[Help Needed] Julie is mixing orange juice. She adds water to the concentrate until she has 2 liters of juice which is 60% concentrate. She then adds some leftover juice which is 75% concentrate. If the result is 65% concentrated juice, how many liters of the stronger mix did she add?

1 liter of the Stronger mix

Step-by-step explanation:

From the above question:

Julie adds water to the concentrate until she has 2 liters of juice which is 60% concentrate.

Hence,

2 liters of juice = 60% concentrate of orange

2 liters of juice = 0.6 of the concentrate of orange

Amount of orange used in liters = 2 × 0.6 = 1.2 liters

She then adds some leftover juice which is 75% concentrate

Liters of left over juice added = unknown and represented by y

hence,

y liters of juice = 75% concentrate

y liters = 0.75 concentrate.

Liters of y added = y × 0.75

= 0.75y

She then adds some leftover juice which is 75% concentrate. If the result is 65% concentrated juice,

Mathematically this means

60% concentrate + 75 % concentrate = 65% concentrate.

In liters

= 1.2 liters + 0.75y liters = 65%(2 liters of orange + y liters of orange)

= 1.2 + 0.75y = 0.65(2 + y)

= 1.2 + 0.75y = 1.3 + 0.65y

Collect like terms

= 0.75y - 0.65y = 1.3 - 1.2

= 0.1y = 0.1

y = 0.1/0.1

y = 1 liter.

Hence, since the liters of the leftover juice = stronger mix is represented by y,

therefore, the liters of stronger mix added = 1 liter.

when functions are first introduced, you will probably have some simplistic "functions" and relations to deal with, usually being just sets of points. these won't be terribly useful or interesting functions and relations, but your text wants you to get the idea of what the domain and range of a function are. small sets of points are generally the simplest sorts of relations, so your book starts with those.

step-by-step explanation:

state the domain and range of the following relation. is the relation a function?

{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}

the above list of points, being a relationship between certain x's and certain y's, is a relation. the domain is all the x-values, and the range is all the y-values. to give the domain and the range, i just list the values without duplication:

domain: {2, 3, 4, 6}

range: {–3, –1, 3, 6}

(it is customary to list these values in numerical order, but it is not required. sets are called "unordered lists", so you can list the numbers in any order you feel like. just don't duplicate: technically, repetitions are okay in sets, but most instructors would count off for this.)

while the given set does indeed represent a relation (because x's and y's are being related to each other), the set they gave me contains two points with the same x-value: (2, –3) and (2, 3). since x = 2 gives me two possible destinations (that is, two possible y-values), then this relation is not a function.

note that all i had to do to check whether the relation was a function was to look for duplicate x-values. if you find any duplicate x-values, then the different y-values mean that you do not have a function. remember: for a relation to be a function, each x-value has to go to one, and only one, y-value.

state the domain and range of the following relation. is the relation a function?

{(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)}

i'll just list the x-values for the domain and the y-values for the range: