D01-26

Official Solution:

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


Let's say that the number of eggs the cook originally purchased was \(x\).

Since the cook paid $12 for them, the price per egg would be \(\frac{12}{x}\), and the price per dozen would be \(12*\frac{12}{x}\). When the cook convinced the seller to add two more eggs, he ended up getting a total of \(x+2\) eggs (which is what we need to determine).

This decreased the price per egg to \(\frac{12}{x+2}\), and decreased the price per dozen to \(12*\frac{12}{x+2}\).

Since the price per dozen went down by one dollar, we have the equation \(12*\frac{12}{x}-12*\frac{12}{x+2}=1\), which simplifies to \(\frac{144}{x}-\frac{144}{x+2}=1\). Rather than solving for \(x\), it's easier to substitute the answer choices to find the correct value of \(x\). Answer choice E fits: if \(x+2=18\), then \(x = 16\) and \(\frac{144}{16}-\frac{144}{18}=9-8=1\).


Answer: E
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a 30 sec solution: think like a test-maker

the testmaker want to trick and trap you right? mean testmaker always includes right answers but to different questions.

here you know that the difference should be 2. hence the whole thing smells like an alegbraic manipulation. when doing algebra people do everything right but forget to look back at what the questions asks. testmaker exploits this.

so we have 2 contenders for the right answer D and E as the difference between them is 2.

if the question asked how many egg did the seller initially give to the cook, the answer would be the lower number - D

Good question. Having both 16 and 18 as answer choices is a bit cruel though. I solved the quadratic for x and got 16. I forgot that the question asked as to determine x+2. If 16 weren't there as a possible answer choice I would have realized what the question was asking.

Need to be able to confidently represent 'price per unit', and answering this question correctly requires such.

memo: price per unit is total price / total quantity; as such, 'price per egg' can be represented as '12 / x', where x = the unknown quantity of eggs. The next step is recognizing that '12 / x' is a generic statement for the price of each egg; hence, 12 * (12/x) = the price per dozen eggs. The fact that the numerator in '12/x' is 12 is a coincidence and should not cause confusion in your attempt to answer the question.

Thus, if given 2 new eggs for free (i.e. x + 2), then the new 'price per dozen' can be represented as '12 * (12 / x +2)'. We can set up an equality here to highlight the fact that this new equation yields a 'price per dozen eggs' (or 12 * 12/x) that is 1 less than the original equation; hence [12 * (12 / x+2)] = [12 * (12/x)] - 1

Solving for x will yield 16; thus, x + 2 = 18, which is the correct answer.

I back-solved and looked for two numbers $1 apart.

Starting with C: He paid $12 for 15 eggs = (4/5) *12 = 9.60 Had he paid $12 for 13 eggs * 12 = 11.something. More than a dollar

E. He paid $12 for 8 eggs = (2/3)*12 = 8 Had he gotten 16... 12/16 = (3/4)*12 = 9
8 and 9 are $1 away. If E had been less than a dollar, answer would have been D.

Answer E.

I think this is a high-quality question.

I think this is a high-quality question and I agree with explanation. 18/12=3/2
16/12=4/3

3/2*4/4=12/8
4/3*3/3=12/9

9-8=1

I think this is a high-quality question and I agree with explanation. very good question ! easy to fall in the trap of forgetting to add 2 to the final result

suppose price of 12 eggs was X
now price of 12 eggs is X-1

In 12 $ earlier he would have got 12/( 12/x) eggs
now he will get 12/(12/(X-1)) Eggs
so 12/(12/(X-1)) - 12/( 12/x) =2;
X(X-1)=72

X=9;
Hence 12/(12/(X-1)) = 18

When you get the answer - do not just jump to the option and tick the answer.
Before selecting the option - CHECK ALWAYYYYYSSSSS - What is the questn asking for - (How many eggs he took HOME) not
How many eggs are there, i.e 16. The answer is - 16+2 = 18 (as he asked for 2 more eggs)

I think this is a high-quality question and I agree with explanation.

I think this is a high-quality question and I agree with explanation.

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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I think this is a high-quality question and I agree with explanation.
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KarishmaB can you pls tell the intuitive way you use of balancing around mean to solve this question ?

Posted from my mobile device

I will start by assuming that he went to buy 12 eggs for $12 i.e. $1 per egg even though 14 is not in the options. (These numbers will help me split the options into 2 sections by doing easy calculations.) If he gets 14 eggs for $12, cost per dozen becomes
\(\frac{12}{14}*12\) which is approx 10.3.
So the cost per dozen has decreased by more than $1. Hence he must have taken home more than 14 eggs in all.

Next, we will try for 16 eggs as the answer i.e. he went to buy 14 eggs. If the reduction is still more than $1, we know the answer is option (E). If the reduction is less than $1, we know the answer is (C).
Actual Cost per dozen = (12/14) * 12
Effective cost per dozen = (12/16) * 12
Their diff =\( \frac{144}{112}\) which is again more than 1.

Hence answer (E)

KarishmaB how did you conclude this? "Hence he must have taken home more than 14 eggs in all.
"? How can we figure this


and also can we do it by the method on your blog which solves average quaetions using deviations from mean? if yes how

Adding 2 eggs in a small number of eggs will decrease the cost per egg more but adding the same two eggs in a larger number of eggs will decrease the cost per egg by a smaller amount. That is why when we see that with 12 eggs, cost reduction is more than $1, we know that number of eggs would be higher.

Also, you can use the deviations from avg here but it will lead to a variable and equations (because neither do we have the number of eggs nor the price per egg so a variable will be required) hence I would much rather use the options.

Say you have n eggs. Cost of each egg is

12/n, 12/n, 12/n ..... (n times)

You add 2 eggs here then per dozen cost decreases by $1 which means that per cost egg decreases by 1/12 of a dollar. It becomes 12/n - 1/12 for each egg.
Hence total reduction in cost of n eggs is n times 1/12 i.e. n/12 and this gets split evenly among the two new eggs i.e. n/24 each.

So
12/n - 1/12 = n/24
and from here we get n = 16 which means 18 eggs were taken home.