SOLUTION: Andy can buy 3 mechanical pencils with 2/7 of his money. He can buy 6 mechanical pencils and 3 pens with the rest of his money. If he only buys pens, how many pens can he buy with

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: Andy can buy 3 mechanical pencils with 2/7 of his money. He can buy 6 mechanical pencils and 3 pens with the rest of his money. If he only buys pens, how many pens can he buy with       Log On


   

Question 1184398: Andy can buy 3 mechanical pencils with 2/7 of his money. He can buy 6 mechanical pencils and 3 pens with the rest of his money. If he only buys pens, how many pens can he buy with all his money?
Found 3 solutions by ikleyn, Theo, greenestamps:
Answer by ikleyn(52770) About Me  (Show Source):
You can put this solution on YOUR website!
.
Andy can buy 3 mechanical pencils with 2/7 of his money.
He can buy 6 mechanical pencils and 3 pens with the rest of his money.
If he only buys pens, how many pens can he buy with all his money?
~~~~~~~~~~~~~~~ 

Let x be the price of a pencil and y be the price of a pen.
Let M be amount of money Andy has.


Then, interpreting the problem literally as it is written, we have these two equations


    3x = %282%2F7%29%2AM           (1)

    6x + 3y = %285%2F7%29%2AM      (2)


Based on the first equation, I can replace 6x in the second equation by  %284%2F7%29M.  So, I do it, and I get


    %284%2F7%29M + 3y = %285%2F7%29M.


It gives 


    3y = %285%2F7%29M - %284%2F7%29M = %281%2F7%29M.


THEREFORE,


    M%2Fy = 3*7 = 21.


But M%2Fy is exactly the number of pens to buy.


HENCE,  Andy can buy  21  pens with all his money.    ANSWER

Solved.



Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
m = price of a mechanical pencil.
p = price of a pen.

3m = 2/7 * x.
6m + 3p = 5/7 * x.
5/7 * x = the rest of his money after he spent 2/7 * x.
if 3m = 2/7 * x, then 6m = 4/7 * x.
6m + 3p = 5/7 * x becomes 4/7 * x + 3p = 5/7 * x.
subtract 4/7 * x from both sides of this equation to get 3p = 1/7 * x.
solve for p to get p = 1/21 * x.
solve for x to get x = 21 * p.

that's your answer.
he could buy 21 pens with his original amount of money.

to confirm, replace x with any value of money.
i chose 105 dollars because that's a multiple of 7 (105 / 7 = 15).
when x = 105, he could buy 3 mechanical pencils with 2/7 of his money.
that means he could buy 3 mechanical pencils for 30 dollars.
that means each mechanical pencil costs 10 dollars.
he could also buy 6 mechanical pencils and 3 pens for 5/7 times 105.
5/7 times 105 = 75 dollars.
6 mechanical pencils cost 60 dollars, leaving 15 dollars for the pens.
3 pens for 15 dollars means that each pen costs 5 dollars.
105 / 5 = 21 pens that can be bought for 105 dollars at 5 dollars apiece.

you get the same answer whether you're working with x or with some value for x.
this is as it should be.
the answer for x is confirmed when i used value for x instead.











Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


Knowing how to set up and solve the problem using formal algebra is useful. Presumably the question is from a student learning algebra; so you should know how to do it, as shown by the other tutors.

But you can get some good mental exercise by solving the problem using logical analysis and some easy mental calculations.

2/7 of his money will buy 3 mechanical pencils; that would leave him with 5/7 of his money.

When he buys 6 mechanical pencils and 3 pens with the remaining 5/7 of his money, the 6 mechanical pencils cost twice as much as 3 of them, which is 2*(2/7)=4/7 of his money.

That means 5/7-4/7=1/7 of his money is used to buy the 3 pens.

And if 1/7 of his money will buy 3 pens, spending all his money on pens only will buy 3*7=21 pens.