Word Problems Flashcards
What does ‘is’ mean?
=
What does ‘was’ mean?
=
What does ‘has been’ mean?
=
What does ‘more’ mean?
+
What does ‘years older’ mean?
+
What does ‘years younger’ mean?
-
What does ‘less’ mean?
-
What does ‘times’ mean?
x
What does ‘less than’ mean?
-
What does ‘fewer’ mean?
-
What does ‘as many’ mean?
x
What does ‘factor’ mean?
x
how to solve age problems
- define variables for the ages in the present day
- represent each age in the future or in the past
- Organize the information from steps 1 and 2 in a matrix, with the columns representing the present, past, or future ages and the rows representing people or objects.
4: Use the information in the matrix as well as the problem stem to create corresponding equations for the present and future or past, and then use those equations to determine the answer.
Difference between variable and fixed costs?
Variable costs increase as more product is sold, but fixed costs do not.
When a company has both fixed and variable costs, the profit equation can be expanded to
profit = revenue - [total fixed costs + total variable costs]
formula for when two people transfer money to each other to equalize hourly wages
When two people earn $n each and $x is transferred from person A to person B to equalize their hourly wages, the formula is:
n-x/person a’s hours = n+x/person b’s hours
Fraction questions to find remaining portion
If you are given fractions, minus the remaining fraction from the whole. after youve done that for all of the portions that were taken, times those all together to find total remaining.
Example: One Sunday morning, a man is leisurely sitting in a coffee shop about to enjoy his full cup of gourmet coffee. During his first hour in the coffee shop, he drinks 1/4 cup of his coffee. During his second hour, he drinks 2/3 of his remaining coffee. During his third hour, he drinks 3/5 of the remaining coffee. What fraction of his coffee remains after three hours?
hour 1: 1-1/4: 3/4
hour 2: 1-2/3: 1/3
hour 3: 1-3/5: 2/5
times: 3/4 * 1/3 * 2/5 = 6/60 = 1/10 of remaining after three hours
general formula for remaining portion
The store began with x cell phones. After selling 1/y of the x cell phones, the store had x - 1/y * x cell phones in stock. simplified, the formula is x(y-1)/y where x is the total stock and y is the portion removed
Compound interest formula
A = P(1 + r/n)^n*t
A= future value
P = initial value [principal]
r = annual interest rate [expressed in decimal form]
t = number of years. if months, put over /12
n = number of compounding periods per year
constant growth formula
F = kn + p
F = final value
p = initial value
k = constant increase during each period
n = number of periods during which the growth occurs
exponential growth formula
final value = initial value( 1 + growth rate)^# of intervals
for growth questions, can one assume growth driver?
No. In order to determine the amount of growth, we must be presented with the growth driver.
When breaking down dry mixture problem, we need to consider three attribuets
- components of the mixture [what the mixture contains]
- the units of each component [e.g. dollars per pound]
- the quantitiy of each component
from there, create a matrix with the info.
- the components should consist of the left hand column
- the units should be the second column
- the quantity should be the third column
- the fourth column should be some sort of total [e.g. price]
- bottom row final mixture
IMPORTANT MATRIX NOTE:
- to find ‘total’ in last column, multiple quantities in rows
- for different solution ‘totals’, sum those and set equal to the final mixture [last row] product that you found earlier
When breaking down wet mixture problem, we need to consider three attributes
- components of the mixture [what the mixture contains]
- the concentration of each component [e.g. dollars per pound]
- the quantitiy of each component
from there, create a matrix with the info.
- the components should consist of the left hand column rows
- the concentations should be the second column
- the quantity should be the third column
- the fourth column should be some sort of total
- bottom row final mixture
IMPORTANT MATRIX NOTE:
- to find ‘total’ in last column, multiple quantities in rows
- for different solution ‘totals’, sum those and set equal to the final mixture [last row] product that you found earlier
‘up to’
≤
less than or equal to
‘more than’
’>’
greater than
at least
≥
greater than or equal to
‘exceed’
’>’
greater than
‘no more’
≤
less than or equal to
‘at most’
≤
less than or equal to
‘as few as’
≥
greater than or equal to
counting objects in a line formula
In general, if you (or anyone) are the mth person counted from the beginning of the line and the nth counted from the end, then the number of people waiting in line is m + n -1
formula for distance
distance = rate * time
formula for time
time = distance / rate
formula for rate
rate = distance / time
average rate for an entire trip
distance 1 + distance 2…etc / time 1 + time 2 … etc
formula for total distances for converging objects
distance object 1 + distance object 2 = total distance objects 1 and 2
in practice: Thorstein is in Easton and Tom is in Weston, two towns that are 240 miles apart. Thorstein and Tom leave their respective towns at the same time and drive toward each other on the same road. If Thorstein drives at a constant speed of 60 miles per hour and Tom drives at a constant speed of 40 miles per hour, how many minutes will it take before Thorstein and Tom meet each other on the road?
Thorstein: distance = 60t
Tom: distance = 40t
60t + 40t = 240
100t = 240
t= 12/5 hours, or 144 minutes
formula for total distances for diverging objects
distance object 1 + distance object 2 = total distance objects 1 and 2
Setting up variable for distance for round trip problems
In a round-trip problem when only the total time traveled is provided, consider letting the time to a destination equal some variable t and the time back equal (total trip time - t).
set each leg of the trip equal to each other to solve
formulas where objects catch up and pass each other
- faster object’s distance = slower objects distance + (difference in starting points + difference in ending points)
- time = change in distance (difference in starting points + difference in ending points) / change in rate (difference in rate of two objects)
what is relative motion
An object travels relatively faster when it is moving along with an outside force than when it is traveling under its own power. Conversely, an object will move relatively slower when it is moving against an outside force than when it is moving under its own power.
Example, if airplane is traveling at 500 MPH and flies along with wind that is blowing at 50 MPH, its speed is 550 MPH relative to ground
formula to find rate, gas/energy used, distance
rate * gallon = distance
distance is directly proportional to rate and time
if rate increases and time remeains constant, distance increases
if time increases and rate remains constant, distance increases
rate is inversly proportional to time and directly proportional to distance
if rate increases time to travel distance decreases and vice versa
if rate increases, distance increases
time is inversly proportional to rate and directly proportional to distance
if time to travel increases rate decreases and vice versa
if time increases, distance increases
Setting up variables for time for two objects when both start a task but one stops and the other must finish the job alone
let the time for the object that stops first represent ‘x’ and the work time for the object that finishes the job be ‘x+y’ with y being the remaining time needed
Basic formula for setting up work rate variables when two objects are working against each other
1/x - 1/y
