In a Battle, 70% of the Combatants Lost One Eye, 80% an Ear, 75% an Arm, 85% a Leg, x% Lost all the Four Limbs.

Question: In a battle, 70% of the combatants lost one eye, 80% an ear, 75% an arm, 85% a leg, x% lost all the four limbs. What is the minimum value of x?

1. 10
2. 12
3. 15
4. 20
5. 25

“In a battle, 70% of the combatants lost one eye, 80% an ear, 75% an arm, 85% a leg, x% lost all the four limbs. What is the minimum value of x?” –  is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "Official Guide for GMAT Review". GMAT Quant section consists of a total of 31 questions. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1

Given in the question that in a battle, 70% of the combatants lost one eye, 80% an ear, 75% an arm, 85% a leg, and x% lost all four limbs.
The goal of this question is to minimize the number of combatants who lost all 4 limbs.
This question can be solved by the double matrix method.

Firstly we’ll start by minimizing the number of combatants who lost an eye and an ear.
From this information we get,

We minimize the number of combatants who lost an eye and ear with each of the following scenarios.

 

So now with regard to losing ears and eyes the minimum number of combatants who lost their eyes, as well as ears, are 50.
Now we’ll minimize the number of combatants who lost an arm and who already lost one eye and one ear we get the following.

 

We minimize the number of combatants who lost an arm and already lost one eye with the following scenario.

So with regard to losing eyes, ears, and arms, the minimum number of unlucky combatants is 25
Finally we minimize the number of combatants who lost a leg and who already lost an eye, an ear, and an arm,

Now finally we minimize the number of combatants who lost all the body parts with the following scenario –

Therefore we get the minimum percentage of combatants who lost all the four limbs are = 10%.

Approach Solution 2:

Given in the question that in a battle, 70% of the combatants lost one eye, 80% an ear, 75% an arm, 85% a leg, and x% lost all four limbs.
The goal of this question is to minimize the number of combatants who lost all 4 limbs.
Given that 75% lost an arm and 85 % lost a leg,
No of people who lost an arm or leg = 75 + 85 - 100 = 60
So 60% lost an arm or leg.

70% lost an eye and 80% lost an ear,
So 50% lost an eye or ear.
Minimum value of x who lost all four limbs is = 60-50 = 10
10% is the correct answer.
Hence, A is the correct answer.

Approach Solution 3:

For the ease of understanding, let us take eye, ear, arm and leg as a, b, c and d respectively.
Now to minimise the value of x:
people with all the four injuries, we have to distribute the injuries one after another in a way that it fills up 100%.
So 70% lost an eye. Next 80% lost an ear.
So the 100-70 or 30 are filled up for ear alone and then the remaining 80-30 of ear injuries are given to those who had already lost an eye.
At the end of this distribution, we have 50% with both injuries, 20% with just eye and 30% with just ear.

Next 75% lost an arm.
So we will fill up those with just one injury first. => So 20% now have injury in eye and arm, while 30% have in eye and arm. But we still have 75-20-30 or 25 arm injuries to fill up, who will go to those with eye and ear.
In the end of this distribution we have 25% with 3 injuries and remaining 75% with 2 injuries.
Next 85% lost a leg.
First we fill up those with two injuries, that is 75%, so 85-75 or 10% are still left and we have everyone with 3 injuries. Thus, the renaming 10% will have to be given to those people who already have 3 injuries.

So x has to be 10.

The concept of distribution is explained above, the method is straightforward
Add all of them => 70+80+75+85=310%
So 100% have three injuries, leading to 300%. So remaining 10% will make 10% of people to have all 4 injuries.

Suggested GMAT Quant Questions:

• How Many Litres of a 90% Solution of Concentrated Acid Needs to be Mixed with a 75% Solution
• For How Many Values of k is 12^12 the Least Common Multiple of the Positive Integers 6^6, 8^8 and k?
• Two Varieties of Sugar are Mixed together in a Certain Ratio
• The Average Age of Chief Executive Officers (CEO’s) in a Large Sample of Companies is 57
• If a Carton Containing a Dozen Mirrors is Dropped which of the following Cannot be the Ratio of Broken Mirrors to Unbroken Mirrors?
• A man can hit a target once in 4 shots. If he fires 4 shots in succession, what is the probability that he will hit his target?
• 12 marbles are selected at random from a large collection of white, red, green and yellow Marbles. 
• If 75 Percent of a Class Answered the First Question on a Certain Test Correctly
• 4 Out Of 15 Apples Are Rotten. They Are Taken Out One By One At Random
• Of the 150 houses in a certain development, 60 percent have air-conditioning, 50 percent have a sunporch, and 30 percent have a swimming pool
• There is a 120 Liter Mixture of Alcohol and Water. The Ratio of Alcohol to Water is 7 : 5
• The Salaries A, B, C are in the Ratio 2 : 3 : 5
• The Price of Raw Materials Has Gone up by 15%, Labour Cost Has Also Increased From 25% of the Cost of Raw Material to 30% of the Cost of Raw Material.
• What is the Largest Power of 3 Contained in 200!
• Given that (P + 2Q) is a positive number, what is the value of (P + 2Q)?
• Three Cars Leave From A To B In Equal Time Intervals
• A Clock Strikes 4 taking 9 seconds. In Order to Strike 12 at the Same Rate, the Time Taken is
• 16 Ounces of Fresh Orange Juice Contains 216 Calories And 16 Ounces of Fresh Grapefruit Juice Contains 174 Calories
• X is Older Than Y, Z Is Younger Than W And V is Older Than Y. Is Z Younger Than X?
• Company C Sells a Line of 25 Products with an Average Retail Price of $1,200