Problem Solving

Math 2: Problem SolvingSteps Read the problem (try to understand it upon the first read)Determine the type of problemIdentify the variablesSet up the equationSolve the equation/sFind the final solutionCheck the solution*Use trial and error only if you think it will be faster*(The examples in the following slides emphasize the equations and not the solutions)Coin ProblemsA1Q1 + A2Q2 + A3Q3 + .... = Total AmountWhere A= value of coins and Q = quantity of coinsExample: Percy has a box in his tent that contains 45 coins. Some are nickels and the rest are dimes. If the total value of the coins is $3.25, how many of each coin does he have? Let x= # of nickels and 45-x = # of dimes5(x) + 10(45-x) =325X = 25 and 45-x = 20 Final Solution: Percy has 25 nickels and 20 dimes in his boxPercent ProblemsPercentage = Rate x Base Where Percentage = Part of the wholeRate = usually with the % or in decimal form Base = whole or principal amount ORSimple InterestInterest = Percentage x Rate x Time Where Principal = amount originally invested or loanedRate = interest rate of the bank or institutionTime= how long the money is investedJesun Burn earned $8,000 last year by selling fish balls. He invested part at 2% simple interest and the rest at 3%. He earned a total of $210 in interest. How much did he invest at each rate?

PercentageRateTimeInterestX2%1 year0.02x8,000-x3%1 year0.03(8,000-x)(Interest at 2%) + (Interest at 3%) = total interest0.02x + 0.03(8,000-x) = $210 X = 3,0008,000-x = 5,000Final answer: $3,000 and $5,000 were invested at 2% and 3% respectively B. Compound InterestAmount after one year = Principal x (1+Rate) What is the interest compounded per annum on a principal of P6,000 when the amount after one year is P7,200? Let x = rate or the interest compounded per annum7,200 = 6,000(1+x)x= 0.2 or 20%Final answer: The interest compouned per annum is 20%Percent of Increase or DecreaseIn decimal, % = new price original price original priceIf the % is positive, the price increasedIf the % is negative, the price increasedA belt costs P650 but Chloe was able to purchase it at a price of P601.25. What was the percentage of the decrease? % = 601.25-650 / 650 = -0.075 or 7.5Final answer: The percentage decrease was 7.5%DiscountDiscount = %Discount x (Original Price)Discounted Price = Original Price DiscountNew Price = (1-%Discount) x Original PriceGustong bumili ni Stark ng pitaka na nagkakahalaga ng P475. Kung may tawad na 20%, magkano ang kanyang babayarin? Discount = 0.20(475) = P95Discounted price = P475-P95 = P380Final answer: Ang babyarin ni Stark at P380CommissionCommission = %Commission x Total EarningsA car agent sells a brand new car for P283,000. If he is given 10% commission, how much will he receive? Commission = 0.1(283,000)= P28,300Final answer: The car agent will receive P28,300 in commissionTaxesTax = %Tax x (Original Price) Price with tax = Original Price + TaxNew Price = (1+%Tax) x Original Price20% tax is imposed on purchasing a pair of rare sandals. If the original price of the sandals is P35,000 how much will one have to pay upon buying? Tax = 0.2(35,000) = P7,000P35,000 + 7,000 = 42, 000 Final answer: One has to pay P42,000 for the shoesMixture ProblemsDry MixA1Q1 + A2Q2 + A3Q3 + .... = Total AmountWet Mix%origQorig + %mixQmix = %final x (Qorig + Qmix)Removal of solution%origQorig - %mixQmix = %final x (Qorig - Qmix)Replacing of a solution%origQorig - %origQorig + %mixQmix = %finalQorig

How many liters of a 25% alcohol solution should Zyrene mix with 20 liters of a 55% solution to get a 30% solution?

OriginalAddedMixture% alcohol55%25%30%Volume20 LxL20 + xLAmt of alcohol0.55(20)0.25x0.30(20+x)0.55(20) + 0.25x = 0.30(20+x)X = 100 litersFinal answer: She should have 100 liters of the 25% solutionAge ProblemsThere is no general formula. Derive the equations based on how they are stated in the problem. Use direct translation of words into math symbols. If x is the present age then The age n years ago is x-n The age m years from now is x+mHelen is 5 years younger than Paris. 10 years from now, Paris age will be 50 years less than twice Helens age. How old is Paris now? Let P = Paris present age; H = Helens present agePresent: H = P-5; 10 years from now: P + 10 = 2(H+10) 50 P + 10 = 2(P-5+10)-50; P = 50 Final answer: Paris is 50 years old now FractionsA fraction is a ratio of 2 numbers. If the value of the graction is 2/3, the numerator : denominator is not only 2:30 but also 1:1.5 or 30:45... Etc. Thus, the numerator may be represented by 2x, the denominator by 3x, and the fraction by 2x/3x. The sum of the numerator and denominator of a fraction is 16. If the value of the fraction is 1/3, what is the denominator? Let x / 3x = original fraction, where x is the common factor. X + 3x = 16; x = 4Hence, 3x = 12 Final answer: The denominator is 12Consecutive IntegersConsecutive integers are one unit apart and can be represented by x, x+1, and x+2 and so onConsecutive even or odd numbers are 2 units apart and can be represented by x, x+2, x+4, and so on Ang sum ng tatlong magkakasunod na even integer ay 84. Ano ang ikalawang integer? x + (x+2) + (x+4) = 84 x = 26; x + 2 = 28 Final answer: Ikalawang even integer = 28 Digit/Number ProblemsA 3-digit number whose hundreds digit is x, tens digit is y, and units digit is z can be expressed as 100x + 10y + z. The reverse of this number is written as 100z + 10y + x. The sum of the digits of a two-digit number is 12. If the digits are reversed, the new number is 54 more than the original number. Find the two-digit number.

Let x = tens digit; y = units digit10x + y = original number10y + x = reversed numberEquation 1: x + y = 12Equation 2: new = original + 5410y + x = 10x + y + 54 y - x = 6 Now...y + x = 12y x = 62x = 6 x = 3 and 3 + y = 12 so y 9 Final answer: the original number is 39

Uniform Motion ProblemsDistance = Rate x Time Motion in Opposite directionsD1 + D2 = Dtotal or R1T1 + R2T2 = Dtotal Motion in Same direction (catch-up)D1 = D2 or R1T1 = R2T2 Round TripDgoing = Dreturning or RgoTgo = RretTret

Mica drives her car and it takes her 30 minutes to reach her work. As she returns, she rides the bus and it takes her 45 minutes to reach her home. The average speed of the bus is 12 miles per hour less than her speed when driving. Find the distance she travels to work. Let x = Micas speed when driving

RateTimeDistanceHome to workX 30 min = hr1/2xWork to homeX-1245 min- hr (x-12) x = (x-12)X = 36 mphD = x = 18 miles Final answer: Mica drives a distance of 18 miles to her work Motion Involving Currentx = rate of the vehicle in still currenty = rate of the currentWith the wind or downstream x + y Against the wind or upstreamx - yIt takes 3 hours for a boat to travel 75 miles downstream. The same boat can travel 60 miles upstream in 4 hours. Find the speed of the boat in still water and the rate of the current. X = rate of the boat in still water Y = rate of the current

RateTimeDistanceDownstreamx+y375Upstreamx-y460Rate = Distance / Timex + y = 75/3x y = 60/4x + y = 25x y = 152x = 40 x = 20 mph The rate of the current is 5 mph, the speed of the boat is 20 mph

Work ProblemsWork done = Rate x TimeWork done together (R1 + R2) T = WWork done in opposition (Rfaster Rslower) T = WAn inlet pipe can fill a tank of water in 6 hours while an outlet pipe can drain the same tank in 8 hours. If a half-full tank is first drained in two hours and is filled while the outlet pipe is still open, how long should the inlet pipe be opened to fill the tank? Time to fill/empty the tankRateActual filling/draining timePart of work doneInlet pipe61/6xx/6Outlet pipe81/8x + 2(x + 2)/ 8x / 6 (x+2) / 8 = 1 / 2 (tank)X = 18 The inlet pipe should be opened for 18 hours

Clock ProblemsUsing the 12 o clock position as a reference point, we subdivide the clock into 60 equal arcs. When the minute-hand moves 60 unites, the hour-hand moves 5 units. As the minute hand moves M units, the hour hand moves 5M/60 = M/12 units. When the hands form an angle, the units between them is angle/6. Arcs = M/12 Angle= Arc/6M = 60H (+-) 2A 11M = no. Of minutes when the 2 hands form the specified angleH = no. Of hoursA = angle the 2 hands form in the final positionUse + when the minute hand is farther than the hour hand in the final positionUse = when the hour hand is farther than the minute hand in the final positionWhen the hands point in the same direction, then A = 0 At what time after 12:00 noon will the hour hand and minute hand of a clock first form an angle of 120 degrees? H = 0 (since the hour starts at 12 noon)A = 120 We use + here since the minute-hand overtakes the hour hand in the final positionM = 60(0) + 2(12) 11 M = 21.818 The hands of the clock will first form 120 degrees the first time at 12:21.818 PM