Introduction to Calculus: Understanding "Derivatives" via the Power Rule

Heto ang karugtong ng ating Calculus series para sa Senior High School. Noong nakaraan, pinag-aralan natin ang Limits . Ngayon naman, tutuklasin natin kung ano ang ibig sabihin ng Derivatives. Sa totoong buhay, ang derivative ay nangangahulugan lang ng rate of change o ang bilis ng pagbabago ng isang bagay . Halimbawa: gaano kabilis lumalaki ang populasyon, o gaano kabilis bumababa ang presyo ng bilihin sa bawat segundo.

Huwag kayong matatakot kapag nakakita kayo ng ganitong simbolo sa exam. Ang tinatawag na Power Rule ang pinakamadali at pinakamahalagang shortcut sa Calculus para makuha ang derivative . Ang batas ng panuntunang ito ay napakasimple lang: Kung mayroon kang variable na X na may exponent o kapangyarihan na N—ibababa mo lang ang exponent sa unahan ng X, tapos ay babawasan mo ng numerong isa ang lumang exponent mo. Ganyan lang po kadali!

Subukan natin ang saktong halimbawa na ito: 'Find the derivative of X cube.' Paano natin kukunin ang derivative nito gamit ang Power Rule? Sundin natin ang dalawang saktong hakbang. 

Step One: Ang exponent na 3 ay kunin mo at ibaba mo sa unahan ng letra ng X. Kaya magiging three X na siya.

Step Two: Ang lumang exponent na 3 ay bawasan mo ng saktong isa. Three minus one, katumbas ay two.

Pagsamahin natin ang ating kalkulasyon: ang pinal na sagot ay three X squared o 3x²! Napakabilis at napakadali lang, 'di ba? Hindi mo na kailangang dumaan sa mahahabang formula. Bago ko ibigay ang susunod nating math short cuts, paki-click muna ang 'Follow' button sa ating Page para lagi kang updated sa ating libreng lecture series araw-araw!"

Introduction to Calculus: Understanding Limits

 

ALS Senior High School Lesson Series: Introduction to Calculus

Lesson 1 - Understanding "Limits" (Ang Konsepto ng Paglapit) 

Pagpasok ninyo ng Senior High School, maririnig ninyo ang salitang Calculus. Marami ang natatakot dito, pero ang totoo, napakapraktikal nito. Ang Calculus ay ang pag-aaral ng pagbabago . Halimbawa: kung gaano kabilis tumakbo ang isang sasakyan, o kung paano lumalago ang isang negosyo buwan-buwan. At ang unang-unang hagdanan para maunawaan ito ay ang tinatawag na LIMITS.

Ano ba ang ibig sabihin ng Limit sa Calculus? Sa simpleng Tagalog, ang Limit ay nangangahulugan ng PAGLAPIT . Isipin mo na sumasakay ka sa isang jeep papunta sa iyong destinasyon. Habang papalapit ka nang papalapit sa landmark mo, hindi mo naman kailangang bumangga doon. Tinitingnan lang natin kung anong numero o halaga ang PINAPALAPITAN ng ating equation. Sa Calculus, sinusuri natin ang gawi ng isang numero kapag nilapitan ito mula sa kaliwa at mula sa kanan.

Heto ang saktong halimbawa na madalas lumabas sa exam. Basahin natin ito ng tama: 

lim 2x + 4

x→x3

The limit of two X plus four, as X approaches three.

Ibig sabihin lang niyan, kapag ang halaga ng X ay pinalapit natin nang pinalapit sa numerong 3, ano namang numero ang lalabas sa buong equation? Ang gintong sikreto rito, gagamit tayo ng Direct Substitution . Palitan mo lang ang letra ng X ng numerong pinalalapitan, na sa sitwasyong ito ay 3.

lim 2(3) + 4 = 6 + 4 = 10

Kalkulahin natin: Two multiplied by three, katumbas ay six. Pagkatapos, i-plus mo ang four. Six plus four, ang saktong sagot ay TEN! Ibig sabihin, ang limit ng ating function habang lumalapit sa 3 ay 10. Ganyan lang kabilis! Bago natin ibigay ang susunod na technique, paki-click muna ang 'Follow' button sa ating Page para lagi kang updated sa ating libreng lecture series araw-araw!"

Mathematics 7 Summative Test 4th Quarter

C 1. It refers to the number of times an observation occur or appears in a data set. 

    C. Frequency

D 2. It is used to present data in a most systematic and organized manner.

    D. table

A 3. It is  the tabular method of organizing data.

    A. frequency distribution table

D 4. It is the difference between the highest and the lowest values.

    D. range

C 5. The following is/are the properties of an effective table.

    I. The date of the survey is indicated.

    IV. The data are arranged systematically in columns and properly labeled.

For items 6 - 10.

    Pelmar opened a pack of marbles. He found out that there are 15 red, 10 blue, 7 orange, 7 yellow, 13 green and 8 purple marbles.

B 6. Which of the following is the correct Frequency Distribution Table of the given data?

    B.  Color        Frequency

          red                 15

          Green            13

         Blue                10

         purple              8

         orange             7

         yellow              7

C 7. Which of the following is the appropriate title of the distribution table?

    C. Frequency Distribution Table of Pelmar's Marbles

D 8. What is the total number of marbles?

    D. 60

C 9. Which color has the highest number of marbles?

    C. red

D 10. Which color has the least number of marbles/

    D. yellow     

For Items 11 - 15.

A certain school conducted a survey of 50 students for their preferred learning modalities for the school year 2020-2021. The results are as follows: 40 enrolled for modular, 8 enrolled for online learning, and 2 enrolled for TV/Radio-Based learning.

A 11. Which of the following is the correct Frequency Distribution Table for the given data?

    A.    Learning Modalities        Number of Students

                Modular                                    40

                Online                                         8

                Radio/TV                                    2

D 12. Which of the following is the appropriate title of the Frequency Distribution Table?

    D. Frequency Distribution Table of the Preferred Learning Modalities for School Year 2020-2021

A 13. What learning modalities has the highest number of enrollees?

    A. Modular

C 14. What learning modality has the lowest number of votes in the survey?

    C. TV/Radio

C 15. What is the difference between the highest and the lowest number of enrollees?

    C. 38

D 16. It is used to present data in a most systematic and organized manner.

    D. table

C 17. It refers to the number of times an observation occurs or appears in a data set.

    C. frequency

A 18. These are generally equal in width and are non-overlapping from each other.

    A. class intervals

C 19. It is the science of collecting, organizing, interpreting and analyzing data.

    C. Statistics

B 20. It is the number  of times can observation or a particular value appears in a data set.

    B. Frequency 

Class mark 21. In a class interval, the middle score is called _______.

Frequency 22. The number of times a particular observation appears in a data set.

B 23. What is the pictorial representation of a statistical data?

    B. Graph

    

A 24. What graph is illustrated below?

(Image from Bar Graphs (mathsisfun.com))

    A. Bar Graph

C 25. Which of the graphs is used to present data that indicate trends over a period of time?

    C. Line Graph

For questions 26 - 28, refer your answer in the illustration below:

                                   

        

D 26. What type of graph is represented in the illustration above?

    

    D. Pie Chart

A 27. What color is the most favored? 

    A. Blue

B 28. What percent of students chose red as the third frequency of colored T-shirts?

    B. 25.2%

C 29. What type of information does a histogram show?

    C. frequency

D 30. Pi (π) is a special symbol whose value is used to find the area of a circle. The following number gives the first 100 digits of the number pi:

 

3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 816 406 286 208 998 628 034 825 342 117 067

Which digit from 0 to 9 occurs most frequent in this number?

    D. 9

D 31. Which kind of graph below is also known as cumulative frequency polygon?

    D. ogive

A 32. What is a cumulative frequency?

    A. It is the running total of frequencies. 

B 33. Which of the following is the correct entry for the cumulative frequency column in the table given below?

Marks                        Frequency                    Cumulative Frequency

0-5                                    2                                            2

5-10                                  10                                        12

10-15                                 5                                         17

15-20                                 5                                         22

    B. 2, 12, 17, 22

For numbers 34-35, refer to the given graph below:

The histogram shows the heights of 25 students in Grade-7 class, grouped into widths of 5 inches.

B 34. How many students are greater than or equal to 150 inches tall?

    B. 12

C 35. How many students are greater than or equal to 140 inches tall but less than 160?

    C. 22

36. What is an ogive graph?

    D. Cumulative Frequency Graph

37. Which occurs the most frequent in the following tongue twister?

    SHE SELLS SEASHELLS BY THE SEASHORE.

    D. S

38. Which of the following is an ogive graph?

    A.

39. Based on the table below, how many days had rained in the past six months?

            Record of the Number of Days with Rain in a Month

           Month                                    Number of Days with Rain

                1                                                            II

                2                                                        IIII - I

                3                                                           IIII 

                4                                                            II

                5                                                             I

                6                                                IIII - IIII - IIII - IIII 

    C. 35

    40. Which of the following is a graphical display of data using bars of different heights that are joined together?

    B. Histogram

The Definition of Differentiation/Ang Kahulugan ng Differentiation

The essence of calculus is the derivative. The derivative is the instantaneous rate of change of a function with respect to one of its variables. This is equivalent to finding the slope of the tangent line to the function at a point.

(Ang kakanyahan ng calculus ay ang derivative. Ang derivative ay ang agarang rate ng pagbabago ng isang function na may kaugnayan sa isa sa mga variable nito. Ito ay katumbas ng paghahanap ng slope ng tangent line sa function sa isang punto.)

We want to find the slope of the tangent line to a graph at the point P. We can approximate the slope by drawing a line through the point P and another point nearby, and then finding the slope of that line, called a secant line. The slope of a line is determined using the following formula (m represents slope) :

(Nais nating hanapin ang slope ng tangent line sa isang graph sa puntong P. Maaari nating tantiyahin ang slope sa pamamagitan ng pagguhit ng linya sa puntong P at isa pang punto sa malapit dito, at pagkatapos ay hanapin ang slope ng linyang iyon, na tinatawag na secant line. Ang slope ng isang linya ay tinutukoy gamit ang sumusunod na formula (ang m ay kumakatawan sa slope):)

Let P = (x,y) and Q := (a,b). Let

Then the slope of the line

What we want to do is decrease the size of Delta-x as much as possible. We do this by taking the limit as Delta-x approaches zero. In the limit, assuming the limit exists, we will find the exact slope of the tangent line to the curve at the given point. This value is the derivative;

(Ang gusto nating gawin ay bawasan ang laki ng Delta-x hangga't maaari. Ginagawa natin ito sa pamamagitan ng pagkuha ng limitasyon habang ang Delta-x ay lumalapit sa zero. Sa limitasyon, sa pag-aakalang umiiral ang limitasyon, makikita natin ang eksaktong slope ng padaplis na linya patungo sa kurba sa ibinigay na punto. Ang halagang ito ay ang derivative;)

There are a few different, but equivalent, versions of this definition. In more general considerations, h is often used in place of Delta-x. Or Delta-y is used in place of 

(Mayroong ilang iba't ibang, ngunit katumbas, na mga bersyon ng kahulugang ito. Sa mas pangkalahatang pagsasaalang-alang, ang h ay kadalasang ginagamit bilang kapalit ng Delta-x. O Delta-y ay ginagamit sa halip ng

This leads to three commonly used ways of expressing the definition of the derivative: (Ito ay humahantong sa tatlong karaniwang ginagamit na paraan ng pagpapahayag ng kahulugan ng derivative:)

--o0o--

Mga nilalaman at larawan, hinango sa:

https://web.mit.edu

Lesson 6 - Solving Quadratic Equations in Taglish by Factoring, Completing the Square, and Using the Quadratic Formula

 Lesson 6 – Quadratic Equations

        Matapos nating mapag-aralan ang tungkol sa linear equations at ang kaugnayan ng mga ito sa pagresolba ng mga problema sa pang-araw-araw na buhay, dadako naman tayo sa quadratic equations.

Ang quadratic equation na may isang variable/unknown ay isang equation kung saan ang pinakamataas na exponent ng variable ay 2.

        Ang karaniwang anyo ng isang quadratic equation na may isang variable na kinakatawan ng x ay: 

        ax² + bx + c = 0 

kung saan ang a, b, at c ay mga constant at ang a ≠ 0.

Ang pinakakaraniwang ginagamit na paraan sa paglutas ng mga quadratic equation ay ang mga sumusunod:

Solving Quadratic Equations by Factoring

1. Ang factoring o pag-factor ay ang pinakamadaling paraan ng paglutas ng mga quadratic equation. Gayunpaman, nangangailangan ito ng kasanayan sa iba't ibang mga pamamaraan ng factoring. Bukod dito, hindi lahat ng quadratic equation ay malulutas sa pamamagitan ng factoring.

        Upang malutas ang isang quadratic equation sa pamamagitan ng factoring, sundin ang mga hakbang na ito:

        a. Ilipat ang lahat ng mga termino sa kaliwang bahagi ng equation upang ang kanang bahagi ay maging katumbas ng zero.

        b. Pasimplehin at i-factor ang quadratic expression na karaniwan na isang trinomial.

        c. Ang bawat isa sa mga factor ay isa na ngayong linear equation. I-equate ang bawat factor sa 0 at lutasin ang bawat linear equation para sa hinahanap na variable.

HALIMBAWA 1

        Solve for x in the equation 12x² - 9  = -10x + 3 

Step 1: Ilipat ang lahat ng terms sa kaliwang panig.

          12x²  - 9 = -10x + 3

  12x² – 9 + 10x - 3 = -10x + 3 + 10x - 3

      12x² +10x – 12 = 0

Step 2: Simplihen at i-factor

12x² + 10x – 12 = 0

6x² + 5x - 6 = 0 

(2x + 3) (3x – 2) = 0

Step 3: I-solve nang magkahiwalay ang dalawang factors.

2x + 3 = 0 at 3x – 2 = 0

      2x = -3       3x = 2

  2𝑥/2 = −3/2    3𝑥/3 = 2/3

𝑥 = −3/2         𝑥 = 2/3

Step 4: State your final answer.

There are two roots, namely −𝟑/𝟐   and 𝟐/𝟑.

Solving Quadratic Equations by Completing the Square

2. Ang completing the square ay ginagawa sa pamamagitan ng pagsunod sa mga hakbang na ito:

        a. Ilipat ang lahat ng constant term sa kanang bahagi ng equation.

        b. Hatiin o i-multiply ang magkabilang panig ng equation upang ang maging coefficient ng x² ay 1.

        c. Tukuyin ang constant na idaragdag sa kaliwang bahagi ng equation upang ito ay maging isang perpektong trinomial. Idagdag din ang constant na ito sa kanang panig ng equation. Ang constant na idaragdag ay ang square ng kalahati ng numerical coefficient ng linear term (bx).

        d. I-factor ang kaliwang bahagi ng equation.

        e. I-extract ang square root ng bawat panig ng equation. Ang square root ng kaliwang bahagi ng equation ay linear na ngayon at ito ay katumbas sa 

± square root sa kanang bahagi ng equation.

        f. Lutasin ang bawat isa sa mga linear na equation para makuha ang mga root.

HALIMBAWA 2

        Solve for x in the equation 3x² – 36x + 42 = 0.

Step 1: Ilipat ang constant sa kanang panig ng equation.

  3x² – 36x + 42 = 0

            3x² -36x = -42

Step 2: Hatiin o i-multiply ang equation upang maging 1 ang coefficient ng x². 

Sa ating problema, magagawa natin ito sa ating equation kung idi-divide natin ito ng 3.

            3x² -36x = -42

            x² – 12x = - 14

Step 3: Complete the square.

Paano ito gagawin?

Ang numerical coefficient ng linear term na -12x ay -12. Ang kalahati ng -12 ay -6 at ang square ng -6 ay 36.

Kaya ang constant na idaragdag sa magkabilang panig ng equation ay 36.

                        x² – 12x = - 14

 x² -12x + 36 = -14 + 36

x² – 12x + 36 = 22

Step 4: I-factor ang kaliwang panig ng equation.

x² – 12x + 36 = 22

(x – 6) (x – 6) = 22

(x – 6)² = 22

Step 5: Kunin ang square root ng bawat panig ng equation.

     (x – 6)² = 22

     

Step 6: Kunin ang value ng x.

   

Step 7: State your final answer.

The roots of the equation are 

Solving Quadratic Equations Using the Quadratic Formula

3. Maaaring gamitin ang quadratic formula upang malutas ang anumang quadratic equation.

        Sundin ang mga hakbang na ito sa paggamit ng quadratic equation.

        a. Ilipat ang lahat ng mga termino sa kaliwang bahagi ng equation upang ang kanang bahagi ay maging katumbas ng zero.

        b. Pasimplehin ang equation.

        c. Gamitin ang quadratic formula kung saan:

a = ang coefficient ng quadratic term (ax²); 

b = ang coefficient ng linear term (bx) ; at 

c = ang constant.

HALIMBAWA 3

Solve for h in the equation 3h² – 15h = -18

Step 1: Ilipat ang lahat ng termino sa kaliwang panig ng equation upang maging zero ang kanang panig.

3h² – 15h = -18

3h² – 15h + 18 = -18 + 18

3h² – 15h + 18 = 0

Step 2: Gawing simple ang equation.

3h² – 15h + 18 = 0

h² – 5h + 6 = 0

Step 3: Tukuyin ang value ng a, b, at c sa equation.

h² – 5h + 6 = 0

a = 1 b = -5 c = 6

Step 4: Kunin ang value (roots) ng h gamit ang quadratic formula.

Thus, the roots of our equation are 3 and 2.

Step 5: Optional. Check the answer.

For h = 3 

              3h² – 15h = -18   

3(3)² – 15(3) ≟ -18          

      3(9) – 45 ≟ -18               

        27 – 45 ≟ -18        

               -18 = -18              

For h = 2

            3h² – 15h = -18

      3(2)² – 15(2) ≟ -18

            3(4) – 30 ≟ -18

               12 – 30 ≟ -18

                      -18 = -18

REMEMBER

1 - A quadratic equation is an equation in which the highest power (exponent) of any of its variables is 2.

2 - The standard form of a quadratic equation with a single variable, x, is 

ax² + bx + c = 0where a, b, and c are constants and a ≠ 0.

3 – There are three ways in solving quadratic equations:  factoring, completing the square, and using the quadratic formula.

4 – You can only use the completing the square method as long as the coefficient of x² is 1.

5 – The formula for the quadratic equation is:

6 - When using the quadratic formula, you should be aware of three possibilities. These three possibilities are distinguished by a part of the formula called the discriminant. The discriminant is the value under the radical sign, b² – 4ac.

     A quadratic equation with real numbers as coefficients can have the following:

        a. Two different real roots if the discriminant b² – 4ac is a positive number                 (greater than zero).

        b. One real root if the discriminant b² – 4ac is equal to 0.

        c. No real root if the discriminant b² – 4ac is a negative number (less than                         zero).

--o0o--

Sanggunian: ALS Module: Equations 2

Lesson 5b - Solving Word Problems Using Two variables and Systems of Equations

 Lesson 5 – Applications of Linear Equations: Part 3

        Sa nakaraang aralin ay nai-apply natin ang ating natutunang konsepto at pamamaraan sa paglutas ng mga word problems na kinasasangkutan ng number, digit, uniform motion, money, age, investment, at mixture problems na may isang variable o unknown.

Sa bahaging ito ay pagtutunan naman natin ang pagsagot ng mga word problems gamit ang dalawang variables at/o dalawang equations.

Number Problems

        May mga word problems na nalulutas sa pamamagitan lamang ng isang variable o unknown at isang equation. Gayunpaman, may mga pagkakataon na mas madali ang pagsagot ng mga ito kung gagamit ng dalawang variables at dalawang sistema ng equations.

Tunghayan natin sa mga sumusunod na halimbawa:

HALIMBAWA 1

The sum of the two numbers is 17 and their difference is 7. Find the two numbers.

Let  x  =  the  smaller number

y = the larger number

Given: x + y = 17 (Equation 1)

y – x  = 7         (Equation 2)

Add the two equations.

x + y = 17

 + y – x = 7

           2y = 24

      y = 12

Substitute the value of y into either Equation 1 or 2 to find x.

    y - x = 7 (Equation 2)

         12 – x  = 7

          12 – 12 – x = 7 – 12

        -x = -5 

                  -1( -x = -5)

         x = 5

Check our answer.

        For Equation 1:

x + y = 17

        5 + 12 ≟ 17

      17 = 17

        For Equation 2:

y - x = 7

        12 - 5 ≟ 7

        7 = 7

        Thus, our smaller number is 5 and the larger number is 7.

REMEMBER

We can solve many problems by translating them into systems of equations and using the following problem-solving guidelines:

1.    Understand the problem. Read it carefully and decide which quantities are unknown. (Unawain ang problema. Basahin itong mabuti at magpasya kung aling mga quantities ang hindi alam o unknown.)

2.    Develop a plan. Represent one of the unknown values by one variable and the second unknown by another variable. (Bumuo ng isang plano. Katawanin ang isa sa mga unknown ng isang variable at ang pangalawa ng isa pang variable.)

3.    Carry out your plan. Study the stated facts until you understand their meanings. Then translate the related facts into equations in two variables. Solve the system of equations. (Isagawa ang iyong plano. Pag-aralan ang mga nakasaad na katotohanan hanggang sa maunawaan mo ang mga kahulugan nito. Pagkatapos ay isalin ang mga kaugnay na katotohanan sa mga equation sa dalawang variable. Lutasin ang sistema ng mga equation.)

4.    If available, check your answers using the derivations you made and not the given equation itself. Write a statement to answer the question being asked in the problem. (Kung mayroon, suriin ang iyong mga sagot gamit ang mga derivasyon na iyong ginawa at hindi ang ibinigay na equation mismo. Sumulat ng isang pahayag upang masagot ang tanong sa problema.)

Age Problems

HALIMBAWA 2

        Olive is 4 years younger than Popeye. Twenty years ago, Popeye’s age was 13 years more than half the age of Olive. How old are they now?

        Let    x    =    Olive's age now

                 y    =    Popeye's age now

Labis na makatutulong kung ilalagay natin sa isang table ang mga ibinigay na impormasyon.

        From the table, we can get a system of two equations in x and y.

        x = y – 4 (Equation 1)

y – 20 = ½(x – 20) + 13 (Equation 2)

Simplifying Equation 2, we will have:

y – 20 = ½(x – 20) + 13 (Equation 2)

y – 20 = 1/2x – 10 + 13

y – 20 = 1/2x + 3

y = 1/2x + 23         (Equation 3

        Substitute the value of x in Equation 1 into Equation 3.

        y = ½(y – 4) + 23         (Equation 3)

        y = 1/2y – 2 + 23

1/2y = 21

        2(1/2y = 21)

y = 42

Substitute 42 as the value of y in Equation 1 to find the value of x.

        x = y – 4 (Equation 1)

        x = 42 – 4

        x = 38

Thus, Olive is 38 years old and Popeye is 42 years old now.

        Check our answer.

For Equation 3:

   y = ½(y – 4) + 23 (Equation 3)

42 ≟ ½(42 – 4) + 23

42 ≟ ½(38) + 23

42 ≟ 19 + 23

42 = 42

For Equation 2: 

y – 20 = ½(x – 20) + 13

        42 – 20 ≟ ½(38 – 20) + 13

        22 ≟ 19 – 10 + 13

        22 = 22

     Since we got equality, our answer is correct.

Digit Problems

HALIMBAWA 3

In a three-digit number, the hundreds digit is twice the units digit. If 396 be subtracted from the number, the order of the digits will be reversed. Find the number if the sum of the digits is 17.

Let h = the hundreds digit

t =  the tens digit

u = the units digit

100h + 10t + u =  the number

h = 2u (Equation 1)

        h + t + u = 17     (Equation 2)

                       (100h + 10t + u) – 396 = 100u + 10t + h

100h + 10t + u – 396 -100u – 10t – h = 396

                                   99h – 99u = 396

                          99h/99 - 99u/99 = 396/99

                                           h – u = 4          (Equation 3)

Substitute h = 2u into Equation 3.

h – u = 4

         2u – u = 4

         u = 4

Substitute  4 as u into Equation 1.

h = 2u

h = 2(4)

h = 8

        Substitute u = 8 and h = 4 into Equation 2 to find the value of t.

h + t + u = 17

4 + t + 8 = 17

  4 – 4 + t 8 – 8 = 17 – 4 – 8

                       t = 5

Substitute these 3 values into our number.

100h + 10t + u =

100(8) + 10(5) + 4 = 

800 + 50 + 4  = 854

Thus, our number is 854.

Let us check our answer by substituting our values into our three equations.

For Equation 1:

h = 2u

8 ≟2 (4)

8 = 8

For Equation 2:

h + t + u = 17

      8 + 5 + 4 ≟ 17

         17 = 17 

For Equation 3:

h – u = 4

8 – 4 ≟  4 

       4 = 4

Since our left and right terms for our three equations are balanced, our answer is correct.

Mixture Problems

HALIMBAWA 4

Tikyo wants to make a 1000 ml of 50% alcohol solution mixing a quantity of a 20% alcohol solution with a 70% alcohol solution. What are the quantities of each of the two solutions he has to use?

        Let     x    =    amount of 20% alcohol

                  y    =    amount of 70% alcohol

Analyzing the problem, we find that two conditions must be met.

Condition 1

liters of 20% solution + liters of 70% solution = liters of 50% solution

x + y = 1000

        Condition 2

        pure alcohol in 20% solution + pure alcohol in 70% solution = pure alcohol in 50% solution

                        0.20x + 0.70y = 0.50(1000)

The system we must solve is:

       x + y = 1000      (Equation 1) 

 0.20x + 0.70y = 0.50(1000)      (Equation 2)

        Simplify Equation 2.

0.20x + 0.70y = 0.50(1000)

0.20x + 0.70y = 500

10(0.20x + 0.70y = 500)

2x + 7y = 5000 (Equation 3)

Multiply Equation 1 by 2.

x + y = 1000

2( x + y = 1000)

2x + 2y = 2000         (Equation 4)

Subtract Equation 4 from  Equation 3.

2x + 7y = 5000         (Equation 3)

    -     (2x + 2y = 2000)         (Equation 4)

            5y = 3000

         5y/5 = 3000/5

              y = 600

Substitute 600 as the value of y into Equation 1.

x + y = 1000

          x + 600 = 1000

        x + 600 – 600 = 1000 – 600

            x = 400

Thus, Tikyo has to use 400 liters of 20% alcohol and 600 liters of 70% alcohol to obtain 1000 liters of 50% alcohol.

        Let us check to confirm our answer.

          x + y = 1000      (Equation 1) 

400 + 600 ≟ 1000

          1000 = 1000

 0.20x + 0.70y = 0.50(1000)    (Equation 2)

      10(0.20x + 0.70y ≟ 500)

2x + 7y ≟ 5000

     2(400)x + 7(600) ≟ 5000

      800 + 4200 ≟ 5000

         5000 = 5000

Since we got equality in our original equations, our answer is correct.

Money Problems 

HALIMBAWA 5

The price of 3 chairs and 2 tables is ₱4500 and the price of 5 chairs and 3 tables is ₱7000. Find the price of 2 chairs and 2 tables.

Let  c =  the price of a chair

t = the price of a table

Given:

3c + 2t = 4500         (Equation 1)

5c + 3t = 7000         (Equation 2)

Subtract Equation 1 from Equation 2.

5c + 3t = 7000         (Equation 2)

  -   (3c + 2t = 4500) (Equation 1)

2c + t   =  2500

  2c – 2c + t = 2500 – 2c

          t = 2500 – 2c  (Equation 3)

Substitute Equation 3 into either Eq’n 1 or 2.

If Equation 2:

5c + 3t = 7000

  5c + 3(2500 – 2c) = 7000

5c + 7500 – 6c = 7000

  -c + 7500 – 7500 = 7000 – 7500

-c = -500

-1(-c = -500) 

                                c = 500

        Substitute 500 for c into Equation 3 to find t.

t = 2500 – 2c  (Equation 3)

t = 2500 – 2(500)

t = 2500 – 1000

t = 1500

Thus, the price of two chairs (2 x 500 = 1000) and two tables (2 x 1500 = 3000) is ₱ 4000 ( 1000 + 3000).

Let us check to confirm our answer.

                  3c + 2t = 4500 (Equation 1)

3(500) + 2(1500) ≟  4500

       1500 + 3000  ≟ 4500

                     4500 = 4500

                                 5c + 3t = 7000 (Equation 2)

5(500) + 3(1500) ≟ 7000

       2500 + 4500 ≟ 7000

                    7000 = 7000

        t = 2500 – 2c  (Equation 3)

1500 ≟ 2500 – 2(500)

1500 ≟ 2500 – 1000

1500 = 1500

Since we got equality in our original equations, our answer is correct.

Investment Problems 

HALIMBAWA 6

Bongbong invested ₱11,000.  Part of his money is invested in bonds which yield 8% and the remainder is invested in the money market which yields 10%.  His total annual income from these investments is ₱1,020.  Find the amount he has invested in each kind of investment.

Let      b     =  amount he invested in bonds at 8% 

                    m         = amount he invested in money market at 10%

                0.08b = annual income from 8% bonds investment

        0.10m =  annual income from 10% money market investment

b + m = 11000     (Equation 1)

       0.08b + 0.10m = 1020 (Equation 2)

        Multiply Equation 2 by 100.

0.08b + 0.10m = 1020         (Equation 2)

   100(0.08b + 0.10m = 1020)

          8b + 10m = 102000 (Equation 3)

        Multiply Equation 1 by 8.

b + m = 11000         (Equation 1)

8(b + m = 11000)

8b + 8m = 88000         (Equation 4)

Subtract Equation 4 from Equation 3.

         8b + 10m = 102000 (Equation 3)

     -   (8b + 8m = 88000) (Equation 4)

           2m = 14000

        2m/2 = 14000/2

             m = 7000

        Substitute 7000 as the value of m into Equation 1 to find b.

b + m = 11000         (Equation 1)

        b + 7000 = 11000

    b + 7000 – 7000 = 11000 – 7000

            b = 4000

Thus, Bongbong invested ₱4000 in bonds at 8% and ₱7000 in money market at 10%.

Let us check to confirm our answer.

b + m = 11000         (Equation 1)

          4000 + 7000 ≟ 11000

                  11000 = 11000

  0.08b + 0.10m = 1020         (Equation 2)

  100(0.08b + 0.10m ≟ 1020)

         8b + 10m ≟ 102000

 8(4000) + 10(7000) ≟ 102000

        32000 + 70000 ≟ 102000

             102000 = 102000

Since we got equality in our original equations, our answer is correct.

Motion Problems 

HALIMBAWA 7

A boat can travel 16 miles up a river in 2 hours.  The same boat can travel 36 miles downstream in 3 hours.  What is the speed of the boat in still water?  What is the speed of the current?

Let  s =  speed of the boat in still water

                v = speed of the current

        Kung hindi pa bihasa sa ganitong klaseng problema, makatutulong nang malaki ang paggawa ng table upang ilagay ang ating mga datos.

    

        Mapapansin na ang rate (s + v) ng boat downstream ay ang speed nito sa kalmang tubig (still water) PLUS ang speed ng current  (o daloy ng tubig)  dahil mas mabilis ang takbo ng bangka kung paibaba o paayon sa daloy ng tubig.  Kung salungat naman sa daloy ng tubig o upstream, magiging mabagal ang takbo ng bangka kaya ibabawas natin ang speed ng current sa kanyang speed sa still water upang makuha ang rate (s - v) nito.

        Mula sa talahanayan o table, narito ang ating nakuhang equation:

16 = 2(s -  v) (Equation 1)

36 = 3(s + v) (Equation 2)

Divide Equation 1 by 2.

16 = 2(s - v) (Equation 1)

       16/2 = 2(s - v)/2

    8 = s - v (Equation 3)

Divide Equation 2 by 3.

36 = 3(s + v) (Equation 2)

       36/3 = 3(s + v)/3

12 = s + v (Equation 4)

        Add Equation 3 and Equation 4.

8 = s - v (Equation 3)

   +    12 = s + v (Equation 4)

         20 = 2s

     20/2 = 2s/2

        10 = s

Substitute 10 for s in any of the equations to find v. Let’s take Equation 4.

12 = s + v (Equation 4)

12 = 10 + v

12 – 10 = 10 – 10 + v

 2 = v

        Thus, the speed of the boat in still water is 10 miles per hour while the speed of the current is 2 miles per hour.

Let us check to confirm if our answer is correct.

16 = 2(s -  v) (Equation 1)

16 ≟ 2(10 – 2)

16 ≟ 2(8)

16 = 16

36 = 3(s + v) (Equation 2)

36 ≟ 3(10 + 2)

36 ≟ 3(12)

36 = 36

Since we got equality in our original equations, our answer is correct.

Pagsasanay 

Solve the following word problems and check your answers:

1.    If one number is three times as large as another number and the smaller number is increased by 19, the result is 6 less than twice the larger number. What is the larger number?

2.    A silver coin is 28 years older than a bronze coin. In 6 years, the silver coin will be twice as old as the bronze coin. Find the present age of each coin.

3.    The tens digit of a two-digit number is twice the units digit. If the digits are reversed, the new number is 36 less than the original number. Find the number.

4.    You need a 15% acid solution for a certain test, but your supplier only ships a 10% solution and a 30% solution. Rather than pay the hefty surcharge to have the supplier make a 15% solution, you decide to mix 10% solution with 30% solution, to make your own 15% solution. You need 10 liters of the 15% acid solution. How many liters of 10% solution and 30% solution should you use?

5.    The cost of admission to a popular music concert was ₱162 for 12 children and 3 adults. The admission was ₱122 for 8 children and 3 adults in another music concert. How much was the admission for each child and adult?

6.    Isko has invested in two savings accounts. One earns 10% and the other earns 15%. He invests ₱200 more in the account that earns 15%. The total interest earned for one year is ₱230. How much is invested in each account?

7.    A steamer goes downstream and covers the distance between two ports in 4 hrs., while it covers the same distance upstream in 5 hrs. If the speed of the stream is 2km/h, find the speed of the steamer in still water.

8.    The present ages of Bongbong and Sara are in the ratio 3:4. Five years from now, the ratio of their ages will be 4:5. Find their present ages.

Note: The above sample word problems were taken from different internet sites. 

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