7) Ratios Flashcards
Rules about ratios
> ratio tell us about the underlying RELATIONSHIP between quantities
> ratio does NOT provide the ACTUAL NUMBERS
Can express ratio in many different forms, but always add ratio MULTIPLIER “x” to BOTH numerator and denominator of the ratio
e.g., ratio of the # of boys to the # of girls in the class is 4/3 —> 4x/3x
Note:
> if you know the value of “x”, then you’ll know the actual number of the quantities
> Need to be given either the SUM of the all the items in the ratio or one individual quantity to be able to solve for x
> also the ratio multiple DOES NOT NECESSARILY NEED TO BE AN INTEGER (sometimes the ratio multiplier has to be an integer in order to produce whole quantities) e.g., cannot have a fraction of a football player
> CREATE A TABLE (individual items as column headers + “Total”)
> you can use ratios to identify the MINIMUM QUANTITY!!! (x>=1 if dealing with integers)
> sometimes you might face challenging problems with MULTIPLE RATIO MULTIPLIERS (remember: one ratio multiplier for the SAME ratio)
When are ratios useful vs not useful?
Not all ratios are considered useful (value of the ratio keeps changing depending the value of the unknowns)
Example of an unhelpful ratio:
(4+m)/m –> depending on the value of m, the value of the ratio also changes
Example of a helpful ratio (FIXED relationship): 5m/m –> 5/1
Comparing quantities using like variables in ratios
e.g., A=2B and B=3C, what is ratio of A to C?
e.g., if T = A + B, and B = 3A, then what is ratio of A to total?
> always try to use COMMON LIKE VARIABLES to express relationship between quantities
e.g., use common like variable “B” to express the relationship between A and C —> “B” will be cancelled out
e.g., you know that total = A + B –> re-express total as A’s
Total = A + 3A = 4A
Note:
> common like variables can bethe TOTAL
Creating a 3 part ratio, given two ratios that share one common variable
e.g., x:y = 3:4 and x:z = 7:11, what is x:y:z?
> find the LCM of the COMMON VARIABLES and combine two ratios into a three-part ratio
e.g., x:y:z = 21:28:33 (x is the common variable that is used to connect the two)
Note:
> one challenging application of this multi-part ratio is figuring out the MINIMUM QUANTITY of parts when putting together the multi-part ratio using LCM!!
Adding and subtracting from one or both quantities to achieve a desired ratio
(1) first figure out the ratio multiplier to get rid of one unknown
(2) then figure out the amount to be added or subtracted from quantities to get to your desired ratio
> don’t forget about adjusting the TOTAL FIGURE too (if required) ***
Adjusting ratios with multiplication and division
Just multiple the entire ratio (expressed in fraction form) by factor (>1 or <1)
e.g., 4/3 * 1/2 = 4/6
What are proportions?
Related concept to RATIOS
> two equal ratios are known as proportions (just adjust the ratio x multiplier)
Direct variation
A direct variation relationship between two variables x and y can be related by the equation y = kx, where k > 0 constant
Buzzwords:
> “y varies directly with x”
> “y is proportional to x”
Examples of direct variation:
> linear relationship (easiest)
> other forms (e.g., y = kx^2)
Inverse variation
e.g., the # of days it takes to paint a house varies inversely with the number of painters. If a house can be painted in 8 days when there are 3 painters on the job, then how many days would be needed to paint the house if there were 4 painters on the job?
Inverse relationship between variables
y = k/x , where k is a constant
To solve, we still need to SOLVE k
Examples of direct variation:
> ,linear inverse relationship (easiest)
> other forms (e.g., y = k/x^2)
What does “combined variation” mean?
When a single variable (y) has direct variation with one variable and inverse variation with another variable and expressed in one equation
e.g., y varies directly with y and varies inversely with z …
y = k1*x * k2/z
—> instead of having separate constants, let “k” represent the COMBINED CONSTANT
y = (kx)/z
also y = kxz
y = k/(xz)
Alternative way to solve direct variation / inverse variation / combined variation problems?
Instead of calculating the ABSOLUTE VALUE, focus on the TRANSFORMATIONS (CHANGES) to the numbers
e.g., You are given
y = (kx)/z and when x = 100 and z = 200, y = 500
What is y when x = 200 and z = 100?
ONE way is to calculate the value of k, then substitute the new values of x and z to solve for y
ALTERNATIVE WAY is to focus on the changes to the individual components and changes to y (original numbers)
y = (kx)/z = 500
x doubles
z halves
k*(2x)/(0.5z)
= 4(kx/z) —-> SUBSTITUTE y for kx/z
= 4y —> we know y = 500
= 4*500
= 2000
Hard ratios question:
At a particular movie, what was the ratio of the number of people who watched the entre movie to the number of people who left the movie before it ended? (assume all audience members were in their seats by the start of the movie)
(1) At the beginning of the movie, with the entire audience seated, the ratio of the # of seats filled to the # of seats not filled was 5 to 6
(2) At the end of the movie, the ratio of the # of seats filled to # of seats not filled was 1 to 3
of people who watched the entire movie / # of people who left during the movie
> figure out either Part to Part or Part to Whole
(1) what this statement tell us is that the movie theater has a UTILIZATION RATE (i.e., not every seat is occupied at the start of the movie)
of people watching the movie = capacity of theatre - empty spots = filled
Filled + not filled = capacity of the theatre
seats filled/# not filled = 5x/6x
Or # seats filled/Total capacity = 5x/11x
> need to know who stayed at the end of the movie
NS
(2) End of the movie, after people have left, we have remaining people who stayed through = # of seats filled
of seats not filled comprises of BOTH people who have left early and originally unfilled spots —-> need to know how many are originally unfilled
NS
(3) WE HAVE TWO SEPARATE RATIOS –> FIND THE LCM of the TOTALS!!!!!
11x and 4x —> 44x
of seats filled at beginning / # seats empty = 20x/24x
of ppl who stay through entire movie / # seats empty at end = 11x/33x
THEREFORE, # of people who LEAVE = # seats filled at beginning - # of ppl who stay through entire movie
= 20x - 11x
= 9x
of ppl who stay through entire movie / # of people who leave
= 11x / 9x —> x’s cancel
= 11/9 SUFFICIENT
What did we learn?
> the common variable CAN BE the TOTAL figure
> you don’t need to know the multiplier in order to calculate a RATIO
Ratio - parts to parts vs parts to whole
For a TWO PART ratio…
IF you know the ratio of part to part –> then you know the ratio of part to whole
Similarly, IF you know the ratio of part to whole –> then you know the ratio of part to part
Given multiple ratios (application of multi-part ratios)
e.g., If I only buy books and food during a semester at school, and for every $16 I buy, $3 goes toward books. If I had spent an additional $36 on books, then for every $18 that I spend, $5 would have gone towards books. How much did I actually spend on books?
(1) set up the table
(2) translate the word problem
(3) Identify any common variables
> in this question, it is the # of books I buy = 3x
(4) identify the relevant ratios
> in this case, we care about 3/16 and 5/18
We need to solve for the ratio multiplier x in order to solve for the quantity 3x
(3x+36)/(16x+36)= 5/18
x = 18
3x = 54
(NOT 3x+36 —> hypothetical!)
