Algebra Demystified 2nd Ed

Chapter 8 linear appliCaTionS 307

the amount of the total concentration, the amount of the second concen-

tration is “Amount of final concentration – x.” We then solve the following

equation:

(Amt of Mixture #1)(%) + (Amt of Mixture #2)(%)

= (Amt of Final Mixture)(%)

•   Solve work problems. Two workers can complete a task. We either know

how long each worker needs to complete the task working alone and are

asked how long it would take them to complete the task if they work

together, or we know how long it would take for them to complete the

task working together and are asked how long it would take one of them

to complete the task alone (with some information on how much faster

one works than the other). We let the variable represent the time we want

to find. We then solve the following equation:

111

Worker#1Time Worker#2TimeTogetherT

+=

iime

If one of the workers works alone for a while and then the other joins in

to complete the task, the numerator of the last fraction is the proportion

of the task that remains when the second worker joins in.

•   Solve distance problems. Distance problems come in many forms, but all of

them are based on the formula d = rt. If two cars/runners/etc. are traveling

in the same direction, the rate at which the distance between them is

either increasing or decreasing is the difference of their two rates. If they

are traveling in opposite directions, the rate at which the distance between

them is either increasing or decreasing is the sum of their rates. We com-

pute this changing rate for r. We usually know d. We use the equation

d = rt to answer the question. For some distance problems, a trip is divided

into two parts and we know the total distance and the total time. If this is

the case, we solve one of the following:

d

r

d

r

1

1

2

2

+=Totaltime or r, t, = r 2 , t 2

where d 1 = r 1 t 1 represents one part of the trip and d 2 = r 2 t 2 represents the

other part.