Rules of Rounding Decimals | Number Properties

The famous rounding song by Joe Crone is pretty much all you need to solve the trickiest of rounding questions on GMAT:

You just slip to the side, and you look for a five.

Well if the number that you see is a five or more,

You gotta round up now, that’s for sure.

If the number that you see is a four or less,

You gotta round down to avoid a mess.

To put it in our own words, when we round a decimal, we drop the extra decimal places and apply certain rules:

– If the first dropped digit is 5 or greater, we round up the last digit that we keep.

– If the first dropped digit is 4 or smaller, we keep the last digit that we keep, the same.

For Example, we need to round the following decimals to two digits after decimal:

(a) 3.857

We drop 7. Since 7 is ‘5 or greater’, we are left with 3.86

(b) 12.983

We drop 3. Since 3 is ‘4 or smaller’, we are left with 12.98

We drop 463. Since 4 is ‘4 or smaller’, we are left with 26.75

(d) 8.9675

We drop 75. Since 7 is ‘5 or greater’, we are left with 8.97

Note example (c) carefully:

When we round 26.75463 to two decimal places, we do not start rounding from the rightmost digit i.e. this is incorrect: 26.75463 becomes 26.7546 which becomes 26.755 which further becomes 26.76 – this is not correct. .00463 is less than .005 and hence should be ignored. You only need to worry about the digit right next to the digit you are keeping. Just slip to the side, and look for a five!

A logical question arises: what happens when we have, say, 2.5 and we need to round it to the nearest integer? 2.5 is midway between 2 and 3. In that case, why do we round the number up, as the rule suggests? Note that a 2.5 is a tie and we have many tie breaking rules that can be used. They are ‘Round half to odd’, ‘Round half to even’, ‘Round up’, ‘Round down’, ‘Round towards 0’, ‘Round away from 0’ etc. We don’t need to worry about all these since GMAT uses only Round up i.e. 2.5 will be rounded up to 3.

Let’s take a look at a question now which uses these fundamentals.

Question: The exact cost price to make each unit of a widget is ${€7.6xy7}$ , where ${x}$ and ${y}$ represent single digits. What is the value of ${y$ ?

Statement 1: When the cost is rounded to the nearest cent, it becomes €7.65.

Statement 2: When the cost is rounded to the nearest tenth of a cent, it becomes €7.65.

Solution: The question is based on rounding. We need to figure out the value of $y$ given some rounding scenarios. Let’s look at them one by one.

Statement 1: When the cost is rounded to the nearest cent, it becomes €7.65.

When rounded to the nearest cent, the cost becomes 7 dollars and 65 cents. $6xy7$ cents got rounded to 65 cents. When will $.6xy7$ get rounded to .65? When $.6xy7$ lies anywhere in the range .6457 to .6547. Note that in all these cases, when you round the number to 2 digits, it will become .65.

Say price is 7.6468. We need to drop 68 but since 6 is ‘5 or greater’, 4 gets rounded up to 5.

Similarly, say the price is 7.6543. We need to drop 43. Since 4 is ‘4 or smaller’, 5 stays as it is.

So $x$ and $y$ can take various different values. This statement alone is not sufficient.

Statement 2: When the cost is rounded to the nearest tenth of a cent, it becomes €7.65

Now the cost is rounded to the tenth of a cent which means 3 places after the decimal. But the cost is given to us as €7.65. Since we need 3 places, the cost must be €7.650 (which will be written as €7.65)

When will $7.6xy7$ get rounded to 7.650? Now this is the tricky part of the question – from $7.6xy7$ , you need to drop the 7 and round up $y$ . When you do that, you get 7.650. This means $7.6xy7$ must have been 7.6497. Only in this case, when we drop the 7, we round up the 9 to make 10, carry the 1 over to 4 and make it 5. This is the only way to get 7.650 on rounding $7.6xy7$ to the tenth of a cent. Hence $x$ must be 4 and $y$ must be 9. This statement alone is sufficient to answer the question.