Chapter 11: Ratios

what are the there equivalent forms that ratios can be written in?

x/y, x:y, x to y

a ratio explains

how many of one thing there are for every amount of something else

part to part ratio explains how

one portion of a group relates to another portion

part to who ratio explains how

how a single portion relates to the entire group

how do you convert a part-to-part ratio to part-to-whole?

add the parts

how do you convert a part-to-whole ratio to part-to-part whole ratio?

subtract the known part from the whole to find the remaining parts

in general, the ratio x to y tells us that, for every x items in one group, there are y items in another group, we can’t make any conclusions about the _______ of those items

quantities

in the ratio ax/bx, x is the ratio multiplier and ax represents

the actual amount of quantity A

given the ratio in the form of A/B = ax/bx for some number x, if we know the actual quantity of either A or B we can find the ratio multiplier (x) by setting up the equations of

A = ax or B= bx

for every 4 boys in the classroom, there are 3 girls. if there are 20 boys in the classroom, how many girls are there?

boys: girls

4x : 3x

4x= 20

x= 5

3(5)= 15

what is a useful ratio?

one in which the terms cancel, leaving a real number that represents the ratio’s value

when a ratio involves addition or subtraction with variables, it’s often not a

useful ratio

how do you solve problems where there are two equations involving three variables?

to find the ratio of any two of those variables, express both in terms of the common third variable, then simplify by canceling to obtain the desired ratio

when you are given two ratios that share a variable, why can you not accurately combine those variables right away?

you have to ensure that the values of the shared variable are the same

how do you combine two ratios that share a common variable into a three part ratio?

locate the common variable between the two ratios given

determine the LCM of the two numbers of the shared variable

when the values of the numbers that share the variable are the same then you can combine the ratios

what is the ratio of x to y to z: x to y is 3 to 4; x to z is 7 to 11

common variable = x

numbers that share the common variable are 3 and 7

the LCM of 3 and 7 is 21

we multiply 3 by 7 and 4 by 7

then we multiply 7 by 3 and 11 by 3

the result= 21:28:33

for problems that provide a ratio of two quantities we must add or subtract from one or both quantities to get a new ratio, what are the steps to solve these kinds of problems?

step 1: determine the quantities in the original ratio

step 2: assign a variable to the number of additional/fewer items required to produce the new ratio

step 3: solve the resulting equation

in a basket the ratio of apples to bananas is 2 to 5. If there are 20 bananas in the basket, how many apples must be added to make the ratio of apples to bananas 3 to 2?

let x be the ratio multiplier

#of apples = 2x

#of banans = 5x

since there are 20 bananas we can set up the equation 5x=20

x=4 and there are 2(4)= 8 apples

let y be the number of apples that must be added to the basket

new # of apples = 8+y

new # of bananas = 20

8+y / 20 = 3/2

cross multiply and you end up with y=22

if we need to manipulate a ratio using multiplication or division, you first have to express the ratio in

fraction form

the ratio of soap to water is 5 to 2, triple the ratio of soap to water

5/2 × 3 = 15/2

proportion

an equation showing the equality of two ratios

when given a problem that deals with proportions, follow three steps

step 1: restate the question

step 2 create the proportion

step 3: solve for the unknown

if martha always uses 9 lemons for 1 jug of lemonade, how many lemons are needed for 3 jugs of lemonade?

restate the question: 1 jug is to 9 lemons as 3 jugs is to how many lemons

1 jug / 9 lemons = 3 jugs / ? lemons

1x=27

x=27

if y varies directly with x, changing x by a factor changes y

by the same factor

the variable y varies directly with x, and when x=5, y=20, what is the value of y when x=8?

y=kx

20=k(5)

20=5k

k=4

y=kx

y=4(8)

y=32

if y varies inversely with x, they are related by the equation

y=k/x where k is a positive constant

if y varies inversely with x, as x increases by a factor, what happens to y?

if decreases by the same factor

when solving an inverse variation problem, first find the value of

k

if y varies directly with x and inversely with z, then what is the equation?

y= kx/z

variable y varies directly with x and varies inversely with z. when x=2 and z=6, the value of y is 9. what is the value of y when x=3 and z=8?

y=kx/z

8=k(2)/6

48=2k

k=24

y=kx/z

y=24(3)/8

y=72/8

y=9

what are joint variation problems?

variable y can vary directly with two others or inversely with two others

how do you set up this equation: y varies inversely jointly with x and z

y=k/xz