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02 Sep 2024, 05:43
Kudos
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x: number of eggs (excluding 2 free eggs)
\(\frac{12}{x}\): original price per egg
\(12 * \frac{12}{x}\): orginal price per dozen of eggs
\(\frac{12}{x+2}\): price per egg (after 2 free eggs)
\(12 * \frac{12}{x+2}\): price per dozen of eggs (after 2 free eggs)
=> \(12 * \frac{12}{x+2} = \frac{12 * 12}{x} - 1\)
=> \(\frac{144}{x+2} = \frac{144}{x} - 1\)
=> \(\frac{144}{x+2} = \frac{144-x}{x}\)
=> \(144x = (144-x)(x+2)\)
=> \(144x = 144x - x^2 + 288 - 2x\)
=> \(x^2 + 2x + 1 = 289\)
=> \((x+1)^2 = 17^2\)
=> x + 1 = 17
=> x + 2 = 18
\(\frac{12}{x}\): original price per egg
\(12 * \frac{12}{x}\): orginal price per dozen of eggs
\(\frac{12}{x+2}\): price per egg (after 2 free eggs)
\(12 * \frac{12}{x+2}\): price per dozen of eggs (after 2 free eggs)
=> \(12 * \frac{12}{x+2} = \frac{12 * 12}{x} - 1\)
=> \(\frac{144}{x+2} = \frac{144}{x} - 1\)
=> \(\frac{144}{x+2} = \frac{144-x}{x}\)
=> \(144x = (144-x)(x+2)\)
=> \(144x = 144x - x^2 + 288 - 2x\)
=> \(x^2 + 2x + 1 = 289\)
=> \((x+1)^2 = 17^2\)
=> x + 1 = 17
=> x + 2 = 18
28 Nov 2024, 19:16
Kudos
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It's more efficient to calculate the price per dozen, given that's what the question is focused around (e.g. price of dozen eggs differs by $1).
In fact, if we arrive at the previous statement \frac{144}{x} = \frac{144}{x-2} + 1
We can then quickly search for factors of 144 that differ by 1. The prime factorization of 144 is 3^2 * 2^4. Eyeballing, 8 and 9 are factors that differ by one, and 16 and 18 are the corresponding answer choices. If we want to find the solution that includes the free eggs, we look for the smaller number (because 2 eggs were free). Answer is 16.
In fact, if we arrive at the previous statement \frac{144}{x} = \frac{144}{x-2} + 1
We can then quickly search for factors of 144 that differ by 1. The prime factorization of 144 is 3^2 * 2^4. Eyeballing, 8 and 9 are factors that differ by one, and 16 and 18 are the corresponding answer choices. If we want to find the solution that includes the free eggs, we look for the smaller number (because 2 eggs were free). Answer is 16.
einstein801
Anyone done it the price per egg way? Am kinda stuck.
Price per egg (initial): y
Price per egg (after +2): y-1
X eggs for $12: price per egg = 12/x
X +2 eggs for $12: price per egg = 12/(x+2)
eqn1: 12/x = y (before)
eqn 2: 12/(x+2) (after) --> sub eqn 1 and solve --> 2y = x +2.
Stuck here - any smarties able to help?
Price per egg (initial): y
Price per egg (after +2): y-1
X eggs for $12: price per egg = 12/x
X +2 eggs for $12: price per egg = 12/(x+2)
eqn1: 12/x = y (before)
eqn 2: 12/(x+2) (after) --> sub eqn 1 and solve --> 2y = x +2.
Stuck here - any smarties able to help?
Bunuel
A chef visited a market to purchase some eggs and paid $12 for them. However, as the eggs were smaller than expected, the chef convinced the seller to add two more eggs to the purchase, free of cost. As a result of this, the price per dozen of eggs decreased by one dollar. How many eggs did the chef purchase at the market, including the two free eggs?
A. 8
B. 12
C. 15
D. 16
E. 18
A. 8
B. 12
C. 15
D. 16
E. 18
