MANILA: Room 206, JPD Building, CM Recto Avenue, Manila CEBU: 4/F J. Martinez Bldg., Osmeña Blvd., Cebu City Telephone Number: (02) 516 7559 (Manila) E-Mail: buksmarquez1 @yahoo.com (032) 254-9967 (Cebu) Review MODULE – MATHEMATICS (Algebra) WORDED PROBLEMS ~ Number Problems ~ Sample of words used to represent mathematical operations: Add add sum increased by more than total in all gain plus Subtract subtract difference decreased by fewer less than diminished by reduced by minus Multiply multiply times product double twice Divide divide quotient per divided equally split into fraction ratio of Sample sentences with corresponding algebraic expression: Statement Five increased by four times a number. Eight less than twice a number. Three times a number, increased by 9. The product of 4, and a number decreased by 7. A number repeated as a factor of 3 times. Three times a number, divided by 10 equal 15. 10 less than the quotient of a number and 2 is zero. 12 more than the product of a number and 2 is 36. A third of sum of a number and two. Algebraic Expression/s 5 + 4x 2x – 8 3x + 9 4(x – 7) x3 3x/10 = 15 (x/2) – 10 = 0 2x + 12 = 36 (x + 2)/3 1. Find the product of two numbers such that twice the first added to the second equals 19 and three times the first is 21 more than the second. 2. The sum of the three numbers is 51. If the first number is divided by the second, the quotient is 2 and the remainder 5; but if the second number is divided by the third, the quotient is 3 and the remainder 2. Determine the numbers. 3. A farmer has 600 plants arranged in rows, but they require an irrigation ditch. He finds that he must take out 5 plants, from each row. He can then make 6 more rows. Find the original number of plants in each row. ~ Clock Problems ~ 4. Determine the time after 2 o’clock will the hands of the clock form the first 90 degrees? 5. At this moment, the hands of a clock in the course of normal operation describe a time somewhere between 4:00 and 5:00 on a standard clock face. Within one hour or less, the hands will have exactly exchanged positions. What time is it now? ~ Age Problems ~ 6. Six years ago, Kazuha was five times as old as Xiao. In five years, Kazuha will be three times as old as Xiao. What is the present age of Xiao? 7. When I am as old as my father is now, I shall be five times as old as my son is now. By then my son will be eight years older than I am now. The combined ages of my father and me are 100 years. How old is my son? ~ Money Problems ~ 8. The selling price of iPhone 13 Pro is double that of its cost. If the iPhone 13 Pro was sold to a costumer at a profit of 25% of the net cost, how much discount was given to the costumer? ~ Motion Problems ~ 9. Two pals A and B run at constant speeds along a circumference track 1350 m in circumference. Running in opposite directions they meet every 3 minutes, while running in the same direction they are together every 27 minutes. Determine their speeds in km/hr. 10. A boatman rows to a place 48 miles distant and back in 14 hours, but find that he can row 4 miles with the stream in the same time as 3 miles against the stream. Find the rate of the stream. ~ Work Problems ~ 11. [NOV2021] An experienced roofer can roof a house in 26 hours. A beginner roofer needs 39 hours to complete the same job. Find how long it takes for the two to do the job together? 12. A certain job can be finished in 20 days by 30 men. On the 8th day of work 10 more men were added. How many days will it take to finish the job? ~ Mixture Problems ~ 13. The Jager-Bomb is a popular party-starting shot. Normally a 4.2-ounce mix contains 35% Jägermeister. Arataki Itto likes to act tough and wants to try a mix with 70% Jägermeister. How many ounces of pure Jägermeister must be added to a normal mix to obtain the desired mix? 14. Vessel A contains a mixture of 12 liters of wine and 18 liters of water. Vessel B contains 9 liters of wine and 3 liters of water. How many liters must be drawn from Vessel A to produce a mixture containing 7 liters of wine and 7 liters of water? ~ Variation Problems ~ Direct Variation: 𝑦=𝑘𝑥 Inverse Variation: 𝑦 = 𝑘/𝑥 Joint Variation: 𝑦 = 𝑘 𝑥𝑧 Combined Variation: 𝑦 = 𝑘 𝑧/𝑥 15. [NOV2021] The intensity of light (in foot-candle) varies inversely as the square of x, the distance in feet from the light source. The intensity of light 2 ft from the source is 80 ft-candles. How far away is the source if intensity of light is 5 ft-candles? SEQUENCES AND SERIES ~ Arithmetic Progression ~ 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 Sum of Arithmetic Progression: 𝑆𝑛 = 𝑆𝑛 = 𝑛 (𝑎 + 𝑎𝑛 ) 2 1 𝑛 [2𝑎1 + (𝑛 − 1)𝑑] 2 1. The sixth term of an arithmetic progression is 17 and the thirteenth term is 38. Find the nineteenth term. 2. What is the sum of all odd integers between 7 and 777? ~ Geometric Progression ~ 𝑎𝑛 = 𝑎1 𝑟 𝑛−1 Sum of Geometric Progression: 𝑆𝑛 = 𝑎1 (1 − 𝑟 𝑛 ) (1 − 𝑟) Sum of Infinite Geometric Progression (only for | r | < 1.0): 𝑎1 𝑆𝑛 = (1 − 𝑟) 3. The number 28, x + 2, 112 form a geometric progression. What is the 10th term? What is the sum of the geometric progression? 4. The first swing of the pendulum is 50 cm. If each swing is 80% of the preceding swing, how far does the pendulum travel before coming to rest? ~ Harmonic Progression ~ 5. The 3rd term of a harmonic progression is 15 and the 9th term is 6. Find the 11th term. 6. Determine the geometric mean of two numbers if the arithmetic mean and harmonic mean are 4 and 9 respectively.
