Ratios Flashcards
what are the three ways to express the ratio of some quantity x to some quantity y
x/y, x:y, and x to y
in addition to information about how quantities are related through a ratio (ex: x:y), what else do we implicitly know?
the relationship between each part of the ratio and the whole (ex: x:y also informs us of x:(x+y) and y:(x+y)
does a ratio provide actual quantities or just relative amounts?
just relative amounts
what is a ratio multiplier?
a quantity x that can be used in a ratio to represent the actual quantities. For example say the ratio of girls to boys in a classroom is 4:3. By using a ratio multiplier, we could say the ratio is 4x:3x, that way if we know x we would know the actual amount, and we can also notice that the actual quantities are multiples of 3 and 4
does the ratio multiplier need to be an integer?
no, it can be fractional (may happen in baking, chemistry, or any problems that require a portion of something)
what is the following ratio and is it a useful ratio? the ratio of managers m to worker n is what?
(m/n) ->no it is not useful because the ratio changes based on the variable values of m and n
how do you construct a three part ratio given two separate 2 part ratios? ex: x:y = 3:4 and x:z = 7:11, what is x:y:z?
Step 1: identify the common variable, in this case x
Step 2: find the LCM of x across the two ratios - here that is 21
Step 3: multiply the two ratios by whatever number is required to result in x being 21 in both examples - in this case make x:y = 7(3:4) = 21:28, and x:z = 3(7:11) = 21:33
Step 4: use these final values to establish the three part ratio since now they are on the same scale - in this case x:y:z = 21:28:33
how do you add or subtract to achieve a desired ratio? imagine there is a ring (containing only diamonds and rubies) at a ratio of 2 diamonds to 3 rubies and a total of 15 stones - how many diamonds must be added so that diamonds and rubies are present in a 5:5 ratio?
Step 1: figure out how many of each stone are present using the total number of stones (15) and setting the ratio up with a ratio multiplier - in this case 2x+3x=15 -> x=3
Step 2: so, the actual number of diamonds to rubies = 6:9
Step 3: let y = the number of additional diamonds needed to achieve the 5:5 ratio and set it up as an equation : (6+y)/9 = 5/5 -> y = 3
how do adjust ratios via multiplication and division? imagine an engine burns fuel and oxygen at ratio of fuel:oxygen = 4:3, and you want to cut the amount of fuel used in half to improve efficiency?
you simply divide by 2, so the ratio becomes fuel:oxygen = 4:6
a direct relationship, or y varying directly with x can be represented using what equation?
y=kx, where k is a constant
ex: y varies directly with x and when x =5, y=20. When x=8 what does y=? Let y=kx since y and x have a direct relationship. So, 20=k*5 -> k=4, then when x=8, y=32.
direct varying relationships can also be established with nonlinear equations, such as y varying directly with x^2 - how would that equation look?
still would be y = k*x^2
direct variation means if x goes up y will go…?
up
inverse variation means if x goes up y will go….?
down
inverse relationship are represented by what general equation?
y = k/x, where k is a constant and where if x increases by a factor, y will decrease by the same factor
what is combined variation?
when a variable varies inversely with one variable and directly with another. For example, y varies directly with x and inversely with z. This would be expressed as y=(kx)/z
