Can somebody help me understand how to figure these story problems out?

Question

A ski shop is marking all their items down 20%. In addition, if the item is more than $100, they will take an additional $25 off. If a pair of skis were purchased at a sale price of $231, what was the price of the original skis?


Allen's diet says he can only have fruit drinks that contain 15 calories per ounce. All he has is a bottle of grape juice that contains 18 calories per ounce, and a bottle of sunkist grape juice that contains 6 calories per ounce. If he wants to drink a 12 ounce glass of grape fruit juice, how many ounces of each bottle should he mix to get a drink that will fit his diet?

I would like to know how to set up variables to represent the quantities and how the equation should be formed. Thanks in advance

Answer 1

To take 20% off of an item, you multiply the price by 100%-20% or 80%. In the same sense, to take 25% off an item, multiply the price by 75%.

$231*(.80)=$184.80

Then take 25% off the answer we just received.

$184.80*(.75)=$138.60

That's your answer. If you want to just set up the problem so that there is only one calculation, combine the two equations. The result would be: Cost=(.75)(.80)(P) where P is the is the initial price of the item. This simplifies to: Cost=.6*P or 60% off of the initial price.

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This second problem is a little more dificult. There are 2 different things that need to be satisfied, amount of fluid and amount of calories, so we set up a system of equations.

Let G and S equal the number of ounces of generic grape juice and sunkist grape juice respectively. The system looks like this:
G+S=12
18G+6S=12*15=180

If you are confused by the second equation, think about if you have a 12 ounce glass with 15 calories per ounce, how many calories are in the total glass. Answer: 180.

Solve this system of equations: (Substitution)
G=12-S
18(12-S)+6S=180
216-180=18S-6S
36=12S
S=3
G=12-3=9

Use 9 grams of generic grape juice and 3 grams of sunkist.

Hope this helps.

Answer 2

OK, for the first one, the skis were obviously more than $100 originally, since they're STILL more than $100. That means that the shop took off 20%, then took off another $25, and that was the final price. So if S is the sale price and P is the original price, then S = P - 0.2P - 25.

Combine the Ps: S = 0.8P - 25. Add 25 to each side: S + 25 = 0.8P. Divide by 0.8: 1.25(S + 25) = P. Plug in S: 1.25(256) = P. Now multiply: 320 = P.

Another, slightly less formal way to do it is to work the problem backward: Since the shop took off 20% and then took off $25, add 25 to find the price after he took off 20% : $256. Then figure if he took off 20% there's 80% left, or 4/5, so 256 is 4/5 of the price. So divide by 4 and multiply by 5 to get the answer. It's the same technique, only described in terms of logic rather than algebra.

Now, the second problem. Call the amount, in ounces, of the first drink A and the amount of the second drink B. Call the number of calories in the combination C. C = 18A + 6B, and A + B = 12. We want C to equal 15*12. So the equations we set up are:

A + B = 12, 18A + 6B = 180.

Now subtract A from each side of the first equation to solve B in terms of A: B = 12 - A. Plug this into the second equation: 18A + 6(12 - A) = 180. Multiply out the parentheses: 18A + 72 - 6A = 180. Combine the A's and subtract 72 from each side: 12A = 108. Divide by 12: A = 9. Plug this back into the first equation: B = 12 - 9; B = 3.

Now check your work: 9*18 is 162, 3*6 is 18, and 162 + 18 is 180, so it works.

Answer 3

The answer to the first depends on whether the $25 discount is applied before or after the 20% discount.

If it's before, then the original price, less $25, times 0.8 (1 - .2) is $231:

(x - 25)(0.8) = 231

If it's after, the original price, times 0.8, less $25, is $231.

x(0.8) - 25 = 231

For part two, let x be the amount of the regular grape juice, and y be the amount of the Sunkist.

Then:

x + y = 12 // he wants 12 ounces altogether

and

x(18) + y(6) = 12(15)

That is, the total calories from the regular grape juice (18 calories times the amount added) plus the total calories from the Sunkist (6 calories times the amount added) is 12 ounces times 15 calories per ounce, or 180 calories.

Answer 4

1) Since the sale price is over $100, the original price must have been over $100 as well. Thus you must first add back the additional $25 they took off at the end to get to $256. Then we know that this is 80% of the original price (100%-20%=80%), so 256=.8x, x= $320.

2) Let g= ounces of grape juice and s= ounces of sunkist grape fruit.
Since his total amount he is drinking is 12 ounces, g+s=12.
Since we want the final drink to have 15 calories per ounce, it must have 15*12=180 total calories.
18g+6s will thus =180.
We now have two equations and two unknowns so we can solve.
g+s=12
18g+6s=180
muliplying the first equation by 6 gives
6g+6s=72
Substracting this from the second equation yields
12g=108
so g=9
plugging this into the first equation gives s=3
so 9 ounces grape juice and 3 ounces sunkist grape juice.

Answer 5

for the first one:
Let x stand for the original price of the skis, and 0.2x be the 20 percent discount. You also know the extra $25 was taken off because it was over $100.
x - (0.2x) -25 = 231
this leaves: 0.8 * (x) = 256
divide out and x= $320.

for the second one:
First off, you know that the max calories will be 12 times 15, or 180 calories.
If you let x stand for number of ounces of one drink and y stand for the number of ounces of the other, you will get
18x + 6y =180
by testing numbers you will find the solution to this is 9 ounces of the 18, and 3 ounces of the 6.

Answer 6

for the first one:

the skis were purchased for $231, so they must've been over $100 to begin with. Since the offer says that if they were over $100 they get $25 off, you must add $25 back on to start figuring out the original price. The price of the ski's are now $256. Now we must add back on the 20% that had been taken off. So, look at it this way: $256 is 80% of X, the original price. So, X is 100%. Make a proportion:
$256 / 80% = $x / 100%
x = 320. The original price was $320.

Answer 7

302.2$ 231$x1.2 will give you the original price without the 20% discount, than add 25$

18 calories + 6 calories= 24 calories /2= 12 calories

1/2 an ounce of 1st juice + 1/2 ounce of second juice will give you an ounce of a 12 calories juice.

Answer 8

$231 x 1.2=$277.20 + 25=$302.20 origionally

231 x 1.2=y
y+25=z

z is your final answer



Hope I answered your question correctly.