[,] → 60 → )

The calculator will display Σ(Xy-caret,1,60) then press [=].

Σ(Xy-caret,1,60) = 11010 ← answer

Sum from 12th to 37th terms,


Another way to solve for the sum is to use the Σ calculation outside the STAT mode. The

concept is to add each term in the progression. Any term in the progression is given by an = a1 +

(n - 1)d. In this problem, a1 = -23 and d = 7, thus, our equation for an is an = -23 + (n - 1)(7).

Reset your calculator into general calculation mode: MODE → 1:COMP then SHIFT → log.

Sum of first 60 terms:

(-23 + (ALPHA X - 1) × 7) = 11010

Or you can do

(-23 + 7 ALPHA X) = 11010 which yield the same result.

Sum from 12th to 37th terms

(-23 + (ALPHA X - 1) × 7) = 3679

Or you may do

(-23 + 7 ALPHA X) = 3679

Calculator Technique for Geometric

Progression

Problem Given the sequence 2, 6, 18, 54, ...

1. Find the 12th term

2. Find n if an = 9,565,938.

3. Find the sum of the first ten terms.

Traditional Solution


Solution by Calculator

Why A·B^X? The nth term formula an = a1r

n – 1 for geometric progression is exponential in form, the variable n

in the formula is the X equivalent in the calculator.

MODE → 3:STAT → 6:A·B^X

X Y

1 2

2 6

3 18

To solve for the 12th term AC → 12 SHIFT → 1[STAT] → 7:Reg → 5:y-caret

12y-caret = 354294 answer

To solve for n, AC → 9565938 SHIFT → 1[STAT] → 7:Reg → 4:x-caret

9565938x-caret = 15 answer

Sum of the first ten terms, AC → SHIFT → log[Σ] → ALPHA → )[X] → SHIFT → 1[STAT] → 7:Reg → 5:y-caret →

SHIFT → )[,] → 1 → SHIFT → )[,] → 10 → )

The calculator will display Σ(Xy-caret,1,10) then press [=].



You may also sove the sum outside the STAT mode

(MODE → 1:COMP then SHIFT → log[Σ])

Each term which is given by an = a1rn – 1.

(2(3ALPHA X - 1)) = 59048 answer

Or you may do

(2 × 3ALPHA X) = 59048

Calculator Technique for Harmonic

Progression

Problem Find the 30th term of the sequence 6, 3, 2, ...

Solution by Calculator MODE → 3:STAT → 8:1/X

X Y

1 6

2 3

3 2

AC → 30 SHIFT → 1[STAT] → 7:Reg → 5:y-caret

30y-caret = 0.2 answer

I hope you find this post helpful. With some practice, you will get familiar with your calculator

and the methods we present here. I encourage you to do some practice, once you grasp it, you

can easily solve basic problems in progression.

If you have another way of using your calculator for solving progression problems, please share

it to us. We will be happy to have variety of ways posted here. You can use the comment form

below to do it.

Calculator Technique for Solving Volume Flow Rate Problems in

Calculus

The following models of CASIO calculator may work with this method: fx-570ES, fx-570ES

Plus, fx-115ES, fx-115ES Plus, fx-991ES, and fx-991ES Plus.

The following calculator keys will be used for the solution

Name Key Operation

Shift

SHIFT

Mode

MODE

Name Key Operation

Stat

SHIFT → 1[STAT]

AC

AC

This is one of the series of post in calculator techniques in solving problems. You may also be

interested in my previous posts: Calculator technique for progression problems and Calculator

technique for clock problems; both in Algebra.

Flow Rate Problem Water is poured into a conical tank at the rate of 2.15 cubic meters per minute. The tank is 8

meters in diameter across the top and 10 meters high. How fast the water level rising when the

water stands 3.5 meters deep.


http://www.mathalino.com/blog/romel-verterra/calculator-technique-clock-problems-algebra



Volume of water inside the tank

Differentiate both sides with respect to time

When h = 3.5 m

answer


Show Click here to show or hide the concept behind this technique

MODE → 3:STAT → 3:_+cX2

X Y

0 0

10 π42

5 π22

AC → 2.15 ÷ 3.5y-caret = 0.3492 answer

http://www.mathalino.com/blog/romel-verterra/calculator-technique-solving-volume-flow-rate-problems-calculus

To input the 3.5y-caret above, do 3.5 → SHIFT → 1[STAT] → 7:Reg → 6:y-caret

What we just did was actually v = Q / A which is the equivalent of for this problem.

Problem Water is being poured into a hemispherical bowl of radius 6 inches at the rate of x cubic inches

per second. Find x if the water level is rising at 0.1273 inch per second when it is 2 inches deep?

Traditional Solution Volume of water inside the bowl

Differentiate both sides with respect to time

When h = 2 inches, dh/dt = 0.1273 inch/sec

answer

Calculator Technique MODE → 3:STAT → 3:_+cX2

X Y

0 0

6 π62

12 0

AC → 0.1273 × 2y-caret = 7.9985 answer

Calculator Technique for Clock Problems in Algebra

The following models of CASIO calculator may work with these methods: fx-570ES, fx-570ES

Plus, fx-115ES, fx-115ES Plus, fx-991ES, and fx-991ES Plus.

Before we go to Calculator technique, let us first understand the movements of the hands of our

continuously driven clock.

For simplicity, let "dial" be the unit of one hand movement

and there are 60 dials in the complete circle as shown in

the figure.

1. When the minute-hand moves 60 dials, the hour hand moves 5 dials. The ratio of the two movements, hour-hand over minute-hand, is 5/60 = 1/12. Thus, if the minute-hand will move x-minutes, the hour-hand moves by x/12 minutes.

2. When the second-hand moves 60 dials, the minute-hand moves 1 dial. The ratio of the two movements, minute-hand over second-hand, is 1/60. Thus, if the second-hand will move x-seconds, the minute-hand moves by x/60 seconds, and the hour-hand also moves by 1/12 of x/60 or x/720 seconds.

3. The relationship of hand-movements can also be translated in terms of degree unit which I found handy in calculator technique for board exam problems. We know that a complete circle is equal to 360° and equal to 60 dials. Thus, 1 dial is equivalent to 360°/60 = 6° and five dials is equivalent to 5(6°) = 30°. Note that 1 dial move of the minute-hand is equivalent to 1 minute of time, and five dials move of the hour-hand is equivalent to 1 hour of time.

Knowing all of the above, we can now develop the calculator technique for solving clock-related

problem. We will solve some example here in order to apply this time saving technique.

Problem What time after 3:00 o'clock will the minute-hand and the hour-hand of the clock be (a) together

for the first time, (b) perpendicular for the first time, and (c) in straight line for the first time?





The following calculator keys will be used.

Name Key Operation

Shift

SHIFT

Mode

MODE

Name Key Operation

Stat

SHIFT → 1[STAT]

AC

AC

The relationship between the movements of the clock hands is linear. We can therefore use the

Linear Regression in STAT mode.

Approach No. 1 Take 3:00 pm as reference point. Initially, the minute-hand of the clock is at 0 dial and the hour-

hand of the clock is advance by 15 dials, thus, coordinates (0, 15). After 1 hour (4:00 pm), the

minute-hand advanced by 60 dials leaving the hour-hand 40 dials, thus, coordinates (60, -40).

MODE → 3:STAT → 2:A+BX

X Y

0 15

60 -40

(a) Together for the first time: The distance between the hands of the clock is zero. We will

therefore find X when Y is zero in our table.

AC → 0 SHIFT → 1[STAT] → 7:Reg → 4:x-caret

0x-caret = 16.36

Thus, time = 3:16.36 pm

(b) Perpendicular for the first time: The hour-hand is behind by 15 dials by the minute hand. Let

us find X when Y is -15.

AC → -15 SHIFT → 1[STAT] → 7:Reg → 4:x-caret

-15x-caret = 32.73


(c) Straight line for the first time: The hour-hand is behind by 30 dials by the minute hand, thus

find X when Y is -30.

AC → -30 SHIFT → 1[STAT] → 7:Reg → 4:x-caret

-30x-caret = 49.09


The above approach works fine but you need to mentally visualize the hands of the clock to get

the proper sign (positive or negative) and value of the coordinates. See for example if the given

time is 10:00 pm, the hands will be in straight line for the first time with the hour hand advancing

the minute hand by 30 dials. Thus Y is +30 and not -30. This mental visualization takes the same

effort as the traditional solution; the only difference is the absence of drawing. Without the

drawing is good already but we can do better than that. The next approach will be more

consistent, the only catch is that you need to memorize the numbers 30 and 330. I think it is not

hard to memorize that numbers.

Approach No. 2 In the first approach, both X and Y are in dial units. In this second approach, the X coordinate

will be in dial and Y coordinates in degrees. Recall that in 1 hour, the hour-hand will move 5

dials equivalent to 30° and the minute-hand moves for 60 dials or 360°. The 1 hour difference is

therefore 360° - 30° = 330° for the hour and minute-hands of the clock. At 3:00 pm, the minute-

hand is at -90° in reference with the hour-hand, thus coordinates (0, -90). After 1 hour, that is at

4:00 pm, the minute hand advanced the right hand by 330° - 90° = 240°, thus coordinates (60,

240)

MODE → 3:STAT → 2:A+BX

X Y Explanation

0 -90 ← -3 × 30

60 240 ← 330 - 90

(a) Together for the first time: The angle between the hands of the clock is zero. Find X when Y

is zero in our table.


0x-caret = 16.36


(b) Perpendicular for the first time: The angle between the hour-hand and minute-hand is 90°.

Let us find X when Y is 90.


90x-caret = 32.73


(c) Straight line for the first time: The angle between the hour-hand and minute-hand is 180°,

thus find X when Y is 180.


180x-caret = 49.09


For me, the second approach is more rapid and easy to implement. I recommend you master just

one and be good at it.

Problem How soon after 5:00 o'clock will the hands of the clock form a (a) 60-degree angle for the first

time, (b) 60-degree angle for the second time, and (c) 150-degree angle?

Solution by Calculator Technique MODE → 3:STAT → 2:A+BX

X Y Explanation

0 -150 ← -5 × 30

60 180 ← 330 - 150

(a) 60-degree angle for the first time AC → -60 SHIFT → 1[STAT] → 7:Reg → 4:x-caret

-60x-caret = 16.36 minutes answer

(b) 60-degree angle for the second time AC → 60 SHIFT → 1[STAT] → 7:Reg → 4:x-caret

60x-caret = 38.18 minutes answer

(c) 150-degree angle AC → 150 SHIFT → 1[STAT] → 7:Reg → 4:x-caret

150x-caret = 54.54 minutes answer

You may also like the calculator technique for arithmetic progression, geometric progression,

and harmonic progression.

Problem

A sphere of diameter 40 cm is cut by two horizontal planes. One plane is 8 cm below the center of the

sphere and the other is 14 cm above the center of the sphere. Determine the volume of the frustum

formed between the cutting planes.



Hello Symbianizers

:hat:

Today, I will teach you the Calculator technique use to solve Math problems.

In this tutorial I will be using CASIO fx-991 ES. So don't ask me on how to do it in other

calculator.

Moreover, I will not cover the CASIO fx-991 ES PLUS although, some of you think its the

same,

I say to you it's different in someway or in some functions especially in STAT MODE.

This is good and very helpful especially in Board Exam and in some school na hindi

kinakailangan ng solution(multiple choice lang).

Sa una nahirapan ako dahil iniisip ko kung paano ko isusulat ang mga equation. Ayun nagawan

ko naman ng paraan. :D

Requirement in this MATH CALCU tutorial:

1. Basic knowledge in math( You cannot understand it if you don't know the basics..Mostly

shorcuts ito kaya you should know the basic..)

2. CASIO FX-991 ES with the manual

3. Read your manual(all the functions and definiton of each will be found in the manual)

4. Brain/Common Sense

You can see links before reply

You can also post your assignments here..Solve natin yan kung kaya..:D

Gamitin niyo ang LINK (You can see links before reply) na ito para itype ang mga equation

niyo..

Then kunin ang link ng image at ipaste dito gamit ang

Sa mga MATH Wizard diyan, tulong po tayo..hehe

Ok lets get it on!

:getiton:

SESSION 1

PART 1 (You can see links before reply)


SESSION 2





SESSION 3



CALCU TECHNIQUE SHARED by OTHER MEMBER(s):

Laplace Transformation (You can see links before reply)

More will be added soon..

:clap:

SiRhOSEven

10th Jun 2012, 08:05

Getting familiar with the functions

1. What are the roots of the polynomial: You can see links before reply^{2}-7x+12=0

A. -3, -4

B. 6, 2

C. 2, 3

D. 3,4

Set Calculator to equation mode: MODE>5>3 for quadratic equation

Mode 3 because our polynomial is a 2nd degree equation..

Input the equation coefficients a,b,c:

1 = -7 = 1 2 = =

And you will get the answer..

2. What are the roots of the polynomial: You can see links before reply^{3}-7x^{2}+14x-8=0

A. 1,2,3

B. 1,2,5

C. 1,2,4

D. 2,3,5

Set Calculator to equation mode: MODE>5>4 for cubic equation

Mode 4 because our polynomial is a 3rd degree equation..

Input the equation coefficients a,b,c:

1 = -7 = 1 4 = -8 = =


3. Which of the following is a possible root of the polynomial: You can see links before

reply^{4}+%205x^{3}+5x^{2}-5x-6=0

A. 3

B. 0

C. -2

D. 4

NOTE: A root is any value that, when substituted to the variable(ie x), will satisfy the

equation.(In our equation 0=0 )

Set the calculator to computation mode: MODE>1

Input ONLY the left side of the equation.

Trial and error, Use the CALCU function..

Pag nagtanong ang calculator X? iinput ang mga choices..

X? 3 =

output is 240..

repeat the step until you get an output of 0..


4. Find the value of x and y in the following equations:



A. -12/19, 16/19

B. -24/19, -22/19

C. 12/19, -16/19

D. 24/19, 22/19

Set calculator to equation mode: MODE>5>1 for two-variable equation.

Input the coefficients of a,b and constant c of the first equation:

2 = 3 = 6 =

Input the coefficients of a,b and constant c of the second equation:

-3 = 5 = 2 =

Press the = to get the value of x and press again to get the value of y..


5. The equation You can see links before reply^{4}-7x^{3}+5x^{2}-7x+2=0 has two rational

roots, both of which are positive. Find the larger of these two roots.

A. 1

B. 2

C. 3

D. 4

Same method with number 3..We will be using CALCU function..The calculator should display

0..


6. What is the remainder of the polynomial You can see links before reply^{3}+4x^{2}-3x+8

when divided by You can see links before reply

A. 208

B. 218

C. 283

D. 305


Input the left side of the equation

Apply the Remainder theorem

Use CALCU function and substitute x=5


7. Solve the values of y in the system of equations:




Set the calculator to equation mode: MODE>5>2

Input the coefficients a,b and constant c of the first, second and third equation..

Press the equal = button twice to get the value of y..


8. Solve for the value of You can see links before reply[3]{-8}

A. 2

B. 2i

C. -2

D. -2i


Input the equation in the calculator(as is)..

and you will get the answer..

9. Solve for the value of You can see links before reply{\sqrt{\sqrt{\sqr

t{\sqrt{\sqrt{55555}}}}}}

A. 1.91

B. 2.19

C. 1.19

D. 2.91

Set the calculator to computation mode

Input the equation in the calculator(as is)


10. Solve for the value of x in the equation: You can see links before reply{\sqrt{x^{2}}}%20-

%20{\sqrt{5x}}%20-2x=-5

A. 1.91

B. 1.19

C. 1.46

D. 2.13

Set the calculator to computation mode

Input the equation and use the SOLVED function to get the value of x..

To get the SOLVED function, SHIFT>CALCU>=

It should display,

X= answer

L-R= 0 (important, must be 0)


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Technique in Math