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Miscellaneous Problems involving EquationsQ.1: The relation between degree Celsius and degree Fahrenheit is given by, F=9/5+32. What Celsius temperature is equivalent to 176° F.F = 95c + 32176 = 95c + 32176 – 32 = 95c144 = 95c144 X 5 = 9C720 = 9C7209 = C80 = CQ.2: Suppose a firms profit for a product is given by, P = 5x/4 – 300.a) How many units must be produced to make a profit of $150?P = 5x4 – 300150 = 5x4 – 300150 + 300 = 5x4450 = 5x4450 X 4 = 5x1800 = 5x18005=x →360=x b) Producing how many units will result in a profit of 0?P = 5x4 – 3000 = 5x4 – 300300 = 5x4300 X 4 = 5x1200 = 5x12005=x →240=xQ.3: A bicycle cost the wholesaler $28, what will the retailer sell it for if the wholesaler‘s markup is 20% of the wholesale selling price and the retailers markup is 30% of the retail selling price?Whole seller Cost = $28 Whole seller Cost = 80% of Whole seller SaleWhole seller Sale price= 28 x 10080 Whole seller Sale price= 35Retail seller Cost = $28Retail seller Cost = 70% of Retail seller SaleRetail seller Sale price= 28 x 10070 Retail seller Sale price= 50Q.4: A wildlife management, the capture – mark, recapture technique is used to estimate the population of fish and birds in an area or to measure the infestation of insects such as Japanese beetles. Suppose 100 individual of the species being studies are caught, marked and released. After one week 100 more are caught. To estimate the total number of individuals, the following relationship is used:Total marked found in 2nd captureTotal in 2nd capture= Total number markedTotal papolationa) If the 2nd capture of 100, it is found that 3 are marked, what is the population?3100= 100Total papolation(3)(Total population) = (100)(100)(3)(Total population) = 10000Total population = 100003 -> 3333.333b) Suppose that 1000 beetles are capture, marked and release. Suppose further in the 2nd capture of 1000, it is found that 63 are marked, what is the population estimated? 631000= 1000Total papolation(63)(Total population) = (1000)(1000)(33)(Total population) = 1000000Total population = 100000063 -> 15873.015Q.5: the equation F = 9/5 C + 32, describe the relation between temperature readings in Fahrenheit and Celsius. What temperature will the two readings be the same?F = 95c + 32C = 95c + 325C = 9C + 325C – 9C = 32– 4C = 32C = -324C = -8Q.6: the equation relating temperature reading in Fahrenheit and Celsius is given by, C = 5/9 (F-32), a) What Fahrenheit reading is equivalent to 0° Celsius?C = 59 (F-32)0 = 59 (F-32)0 x 9= 5(F-32)0 = 5(F-32)05 = F-320 = F-32F = 32a) What Fahrenheit reading is equivalent to 100° CelsiusC = 59 (F-32)100 = 59 (F-32)100 x 9= 5(F-32)900 = 5F-160900 + 160 = 5F1060 = 5F10605 = FF = 212Q.7: It has been noted that for adults over 5 feet tall in the Northeast United States, the weight is related to their heights according to, 3w + 110 = 11(h – 20) where w is measured in pounds and h is measure in inches. Use the formula to answer:a) Find the weight of an adult whose height is 5 feet 6 inches (66 Inches).3w + 110 = 11(h – 20)3w + 110 = 11(66 – 20)3w + 110 = 11(46)3w + 110 = 5063w = 506 – 110 → 3w = 396W = 3963 → W = 132 b) Find the height of an adult whose weight is 160 pounds.3w + 110 = 11(h – 20)3(160) + 110 = 11(h – 20)480 + 110 = 11(h – 20)590 = 11(h – 20)59011 = h – 2053.64 = h – 2053.63 + 20 = h → H= 73.64 (6 feet 1 inches)Q.8: A 90 minute’s recording tape is 10.125 inches long. How long is a 60 minutes tape?90 minutes tape length = 10.125 inches1 minutes tape length = 10.12590 inches1 minutes tape length = 0.1125 inches60 minutes tape length = 0.1125 x 60 60 minutes tape length = 6.75 inchesQ.9: the manager of a hard ware store marks all items so that the mark-up is 25% of the cost. How much would a saw sale for the cost the manager $25.50?Total mark up = 25% of costCost = $25.50Saw sale = total mark-up + costSaw sale = 0.25(25.50) + 25.50Saw sale = 6.375 + 25.50Saw sale = $31.875Q.10: A firm produces two types of widgets. “A” & “B”. Each unit of “A” requires 3 work hour and each unit of “B” require 4 work hours. If there are 430 work hours available per week, how many units of “A” can be produce if 40 units of “B” are produced?Works hour available per week = 430Work hour require unit A = 3Work hour require unit A = 4Suppose:Number of unit A = xNumber of unit B = ySoUnit A production + Unit B production = Total available hours3x + 4y = 430Putting values3x + 4(40) = 4303x + 160 = 4303x = 430 – 160 -> 270X = 2703 X = 90Q.11: the hill crest youth organization determines the number of youths it has served by adding its active registration and the number of dropouts. If the dropout rate is 13 the total served, find the total numbers of youth served when there are 6000 youths actively registered. Number of youth served = Active register + DropoutSuppose:Number of youth served = xSo,Number of youth served = Active register + Dropoutx = 6000 + 13 x3 x = (3)(6000) + x3x = 18000 + x3x- x = 180002x = 18000x = 180002x=9000Q.12: A city has population “P” at the beginning of the year. The birth rate during the year is 10 per thousand and death rate during the year is 12 per thousand of year’s original population. During the year, 360 people moved into city and 190 moved away. If the population at the end of the year is 30,110, what was the population at the beginning of the year?Population at begining=xSo,Population at beginning + birth rate – death rate +moved in – moved out = Population at endx+101000x-121000x+360-190=30110x1+ 101000- 121000=30110-360+190x1000+10-121000=29940x9981000=29940x= 29940(1000)998→ 29940000998x= 30,000Q.13: A retailer wants a 30% mark-up on the selling price of an item that cost him $214.90, what selling price should be charged?Retailer Cost = $214.90Retailer Cost = 70% of Retailer SaleRetailer Sale = 214.9 x 10070 Retailer Sale = 30.7Q.14: A retailer sells a house and receives a 5% commission on its selling price. If the retailer commission is $4250, what is the selling price?Commission = $4250Commission = 5% of Selling price4250 = 5100S4250 x 1005=S 85000 = SQ.15: A total price of a new car (including 6% the sale tax) is $ 10,033. How much of this is tax?x=9000Total price of car = 10,033Sale tax = 6% of selling priceLetSelling price = xSelling price + tax = Total Pricex+ 6100x=10033x1+ 6100=10033x100+6100=10033 → x106100=10033x= 10033(100)106→ 9465.1SoTax = 6% of selling priceTax = 61009465.1 -> 567.9Q.16: A toaster cost the ace department store $22.74, if the store marks the toaster up 40% of the selling price, what is the selling price?Toaster Cost = $22.74Toaster Cost = 60% of Toaster SaleToaster Sale = 22.74 x 10060 Toaster Sale = 37.9Q.17: A room air conditioner cost the wholesaler $154.98, if wholesaler mark-up is 10% of the wholesaling price, if retailer mark-up is 30% of the retail selling price, for what does the retailer sell the air conditioner?Whole seller Cost = $154.95Whole seller Cost = 90% of Whole seller SaleWhole seller Sale price= 154.98 x 10090 Whole seller Sale price= 172.2Retail seller Cost = $172.2Retail seller Cost = 70% of Retail seller SaleRetail seller Sale price= 172.2 x 10070 Retail seller Sale price= 246Q.18: An electric mixture retails for $48.54, which include a mark-up of 40% of the retailer and 20 % of the wholesaler. If these mark-up are on the base of selling price:a) What did the mixer cost the retailer?Retail seller Cost = ?60% of Retail seller Sale = Retail seller Cost 60100 x 48.54 = Retail seller Cost29.124 = Retail seller Costb) What did the mixer cost the whole seller?Whole seller Cost = ?80% of Whole seller Sale price= Whole seller Cost 80100 x 29.124 *= Whole seller Cost23.29 = Whole seller Cost*whole seller sell is also a retail seller costQ.19: A dinner set costs the wholesaler $189, and mark-up is 10% of the selling price, if the retailer mark-up is 30% of his selling price for the item, what is the retailer selling price?Whole seller Cost = $189Whole seller Cost = 90% of Whole seller SaleWhole seller Sale price= 189 x 10090 Whole seller Sale price= 210Retail seller Cost = $210Retail seller Cost = 70% of Retail seller SaleRetail seller Sale price= 210 x 10070 Retail seller Sale price= 300Q.20: A wholesaler wants to establish a list price for an item so that his selling price gives a trade discount of 10% of the list price. If he makes a mark-up of 20% of this selling price and if his cost is $160, what list price should be established? Discount= 10% of list priceMark-up = 20% of selling priceCost = $160List Price = ?LetList price = xWe know that (Discount + Selling price = List Price)80% of selling price = Cost80100 x S=160S = 160 x 10080→200NowDiscount + Selling price = List Price10% of x + 200 = xx (1- 0.1) = 200x = 2000.9→ 222.23Q.21: To earn an A in a course, a student must get at least a 90 average on four tests and final examination with the final exam. Weighted twice that of any test. If the four test scores are 93, 69, 89 and 97, what is the lowest score the student can earn on the final exam. And still get an A in the course?SupposeNumber in final = xSo93+69+89+92+2x6=902x + 348 6 = 902x + 348 = 90 x 62x + 348 = 5402x = 540 – 3482x = 192x=1922 x=96Q.22: Suppose a professor count the final examination as being equal to each of the other test in her course, and she will also change the lowest test score to match the final examination score if the final exam score is higher. If a student’s four test scores are 83, 67, 52 and 90, what is the lowest score the student can earn on the final examination and still obtain at least an 80 average for the course?SupposeMarks obtained in final = xSo83+67+x+90+x5=802x + 240 5 = 802x + 240 = 80 x 52x + 240 = 4002x = 400 – 2402x = 160x=1602 x=80 Q.23: An antique collector sold two pieces of furniture for $480 each. For one of them, this represent 20% loss, for the other it was a 20% profit (based on the cost of each item). How much did he make or lose on the transaction?1st pieceSelling price = $480Profit = 20% of costSoSelling price = cost + Profit480 = C + 20100C480 = C (1+ 20100) 480 = C ( 100+20100)480 =C 120100480 x 100120 = C400 = C2nd pieceSelling price = $480Loss = 20% of costSoSelling price = cost - Loss480 = C - 20100C480 = C (1- 20100) 480 = C ( 100-20100)480 =C 80100480 x 10080 = C600 = CTotal cost = 400 + 600 = 1000Total selling price = 480 + 480 = 960So this represents loss of $40Q.24: A women making $2000 per month has hersalary reduced by 10% due to sluggish sales. One year later, after a dramatic improvement in sales, she is given a 20% raise over her reduce salary. Find her salary after the raise. What present change in this from $2000 per month?Data from question:Original salary = 2000Reduction = 10% -> 10100 x 2000=200Reduced salary = 2000 – 200 = 1800After reduction raise the salary. So,Raise in salary = 20% -> 20100 x 1800=360New salary = 1800 + 360 = 2160Change:Change in salary = 2160 – 2000 = 160Percentage change in salary = 160200 x 100Percentage change in salary = 8%Q.25: A car dealer purchase 20 automobile for $8000 each. If he sells 16 of them at profit of 20%, for how much he sell the remaining 4 to obtain an average profit of 18%?Cost of 1 car = 8000Cost of 20 car = 8000 x 20 = 160,000Profit = 18%Profit = 18100 x 1,60,000=28,800Selling price of 20 cars = 160000 + 28800 = 188,800Cost of 16 car = 8000 x 16 = 128,000Profit = 20%Profit = 20100 x 1,28,000=25,600Selling price of 16 cars = 128000 + 25,600 = 153,600Selling price of 4 cars = 188,000 - 153,600 = 35200Q.26: One safe investment pays 10% per year and a more risky investment to have an income of $20000 per year from her investment. How much should she invest at each rate?Profit on safe investment is = 10%Profit on Risky investment is = 18%Total investment = 145,600Expected income = 20,000SupposeSafe investment = x & Risky investment = yx + y = 145,600 -> x = 145,600 – y -------- (1)10100x+ 18100y=20000 -> 10x + 18y = 2000000------- (2)10(145,600 – y) + 18y = 200000014, 56,000 - 10y + 18y = 20000008y=2000000-1456000 = 544000y= 5440008 , 68000x=145600-68000 = 776010100 x 7760=776 18100 x 68000=12240So 776 + 12240 = 20,000Q.27: A bank loaned $118,500 to a company for the development of two products. If the loan for product A was for $34,500 more than that for product B. How much was loaned for each product?Total Loan = 118,500Loan at product B = xLoan at product A = x+34,500Solutionx+ x+34,500=118,5002x = 118,500 – 34,500 = 84,000x= 84,0002=42,000SoLoan at product B = 42,000Loan at product A = 42,000+34,500Loan at product A = 76,500Q.28: A women has $23,500 invested in two rental properties. Her total income from the two properties is $ 2550. How much is her income from each property if she earns 10% and 15% respectively?Profit on safe investment is = 10%Profit on Risky investment is = 15%Total Amount = 23,500SupposeBorrowed from bank = x & Borrowed from insurance = yx + y = 23500 -> x = 23500 – y -------- (1)10100x+ 15100y=2550 -> 10x + 18y = 255000------- (2)10(23500 – y) + 15y = 255000235000 - 10y + 15y = 2550005y=255000-235000 = 20000y= 20,0005 , 4000x=23500-4000 = 19,50010100 x 19500=1950 15100 x 4000=600So 1950 + 600 = 2550Q.29: Mr. jack borrowed money from his bank and on his insurance to start a business. His interest rate on the bank loan 10% and his rate of interest on the insurance is 12%. If the total amount borrowed is 10,000 and his total yearly interest payment is $1090, how much did he borrow from the bank?Profit on safe investment is = 10%Profit on Risky investment is = 12%Total Amount = 10,000SupposeBorrowed from bank = x & Borrowed from insurance = yx + y = 10000 -> x = 10000 – y -------- (1)10100x+ 12100y=1090 -> 10x + 12y = 109000------- (2)10(10000 – y) + 12y = 109000100000 - 10y + 12y = 1090002y=109000-100000 = 9000y= 90002 , 4500x=10000-4500 = 550010100 x 5500=550 12100 x 4500=540So 550 + 540 = 1090Q.30: Each ounce of substance A supplies 5% of the required nutrition a patient needs. Substance B supplies 12% of the required nutrition per ounce. If digestive restrictions require that the ratio of substance B be 35. How many ounce of each substance should be in the diet to provide 100% of the required nutrition?Supposex Ounce = Substance Ay Ounce = Substance B5100x+ 12100y=100100 -> 5x + 12y = 100------- (1)xy=35 -> 5x = 3y -> 5x - 3y=0 -------- (2)For elimination of x we subtract equation 2 from 15x + 12y = 1005x - 3y = 0 15y = 100y = 10015 = 2035x + 3y -> 5x + 3203 -> 5x = 20x = 205=4Q.31: A glass of skim milk supplies 0.1mg of iron and 8.5g of protein. A quarter pound of red meat provides 3.4mg of iron and 22g of protein. If a person on a special diet is have 7.15mg of iron and 73.75g of protein, how many glasses of skim milk and how many quarter-pound serving of meat would provide this?IronProteinMilk0.1mg8.5gMeat3.4mg22gRequired7.15mg73.75gSupposeMilk = x & Meat = y0.1x + 3.4y = 7.15 -> x = 71.5 – 34y -------- (1)8.5x+22y=73.75 -------- (2)8.5(71.5 – 34y) + 22y = 73.75607.75 - 289y + 22y = 73.75607.75 - 276y = 73.75607.75 - 73.75 = 276y→ 534 = 276yy= 534276 , 1.934x = 71.5 – 34y -------- (1)So x = 71.5 – 341.934x = 71.5 – 65.756 x = 5.744Q.33: A biologist has a 40% solution and a 10% solution of the same nutrient. How many cubic centimeters of each solution should be mixed to obtain 25c.c. of a 28% solution? 40100x+ 10100y=28100 x 25→ 40x + 10y = 700------- (1)x+y=25→x=25-y -------- (2)40(25 – y) + 10y = 7001000 - 40y + 10y = 7001000 - 30y = 7001000 - 700= 30y→ 300 = 30yy= 30030 , 10x=25-y ------- (1)So x = 25 – 10x = 15Q.34: A nurse has two solutions that contains different concentrations of a certain medication. One is a 20% concentration and the other is a 5% concentration how many cubic centimeters of each should be mix to obtain 10c.c of a 15.5% solution?20100x+ 5100y=13.5100 x 10→ 20x + 5y = 135------- (1)x+y=10→x=10-y -------- (2)20(10 – y) + 5y = 135200 - 20y + 5y = 135200 - 15y = 135200 - 135= 15y→ 65 = 15yy= 6515 , 4.334x=10-y ------- (1)So x = 10 – 4.334x = 5.666Q.35: each ounce of a substance supplies 5% of the required nutrition a patient’s needs. Substance B supplies 15% of the required per ounce and substance C supplies 12% of required nutrition per ounce. If digestive restrictions require that substance A and C be given in equal amounts. Find the number of ounce of each substance that should be in the meal to provide 100% of nutrition?xy = 35→ 5x = 3y→5x-3y=0 -------- (1)x=z -------- (2)5100x+ 15100y + 12100z=100100 -> 5x+ 15y+12z = 100--(3)5x+ 15y+12(x) = 100 12x+5x+15y=100→17x+15y=100 ------ (4)Multiply equation 1 with 5 -> 25x-15y=0 ------ (5)Now we add equation 4 & 5 for eliminate y17x + 15y = 10025x - 15y = 0 42x = 100x = 10042 = 2.38 also x=z So z=2.385x = 3y→52.38=3y→11.9=3yy = 11.93=3.96Q.36: Medication A is given every 4 hour and medication B is given twice each day. The total intake of two medications is restricted to 50.6mg per day, for a certain patient. If the ratio of the dosage of A to the dosage of B is 5 to 8, find the dosage for each administration of each medication.6x + 2y = 50.6 -------- (1)8x=5y→8x-5y=0 -------- (2)First we multiply equation 1 with 530x + 10y = 253 -------- (3)First we multiply equation 2 with 1016x-10y=0 -------- (4)Now we add equation 3 & 4 for eliminate y30x + 10y = 25316x - 10y = 0 46x = 253x = 25346 = 5.58x=5y85.5=5y→44=5yy = 445 = 8.8Q.38: A flower bed 32 feet by 42 feet is bordered by a walk of uniform width. The area of the walk is 656 feet. Find the width of the walk.Formula for solve is:Area of walk = Area of outer rectangular - Area of Inner rectangular256 = (42 + 2x) (32 + 2x) – 42 x 32 256 = (42 x 32) +84x + 64x + 4x2 – (42 x 32)256 = (42 x 32) +84x + 64x + 4x2 – (42 x 32)4x2 + 148x - 256 = 0x2 + 37x - 164 = 0 (Divided by 4)a = 1 , b= 37 , c= -164x= -37±372-4(-164)2x= -37±369+6562x= -37±20252→ x= -37±4522x= -37±452x= -37+452x= 82=4x= -37-452x= -822= -41Q.39: the flower bordered around a rectangular plot of grass is of uniform width and its area is three-fifth the area of the grass. If the plot of the grass is 18 by 30 feet, what is the width of the border?Area of walk = Area of outer rectangular - Area of Inner rectangular324 = (30 + 2x) (18 + 2x) – 30 x 18 324 = (30 x 18) +60x + 36x + 4x2 – (30 x 18)324 = (30 x 18) +60x + 36x + 4x2 – (30 x 18)4x2 + 96x - 324 = 0x2 + 24x - 81 = 0 (Divided by 4)a = 1 , b= 24 , c= -81x= -24±242-4(-81)2x= -24±576+3242x= -24±9002→ x= -24±3022x= -24±302x= -24+302x= 62=3x= -24-302x= -542= -27 ................
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