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Question 254132: Cones cost $1.10 and sundaes cost $2.35. A total of 172 cones and sundaes cost $294.20. How many cones where purchased. Tell if mutliple answers, infinite answer or no solution. My daughter's class working on slopes.
Found 3 solutions by palanisamy, dabanfield, PRMath:
Answer by palanisamy(496)   (Show Source):
You can put this solution on YOUR website! Given, Cones cost $1.10 and sundaes cost $2.35
Let the number of cones bought = x
Let the number of sundaes bought = y
Given, x+y = 172 ...(1)
Total cost 1.10x + 2.35y = 294.20
Multiplying by 100, we get
110x+235y = 29420 ...(2)
(1)*110 => 110x+110y = 18920 ...(3)
(2)-(3) => 125y = 10500
y = 10500/125
y = 84
Substituting in (1), we get
x+84 = 172
x = 172-84
x = 88
Therefore, the number of cones bought = 88
and the number of sundaes bought = 84
Answer by dabanfield(803) (Show Source):
You can put this solution on YOUR website! Cones cost $1.10 and sundaes cost $2.35. A total of 172 cones and sundaes cost $294.20. How many cones where purchased. Tell if mutliple answers, infinite answer or no solution. My daughter's class working on slopes.
Let s be the number of sundaes purchased. Then the number of cones purchased must be 172-s.
Then we have:
2.35*s + (172-s)*1.10 = 294.20
Muliplying both sides by 100 and simplifying we have:
235*s + 172*110 - 110*s = 29420
125*s = 29420 - 18920 = 10500
s = 10500/125 = 84
The number of cones then is 172-84 = 88
There is one solution.
Answer by PRMath(133) (Show Source):
You can put this solution on YOUR website! Cones cost $1.10 and sundaes cost $2.35. A total of 172 cones and sundaes cost $294.20. How many cones were purchased.
This problem can be solved by making up two equations. When you see the first equation, you are going to say: "Ohhhh yeahhh!" Here goes:
You know that there are a TOTAL of 172 Cones AND Sundaes. So, to represent that as an equation you'd say: C + S = 172. ("C" for Cones and "S" for Sundaes).
Now for the second equation. Each cone costs $1.10. For example, if you had 3 cones, it would be 3 x 1.10. So, it makes sense, then, that you multiply $1.10 times however many cones there are, right? Similarly, you would multiply 2.35 times each sundae. The whole amount of money spent on cones and sundaes was 294.20.
Soooo our 2nd equation is: 1.10(C) x 2.35(S) = $294.20
Together, the equations look like this:
C + S = 172
1.10(C) x 2.35(S) = $294.20
Now what? Well, the first equation of C + S = 172 can be rewritten if you solve for C or S. (You can solve for either). I'm going to solve for C, but as I said, you could easily solve for S. In any event, let's solve for "C":
C + S = 172
C = 172 - S (I solved for C by subtracting S from both sides of the equation)
Now that we know what C equals, we can SUBSTITUTE that information (C = 172 - S) for "C" in the 2nd equation. Then we solve things. Here, try this:
1.10(C) + 2.35(S) = $294.20 (our original equation)
1.10 (172 - S) + 2.35S = 294.20 (now distribute the 1.10 to the 172 and the S)
189.2 - 1.10S + 2.35S = 294.20 (now combine like terms of: -1.10S + 2.35S)
189.2 + 1.25S = 294.20
1.25S = 294.20 - 189.2 (subtract 189.2 on both sides to isolate the S)
1.25S = 105 (now divide both sides by 1.25 to further isolate the S)
S = 84
Ok, now we know there were 84 sundaes sold. Let's think about our original equation of: C + S = 172. Let's fill in our sundae info:
C + S = 172
C + 84 = 172
C = 172 - 84 (subtract 84 from both sides of the equation to isolate "C")
C = 88
Does this work out? Let's check:
There are a TOTAL of 172 Cones AND Sundaes
88 Cones + 84 Sundaes = 172 total. Therefore, the first part checks out.
Cones cost $1.10 and sundaes cost $2.35. The whole amount of money spent on cones and sundaes was 294.20. SO..... let's fill in that info:
88(1.10) + 84(2.35) =
96.8 + 197.4 = 294.20
Ok, so that checks out, too.
This type of system is a CONSISTENT system and has one solution.
Here is what your daughter needs to know about systems:
If there is NO solution, the system is INCONSISTENT. That type of graph would show parallel lines.
If there are an INFINITE number of solutions, then the equations on a graph would be the SAME. The equations are DEPENDENT, since they are equivalent.
Finally, if you were to graph the info we have above, the lines on the graph would intersect. The system, therefore is CONSISTENT and has ONE solution. The system is INDEPENDENT.
I hope this helps you. :-)
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