<PRE><B>Question 1011333</B>
cost of a small bottle is y dollars.
cost of a large bottle is (y+1) dollars.


{{{36/y}}} is the number of small bottles.
this comes from the number of small bottle times the cost of each small bottle = 36 dollars.


{{{36/(y+1)}}} is the number of large bottles.
this comes from the number of large bottle times the cost of each large bottle = 36 dollars.


when she spends 36 dollars on small bottles, she has 25 more bottles than when she spends 36 dollars on large bottles.


that leads to the equation {{{36/y = 36/(y+1) + 25}}}


that equation says that the number of small bottles is 25 more than the number of large  bottles.


you have the equation of:


{{{36/y = 36/(y+1) + 25}}}


multiply both sides of that equation by y * (y+1) to get:


{{{(36 * (y+1)) = (36 * y) + (25 * (y * (y+1)))}}} which becomes:


{{{(36 * (y+1)) = (36 * y) + (25 * (y^2+y))}}}


simplify to get:


{{{36y + 36 = 36y + 25y^2 + 25y}}}


subtract the left side of the equation from both sides of the equation to get:


{{{0 = 36y + 25y^2 + 25y - 36y - 36}}}


the 36y and the -36y will cancel out when you combine like terms.


combine like terms to get:


{{{0 = 25y^2 + 25y - 36}}}


that&#39;s the same as:


{{{25y^2 + 25y - 36 = 0}}}


you would factorize 25y^2 + 25y - 36 to get:


(5y+9) * (5y-4) = 0


this could be done using factoring techniques or through the use of the quadratic formula.


when you solve the equation of 25y^2 + 25y - 36, you set it equal to 0 and factorize it and then solve for y.


each of the factors is set to 0 to solve for y.


5y+9 = 0 gets you y = -9/5


5y-4 = 0 gets you y = 4/5


y has to be positive so y = 4/5.


that&#39;s the cost of a small bottle.


the cost of a large bottle would be 4/5 + 1 = 9/5.


4/5 of a dollar = $0.8 dollars = 80 cents.


9/5 of a dollar = $1.8 dollars = 1 dollar and 80 cents.


the cost of a small bottle = 80 cents.
the cost of a large bottle = 1 dollar and 80 cents.


36/y = number of small bottles = 36/.8 = 45 small bottles.


36/(y+1) = 36/1.8 = 20 large bottles.


45 small bottles is 25 more than 20 large bottles.


the equation of 36/y = 36/(y+1) + 25 is good.


all of the questions have been answered.


1.
we derived 25y^2 + 25y - 36 from 36/y = 36/(y+1) + 25
2.
factors are (5y+9) * (5y-4)
3.
we got y = -9/5 and y = 4/5 where y = 4/5 was the only valid solution.
4.
the cost of a small bottle is $0.8 and the cost of a large bottle is $1.8.





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