Question 362970: Please help me find an algebraic explanation for this problem. I have the solution, but I don't understand it.
How is it that when I buy yellow bananas at three schillings a bunch and the same amount of red ones at four shillings a bunch, I would get two more bunches for the same amount if I divided the money evenly between the yellow and red bananas.
Solution:
The amount of money is 336 shillings.
This would buy 48 bunches each of red bananas and yellow bananas, 96 in all, but 168 shillings would buy 56 bunches of yellow and the other 168 would buy 42 bunches of red, 98 in all.
The difference between three-sevenths and one-half, which is one-fourteenth, is an amount which would buy two more at three shillings than it would at four, which amount is 24 shillings.
Therefore if one-fourteenth of the money is 24 shillings, the whole amount must be 336.
If you are not completely baffled now, I thank you profusely for your help!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! it was a nosebleed, but i think i found the solution.
you have 2 equations to work with.
they are:
3x + 4y = z
3a + 4b = z
you know that x = y, so you can replace y with x to get:
3x + 4x = z
3a + 4b = z
you can simplify this to get:
7x = z
3a + 4b = z
since z equals to 7x, you can substitute for z to get:
3a + 4b = 7x
you know that a + b = x + y + 2
since x = y, this becomes:
a + b = 2x + 2
in your equation of 3a + 4b = z, you know that 3a = z/2 and you know that 4b = z/2 because you were given that the price for yellow bananas was equal to the total divided by 2, and the price for red bananas was equal to the total divided by 2.
you have 3a = z/2 and you have 4b = z/2
you know that z = 7x, so you can substitute to get:
3a = 7x/2 and 4b = 7x/2
you can solve for a to get a = 7x/6 and you can solve for b to get b = 7x/8
you know that a + b = 2x + 2
substitute for a and b in this equation to get:
7x/6 + 7x/8 = 2x + 2
multiply both sides of this equation by 48 to get:
56x + 42x = 96x + 96
simplify to get:
98x = 96x + 96
subtract 96x from both sides of the equation to get:
2x = 96
divide both sides of the equation by 2 to get:
x = 48
you now know that x = 48, and you can substitute in the first equation to get:
3*48 + 4*48 = 336
you can now substitute in the second equation to get:
3*a + 4*b = 336
you now know that a = 7x/6 which makes a = 7*48/6 which makes a = 7*8 = 56
you now know that b = 7x/8 which makes b = 7*48/8 which makes b = 7*6 = 42
you get a = 56 and b = 42 which is the answer you are looking for.
you confirm by substituting in the original equations.
x = y = 48
3x + 4y = 336
3*48 + 4*48 = 336
144 + 192 = 336
336 = 336 (good)
a = 56 and b = 42
3*56 + 4*42 = 336
168 + 168 = 336
336 = 336 (good)
end of nosebleed.
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