A dealer offers sells half of the eggs that he has and another half an egg to...

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Question Id : 4015 | Context :NIMCET 2019

Question

A dealer offers sells half of the eggs that he has and another half an egg to Anurag. Then he sells half of the balance eggs and another half an egg to Deepak. Then he sells half of the balance eggs and another half an egg to Sivani. In the end he is left with just 7 eggs and he claims that he never broke an egg. How many eggs did he start with?

🎥 Video solution / Text Solution of this question is given below:

Solution:

Let the initial number of eggs =

x <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>

.

First sale (to Anurag):
Sells half of the eggs plus half an egg:

Eggs sold=x2+12<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>Eggs sold</mtext><mo>=</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>

Remaining eggs:

x−(x2+12)=x2−12<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>x</mi><mo>−</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>

Second sale (to Deepak):
Sells half of remaining eggs plus half an egg:
Remaining eggs:

(x2−12)−(12(x2−12)+12)=x4−54<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>−</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mfrac><mi>x</mi><mn>4</mn></mfrac><mo>−</mo><mfrac><mn>5</mn><mn>4</mn></mfrac></math>

Third sale (to Shivani):
Sells half of remaining eggs plus half an egg:
Remaining eggs:

(x4−54)−(12(x4−54)+12)=x−98<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><mi>x</mi><mn>4</mn></mfrac><mo>−</mo><mfrac><mn>5</mn><mn>4</mn></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>−</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><mi>x</mi><mn>4</mn></mfrac><mo>−</mo><mfrac><mn>5</mn><mn>4</mn></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>9</mn></mrow><mn>8</mn></mfrac></math>

Given that after all sales, 7 eggs are left:

x−98=7<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mrow><mi>x</mi><mo>−</mo><mn>9</mn></mrow><mn>8</mn></mfrac><mo>=</mo><mn>7</mn></math>

Multiplying both sides by 8:

x - 9 = 56 <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>x</mi><mo>-</mo><mn>9</mn><mo>=</mo><mn>56</mn></math>

x = 65 <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>x</mi><mo>=</mo><mn>65</mn></math>

Verification:
Since dealer never broke an egg, and he must sell whole number of eggs each time:

Checking for

x = 65 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>65</mn></math>

:
- After first sale: Remaining eggs = 32
- After second sale: Remaining eggs = 16
- After third sale: Remaining eggs = 7 (Not matching without breaking eggs). ❌

Checking for

x = 63 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>63</mn></math>

:
- After first sale: Sells

632+12=32<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>63</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mn>32</mn></math>

eggs, Remaining = 31 eggs ✅
- After second sale: Sells

312+12=16<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>31</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mn>16</mn></math>

eggs, Remaining = 15 eggs ✅
- After third sale: Sells

152+12=8<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>15</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mn>8</mn></math>

eggs, Remaining = 7 eggs ✅

Thus, the dealer originally had:

63 eggs <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><menclose notation="box"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mn>63</mn><mtext> </mtext><mtext>eggs</mtext></mrow></mstyle></mrow></menclose></math>

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