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Question 1113274: Lars needs to purchase AT LEAST 35 party decorations. The Party Palace charges $0.50 per decorative streamer and $0.25 per balloon, including tax. Which combination of streamers and balloons can Lars purchase with $11.50 at the Party Palace?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! lars needs at least 35 decorations.
the cost is .5 per decorative streamer and .25 per balloon.
the amount of money available to purchase is 11.50.
let d = the number of decorative streamers and let b = the number of balloons.
your cost equation becomes .5 * d + .25 * b = 11.5
your number of units equation becomes d + b >= 35
you do not have any restrictions on how many decorative streamers can be purchased or how many balloons can be purchased.
consequently, you could purchase all decorative streamers or you could purchase all balloons or any combination of some streamers and some balloons.
if you bought all decorative streamers, then you could buy 11.50 / .5 = 23 items.
obviously, you can't buy all streamers.
if you bought all balloons, then you could buy 11.50 / .25 = 46
assuming you wanted to buy just 35 items.
then you have a system of equations that would be like:
d + s = 35
.5 * d + .25 * s = 11.5
you can solve this by substitution.
solve for d in the first equation to get d = 35 - s
replace d with 35 - s in the second equation to get .5 * (35 - s) + .25 * s = 11.5
simplify to get 17.5 - .5 * s + .25 * s = 11.5
combine like terms to get 17.5 - .25 * s = 11.5
subtract 17.5 from both sides of this equation to get -.25 * s = -6
divide both sides of this equation by -.25 to get s = -6 / -.25 = 24
since d + s = 35, and since s = 24, then d must be equal to 35 - 24 = 11
your tentative solution is d = 11 and s = 24
your cost equation of .5 * d + .25 * s becomes .5 * 11 + .25 * 24.
this results in a total cost of 11.5
you can buy exactly 35 party decorations if you buy 11 decorative streamers and 24 balloons.
if you wanted to buy more party decorations than 35, then you can replace 1 decorative streamer for every 2 balloons, and the cost would still be 11.5
for example:
10 streamers and 26 balloons equals 36 total party decorations and would cost 10 * .5 + 26 * .25 = 11.5
9 streamers and 28 balloons equals 37 total party decorations and would cost 9 * .5 + 28 * .25 = 11.5
if you let x equal the number of decorative streamers and y equal the number of balloons, you can probably graph the equation.
the equation to graph would be .5 * x + .25 * y = 11.5
the graph would look like this.
the red line on this graph shows you all possible combinations of x and y that would give you a total cost of 11.50.
the blue line on the graph tells you that the total of x and y is equal to 35 for any combination of x and y.
the red line has to be on or above the blue line for the total party decorations to be greater than or equal to 35.
bottom line is you have several possible answers for the number of party decorations that you can buy if you want the total cost to be exactly 11.50 and you want the number of party decorations to be greater than or equal to 35 in total and you have no restrictions on the number of streamers or the number of balloons.
the possible combinations, as shown on the graph, are shown below in tabular form:
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