Question 1102847: the members of a gardening club decided to spend $36 on buying garden tools and they found that a rake and a travel together cost $7. If they spent all the money on rakes alone, they could buy 3 more tools than they could if they spent all the money on trowels.
a) By taking the cost of atrowel as $q, write down expression in terms of q for i) the cost of a rake
ii) the number of rakes which could be bought for $36.
b) write down an equation which q must satisfy, and show that it reduces to q^2+17q-84=0.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
let R = cost of a rake.
let Q = cost of a trowel.
you are given that the cost of a rake plus the cost of a trowel = 7.
that gets you R + Q = 7
solve for the cost of a rake to get R = 7-Q
let x equal the number of rakes that can be bought for 36 dollars.
that gets you x * R = 36
since R = 7-Q, that gets you x * (7-Q) = 36
solve for x to get x = 36 / (7-Q)
x is the number of rakes that you can buy for 36 dollars when the cost of a rake is equal to 7-Q.
if you buy all rakes or all trowels, you can buy 3 more rakes than trowels.
this means you can buy either x rakes or (x-3) trowels for 36 dollars.
since the cost of a rake is (7-Q) and the cost of a trowel is Q, you get x * (7-Q) = 36 and you get (x-3) * Q = 36
since they're both equal to 36, then they're both equal to each other and you get x * (7-Q) = (x-3) * Q.
since x * (7-Q) = 36, then x * (7-Q) = (x-3) * Q becomes 36 = (x-3) * Q
since x = 36 / (7-Q), then 36 = (x-3) * Q becomes 36 = ((36 / (7-Q)) - 3) * Q
multiply both sides of this equation by (7-Q) to get 36 * (7-Q) = (36 - 3 * (7-Q) * Q
simplify this to get 252 - 36Q = (36 - 21 + 3Q) * Q
simplify this further to get 252 - 36Q = 36Q - 21Q + 3Q^2
re-order the terms in descending order of degree to get -36Q + 252 = 3Q^2 + 36Q - 21Q
combine like terms to get -36Q + 252 = 3Q^2 + 15Q
add 36Q to both sides of this equation to get 252 = 3Q^2 + 15Q + 36Q
combine like terms to get 252 = 3Q^2 + 51Q
subtract 252 from both sides of this equation to get 0 = 3Q^2 + 51Q - 252
factor a 3 out of the expression on the right side of the equation to get 0 = Q^2 + 17Q - 84
this is the same as Q^2 + 17Q - 84 = 0
that is the equation you are looking for.
if you factor this equation, you will get (Q + 21) * (Q - 4) = 0
solve for Q to get Q = 4 or Q = -21.
since Q can't be negative, then Q = 4.
you were given that R + Q = 7
when Q = 4, you get R = 3
the price of a rake is 3 and the price of a trowel is 4.
36/3 gets you 12 rakes.
36/4 gets you 9 trowels.
12 is 3 greater than 9 which means the number of rakes you can buy with 36 dollars is 3 more than the number of trowels you can buy with 36 dollars.
all the requirements of the problem are satisfied.
the answers to your problem are:
ai) the cost of a rake is equal to 7-Q.
aii) the number of rakes which could be bought for $36 is equal to 36/(7-Q).
b) the equation is x * (7-Q) = (x-3) * Q, where x is the number of rakes and (x-3) is the number of trowels.
the equation was reduced to Q^2 + 17Q - 84 = 0 as shown above.
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