Solving Linear Equations - Age Problems

1 Linear Equations - Age ProblemsObjective: Solve age Problems by creating and Solving a Linear application of Linear Equations is what are called age Problems . When we aresolving age Problems we generally will be comparing the age of two people bothnow and in the future (or past). Using the clues given in the problem we will beworking to find their current age. There can be a lot of information in these prob-lems and we can easily get lost in all the information. To helpus organize andsolve our problem we will fill out a three by three table for each problem. Anexample of the basic structure of the table is belowAge NowChangePerson 1 Person 2 Table of Age TableNormally where we see Person 1 and Person 2 we will use thename of theperson we are talking about.

2 We will use this table to set up the is 20 years younger than Brian. In two years Brian will betwice as old asAdam. How old are they now?Age Now+ 2 AdamBrianWe use Adam and Brian for our personsWe use+ 2for change because the second phraseis two years in the futureAge Now+ 2 Adamx 20 BrainxConsider the Now part,Adam is 20 yearsyouger than are given information aboutAdam,not Brian show Adamis 20 years younger we subtract 20,Adam isx Now+ 2 Adamx 20x 20+ 2 Brianxx+ 2 Now the+ 2column is filled is done by adding2to both Adam sand Brian snow column as shownin the Now+ 2 Adamx 20x 18 Brianxx+ 2 Combine like terms in Adam sfuture age: 20+ 2 This table is now filled out and we are ready to tryand 2 AOur equation comes from the future statement:Brian will be twice as old as meansthe younger,Adam,needs to be multiplied by2.

3 (x+ 2) = 2(x 18)ReplaceBandAwith the information in their futurecells,Adam(A)is replaced withx 18 and Brian(B)is replaced with(x+ 2)This is the equation to solve!x+ 2 = 2x 36 Distribute through parenthesis x xSubtractxfrom both sides to get variable on one side2 =x 36 Need to clear the 36+36+36 Add 36 to both sides38=xOur solution forxAge nowAdam38 20=18 Brian38 The first column will help us answer the thex swith 38 and is 18 and Brian is 38 Solving age Problems can be summarized in the following five steps. These fivesteps are guidelines to help organize the problem we are trying to Fill in the now column. The person we know nothing about Fill in the future/past collumn by adding/subtracting the change to thenow Make an equation for the relationship in the future.

4 This is independent ofthe Replace variables in equation with information in futurecells of table5. Solve the equation forx, use the solution to answer the questionThese five steps can be seen illustrated in the following is 12 years older than David. Five years ago the sum of their ages was old are they now?Age Now 5 CarmenDavidFive years ago is 5in the change Now 5 Carmenx+12 DavidxCarmen is 12 years older than don tknow about David so he isx,Carmen then isx+12 Age Now 5 Carmenx+12x+12 5 Davidxx 5 Subtract5from now column to get the change2 Age Now 5 Carmenx+12x+ 7 Davidxx 5 Simplify by combining like terms 12 5 Our table is ready!C+D=28 (x+ 7) + (x 5) =28 ReplaceCandDwith the change + 7 +x 5 =28 Remove parenthesis2x+ 2 =28 Combine like termsx+xand7 5 2 2 Subtract2from both sides2x=26 Noticexis multiplied by222 Divide both sides by2x=13 Our solution forxAge NowCaremen13+12=25 David13 Replacexwith 13 to answer the questionCarmen is 25 and David is 13 Sometimes we are given the sum of their ages right now.

5 These Problems can betricky. In this case we will write the sum above the now columnand make thefirst person s age nowx. The second person will then turn into the subtractionproblem total x. This is shown in the next sum of the ages of Nicole and Kristin is 32. In two years Nicole will be threetimes as old as Kristin. How old are they now?32 Age Now+ 2 NicolexKristen32 xThe change is+ 2for two years in the futureThe total is placed above Age NowThe first person second becomes 32 xAge Now+ 2 Nicolexx+ 2 Kristen32 x32 x+ 2 Add2to each cell fill in the change columnAge Now+ 2 Nicolexx+ 2 Kristen32 x34 xCombine like terms 32+ 2,our table is done!N= 3 KNicole is three times as old as Kristin.(x+ 2) = 3(34 x)Replace variables with information in change cellsx+ 2 =102 3xDistribute through parenthesis+ 3x+ 3xAdd3xto both sides so variable is only on one side4x+ 2 =102 Solve the two step equation3 2 2 Subtract2from both sides4x=100 The variable is multiplied by444 Divide both sides by4x=25 Our solution forxAge NowNicole25 Kristen32 25= 7 Plug 25 in forxin the now columnNicole is 25 and Kristin is7A slight variation on age Problems is to ask not how old the people are, butrather ask how long until we have some relationship about their ages.

6 In this casewe alter our table slightly. In the change column because we don t know the timeto add or subtract we will use a variable,t, and add or subtract this from the nowcolumn. This is shown in the next is 26 years old. Her daughter is 4 years old. In how many years will Louisbe double her daughter s age?Age Now+tLouis26 Daughter4As we are given their ages now,these numbers go intothe change is unknown,so we write+tforthe changeAge Now+tLouis2626+tDaughter44 +tFill in the change column by addingtto each person table is now 2 DLouis will be double her daughter(26+t) = 2(4 +t)Replace variables with information in change cells26+t= 8 + 2tDistribute through parenthesis t tSubtracttfrom both sides26= 8 +tNow we have an8added to thet 8 8 Subtract8from both sides18=tIn 18 years she will be double her daughter sageAge Problems have several steps to them.

7 However, if we take the time to workthrough each of the steps carefully, keeping the information organized, the prob-lems can be solved quite View Note:The oldest man in the world was Shigechiyo Izumi fromJapan who lived to be 120 years, 237 days. However, his exact age has been and Intermediate Algebra by Tyler Wallace is licensed under a Creative CommonsAttribution Unported License. ( ) Practice - Age Problems1. A boy is 10 years older than his brother. In 4 years he will betwice as old ashis brother. Find the present age of A father is 4 times as old as his son. In 20 years the father will be twice as oldas his son. Find the present age of Pat is 20 years older than his son James. In two years Pat will be twice as oldas James. How old are they now?

8 4. Diane is 23 years older than her daughter Amy. In 6 years Diane will be twiceas old as Amy. How old are they now?5. Fred is 4 years older than Barney. Five years ago the sum of their ages was old are they now?6. John is four times as old as Martha. Five years ago the sum oftheir ages was50. How old are they now?7. Tim is 5 years older than JoAnn. Six years from now the sum oftheir ages willbe 79. How old are they now?8. Jack is twice as old as Lacy. In three years the sum of their ages will be old are they now?9. The sum of the ages of John and Mary is 32. Four years ago, John was twiceas old as Mary. Find the present age of The sum of the ages of a father and son is 56. Four years ago the father was 3times as old as the son. Find the present age of The sum of the ages of a china plate and a glass plate is 16 years.

9 Four yearsago the china plate was three times the age of the glass the5present age of each The sum of the ages of a wood plaque and a bronze plaque is 20years. Fouryears ago, the bronze plaque was one-half the age of the wood plaque. Findthe present age of each A is now 34 years old, and B is 4 years old. In how many years will A betwice as old as B?14. A man s age is 36 and that of his daughter is 3 years. In how many years willthe man be 4 times as old as his daughter?15. An Oriental rug is 52 years old and a Persian rug is 16 yearsold. How manyyears ago was the Oriental rug four times as old as the PersianRug?16. A log cabin quilt is 24 years old and a friendship quilt is 6years old. In howmay years will the log cabin quilt be three times as old as the friendshipquilt?

10 17. The age of the older of two boys is twice that of the younger; 5 years ago itwas three times that of the younger. Find the age of A pitcher is 30 years old, and a vase is 22 years old. How many years ago wasthe pitcher twice as old as the vase?19. Marge is twice as old as Consuelo. The sum of their ages seven years ago was13. How old are they now?20. The sum of Jason and Mandy s age is 35. Ten years ago Jason was doubleMandy s age. How old are they now?21. A silver coin is 28 years older than a bronze coin. In 6 years, the silver coinwill be twice as old as the bronze coin. Find the present age ofeach A sofa is 12 years old and a table is 36 years old. In how manyyears will thetable be twice as old as the sofa?23. A limestone statue is 56 years older than a marble 12 years, thelimestone will be three times as old as the marble statue.