This GMAT practice question is a very interesting problem solving question from descriptive statistics. It covers the following statistical concepts: Mean, Median, Mode, and Range.

#### Question: An analysis of the monthly incentives received by 5 salesmen : The mean and median of the incentives is $7000. The only mode among the observations is $12,000. Incentives paid to each salesman were in full thousands. What is the difference between the highest and the lowest incentive received by the 5 salesmen in the month?

- $4000
- $13,000
- $9000
- $5000
- $11,000

#### Explanatory Answer

Video explanation will be added soon#### Use these hints and try

- Mean and Median = $7000. So, find the third highest incentive.
- Only one mode; mode = $12,000
- Use hint 1 and hint 2 to find how many have got $12,000
- Now compute the sum of incentives got by those who got neither $7000 nor $12000

#### Step 1: Data given

- The arithmetic mean of the incentives is $7000.
- The median of the incentives is also $7000.
- There is only one mode and the mode is $12,000.

#### Step 2: Decoding Mean and Median

Let their incentives be a, b, c, d, and e such that a ≤ b ≤ c ≤ d ≤ e

Therefore, the median of these values is 'c'.

The median incentive is $7000. So, c = $7000.

Essentially, the incentives are __ __ 7000 __ __

The arithmetic mean of the incentives is $7000.

So, the sum of their incentives a + b + c + d + e = 5 * 7000 = $35,000

#### Step 3: Decoding Mode

There is only one mode amongst these 5 observations.

The mode is that value that appears with the maximum frequency.

Hence, $12,000 is the incentive received by the most number of salesmen.

So, the incentives are __ __ 7000, 12000, 12000

#### Step 4: Putting it all together

The incentive that c has got is $7000

The incentive received by d and e are 12,000 each

Therefore, c + d + e = 7000 + 12,000 + 12,000 = $31,000

Hence, a + b = 35,000 - 31,000 = $4000

As there is only one mode, the incentives received by a and b have to be different

So, a received $1000 and b received $3000.

Maximum incentive : $12,000

Minimum incentive : $1000

Difference between maximum and minimum incentive : $11,000

#### Complete Explanation

- The arithmetic mean of the incentives is $7000.
- The median of the incentives is also $7000.
- There is only one mode and the mode is $12,000.

Let their incentives be a, b, c, d, and e such that a ≤ b ≤ c ≤ d ≤ e

Therefore, the median of these values is 'c'.

So, c = $7000.

The sum of their incentives a + b + c + d + e = 5 * 7000 = $35,000

There is only one mode amongst these 5 observations.

The mode is that value that appears with the maximum frequency.

Hence, $12,000 is the incentive received by the most number of salesmen.

The incentive that c has got is $7000

So, the incentive received by d and e has to be greater than or equal to $7000

But the mode is $12,000

So, d and e should have got $12,000 each.

Therefore, c + d + e = 7000 + 12,000 + 12,000 = $31,000

Hence, a + b = 35,000 - 31,000 = $4000

As there is only one mode, the incentives received by a and b have to be different

So, a received $1000 and b received $3000.

Maximum incentive : $12,000

Minimum incentive : $1000

Difference between maximum and minimum incentive : $11,000

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