Chapter 7: General Word Problems

what symbol does the word “is” equate to?

=

what symbol does the word “was” equate to?

=

what symbol do the words “has been” equate to?

=

what symbol does the word “more” equate to?

+

what symbol do the words “years older” equate to?

+

what symbol do the words “years younger” equate to?

-

what symbol does the word “less” equate to?

-

what symbol does the word “times” equate to?

x

what symbol do the words “as many” equate to?

x

what symbol does the word “factor” equate to?

x

what are the steps to solve an age problem?

1. define variables for the ages in the present day

2. represent each age in the future or the past

3. organize the information from steps 1 and step 2 in a matrix, with the columns representing the present, past, or futures age and the rows representing people or objects

4. use the information in the matrix as well as the problem stem to create equations and then use those equations to determine the answer

how do you solve a line-up question with one named object ?

1. mark the reference object, let x= the number of people behind it, let y=number of people in front of it. the total number of people in the line is then x+y+1

2. translate the given information

3. to find the total number of objects in the line, use x+y+1 and the given information from step two

how do you solve length and weight problems?

1. assign a variable to each segment’s length/weight

2. Use information from the problem stem to create the equation(s) required to answer the question

how would you solve this problem: at a cafe, 3 sandwiches and 2 cups of soup cost $70.40 and 3 cups and soup and 2 sandwiches is $61.60. how much does one sandwich cost?

3S + 2P= 70.40

2S + 3P = 61.60

then solve the system of equations

how do you solve word problems with variables in the answer choices?

1. assign a variable to the value the question asks for

2. use the given information to create one or more equations involving the variable

3. solve the equation(s) for the desired variable

how do you solve price per item problems?

total cost / number of items purchased

what is the equation to solve profit and loss problems?

profit = (total revenue) - (total fixed costs + total variable costs)

how do you solve splitting the cost problems?

original cost per person= total price/ # of people

altered cost per person total price/ # of people

(altered cost per person) = (original cost per person) + (altered cost)

when presented with two different ways a worker can be paid for a job…

create an algebraic expression for each way the salary can be calculated and answer the question

how do you solve the following problem: to rent a car, Company A charges $50 for the first day and $40 for each additional day. Company B charges $35 per day for the first two days and $50 for each additional day. If a customer rents a car for more than two days, after how many days will the total cost be the same for both companies?

Company A= 50 + 40(D-1)

Company B= 35 +35+ 50(D-2)

50+40(D-1) = 35+35+50(D-2)

when solving word problems in which fractional amounts are successively subtracted from a quantity

always subtract the fractional amount form the quantity that remains after the previous deduction

how do you solve this problem: if Ari ate 1/3 of an entire pizza, and Helga ate ¾ of the remaining pizza, what fraction of the pizza remains uneaten

1-1/3= 2/3

¼ X 2/3 = 2/12 = 1/6 of the original pizza remains

when solving word problems in which whole values are broken up into fractional components…

the sum of the fractional components (and any remaining parts) must equal the original whole

how would you solve this problem: Al started the day with a bowl of candies. he ate 1/6 for breakfast, half of the remainder for lunch, and 12 for dinner, leaving 13 candies. how many candies were in the bowl originally?

1-1/6 = 5/6

1-1/2=1/2

½ x 5/6 = 5/12t

5/12t-12 = 13

5/12t = 25

t=60

what is the simple interest formula?

interest = principal x rate x time

when inputting the time into the simple interest formula, it has to be displayed in terms of

years

what is the compound interest formula?

A = P (1+r/n)^nt

if an investment is compounded semi-anually, what is the value of n in the compound interest formula?

2

what is the linear growth formula?

F_n= kn + p

k= the growth constant

n= number of growth periods

p= original value prior to growth

in exponential growth problems, we can determine the corresponding amount at the end of any growth period by

calculating the product of the initial amount and the growth factor raised to an exponent that matches the growth period

what is the exponential growth formula?

(initial value) x (growth factor)^growth period

what is the difference between exponential growth and linear growth?

exponential growth is described as growth by a constant factor or multiple, while linear growth is described as growth by a constant amount

when is the greatest decrease in an exponential decay problem?

occurs during the first period of decay

if N = bcd, where bcd are digits what is the value of N?

N= 100b + 1c + d

consecutive integers

integers that follow each other in order such that the difference between any integer and the integer that precedes it is 1

algebraically, how can 3 consecutive integers be expressed?

x, (x+1), (x+2) = 3x+3

how do you solve this problem: the sum of 4 consecutive integers is 506 what is the smallest integer?

x + x+1 + x+2 +x+3 = 506

4x + 6= 506

4x= 500

x=125

the difference between any two consecutive even integers is

2

how are consecutive odd integers expressed?

x, (x+2), (x+4)…

consecutive multiples of a number

multiples that follow one another, with the difference between any two consecutive multiples always equal to that number

what are four consecutive multiples of 5?

x + (x+5) + (x+10) + (x+15) = 4x +30

how do you solve dry mixture problems?

1. define the attributes (components, units, and quantities) involved

2. create a matrix with the information in step 1 and solve the problem

if one ingredient is heavier than another ingredient, then the price per pound of the mixture will be close to

the price per pound of the heavier ingredient

wet mixture problems: when two or more liquids, each with a specific concentration of some ingredient, are combined to create a final mixture, how do you solve this problem?

1. define the attributes (components, concentrations, and volumes) involved

2. create a matrix with the information in step 1 and solve the problem

what is a tip for solving word problems with inequalities?

look for the comparison keyword (greater than, at least, no more than, etc) this word drives the inequality

to so for word problems in which quantities are expressed as variables, and the answer choices are algebraic expressions, what are the steps?

1. assign easy-to-work-wth numbers to the variables in the question stem

2. use the assigned numbers to calculate the value the stem asks for

3. use the assigned numbers to evaluate each answer choice and eliminate choices that do not match the value calculated in step 2