Question on Rounding of Decimals | Number Properties

Let’s take a look at a question which shows us how GMAT tests rounding of numbers.

A set has 21 positive decimals, none of which is an integer, and the sum of the 21 decimals is ${N}$ .

The estimated sum of the 21 decimals, ${M}$ , is defined as follows. Each decimal whose tenths digit is even is rounded up to the nearest integer, and each decimal whose tenths digits is odd is rounded down to the nearest integer. If ${\frac{1}{3}rd}$ of the decimals have a tenths digit that is even, which of the following is a possible value of ${M - N}$ ?

I. –13

II. –14

III. 0

IV. 4

GMAT is known to test your concept in this way. It is an example of how the higher level questions are created using the most basic concepts.

There is a lot of information in the question stem and a lot of variables are explained. Let’s review the given data in our own words.

The set has 21 decimals. The sum of all the decimals is $N$ .

7 decimals have even tenths digit. They will be rounded up.

14 decimals have odd tenths digit. They will be rounded down.

The sum of rounded numbers is $M$ .

$M - N$ can take many values so how do we figure which ones it cannot take? We need to find the minimum value $M - N$ can take and the maximum value it can take. That will help us figure out the values that $M - N$ cannot take. Note that $M$ could be greater than $N$ and it could be less than $N$ . So $M - N$ could be positive or negative.

Step I: Getting Minimum Value of $M - N$

Let’s try to make $M$ as small as possible. For that, we need to do two things:

1. When we round up the decimals (even tenths digit), the difference between actual and estimate should be very small. The estimate should add a very small number to round it up so that $M$ is not much greater than $N$ . Say the numbers are something similar to 3.8999999 (the tenths digit is the largest even digit) and they will be rounded up to 4 i.e. the estimate gains about 0.1 per number.

Since there are 7 even tenths digit numbers, the estimate will be approximately $.1 \times 7 = 0.7$ more than actual.

2. When we round down the decimals (odd tenths digit), the difference between actual and estimate should be as large as possible (to get largest negative values). Say the numbers are something similar to 3.999999 (tenths digit is the largest odd digit) and they will be rounded down to 3 i.e. the estimate loses approximately 1 per number. Since there are 14 such numbers, the estimate is $1 \times 14 = 14$ less than actual.

Overall, the estimate will be approximately $14 - 0.7 = 13.3$ less than actual.

Minimum value of $M - N = - 13.3$

Step II: Getting Maximum Value of $M - N$

Now let’s try to make $M$ as large as possible. For that, we need to do two things:

1. When we round up the decimals (even tenths digit), the difference between actual and estimate should be very high. Say the numbers are something similar to 3.000001 (tenths digit is the smallest even digit) and they will be rounded up to 4 i.e., the estimate gains 1 per number. Since there are 7 even tenths digit numbers, the estimate will be approximately $1 \times 7 = 7$ more than actual.

2. When we round down the decimals (odd tenths digit), the difference between actual and estimate should be very little. Say the numbers are something similar to 3.1 (tenths digit is the smallest odd digit). They will be rounded down to 3 i.e., the estimate loses approximately 0.1 per number. Since there are 14 such numbers, the estimate is approximately $0.1 \times 14 = 1.4$ less than actual.