GED Mathematical Reasoning: Comparing and Rounding Numbers

Inequality symbols

< means “less than”
\leq means “less than or equal to”
> means “greater than”
\geq means “greater than or equal to”

 

Rounding

It is often useful to estimate when an exact answer is not necessary or when you are attempting to evaluate the reasonableness or accuracy of a result. Here are some phrases to illustrate the use of an estimate:

  • When you ask someone for directions, she might say: “It’s about ten miles down on the right side.”
  • When a friend tells you how much he paid for his new car, he might say: “I paid close to $12,000.”
  • When a blogger references her website, she might say: “It gets, on average, 100 hits a day.”
  • When you consider your monthly salary, you may think: “I make approximately $3,000 per month.”

We can use rounding as a tool to calculate estimates, like these examples. To do so, we follow a set of steps to round individual values. And then use those rounded values to create an estimate result. The benefit is that using rounded values often makes our calculations easier because rounded values usually contain more zeros.

Here are the steps for rounding. By the way, when you are formally asked to round a number, a place value will be provided.

  1. Identify the digit in the place value given – underline it
  2. Locate the digit directly to the right of that place value
  3. If the digit in step 2 is 0, 1, 2, 3, or 4 (in other words, less than 5) we “round down” by keeping the underlined place value digit the SAME and changing all digits to the right of the underlined digit to zero. If the digit in step 2 is 5 or greater (in other words 5, 6, 7, 8 or 9), we “round up” by increasing the underlined place value digit by 1 and changing all digits to the right of the underlined digit to zero.

 

Example 1

What digit is in the thousands place:  591,045 ?

 

Let’s compare our number 591,045 to the place value chart and determine which digit falls in the thousands spot.

Millions Thousands Ones
hundred millions ten millions millions hundred thousands ten thousands thousands hundreds tens ones
5 9 1 0 4 5

So we see that the digit in the thousands place is a one.

 

Example 2

Put the following numbers in order from least to greatest: 910, 1302, 1349, 891

 

To compare numbers, start with the largest place value and compare the digits – the greater the digit the greater the number. Also note that numbers containing fewer digits are smaller. This means, for example two, that 910 and 891 are smaller than 1302 and 1349 since they have fewer digits.

So let’s begin by comparing 910 and 891 to determine which number is the smallest of the four. When we compare the largest place value, the hundreds place, we see that 891 is smaller than 910, since 8 is less than 9. Therefore 891 is the smallest of the four numbers. And the next largest number is 910.

Now let’s compare 1302 and 1349 to determine which is third in line. When we compare the largest place value, the thousands place, we see that both numbers contain a one. Since the digits in the thousands place are the same, we are not able to determine which number is larger so we move to the hundreds place. The digits in the hundreds place are also the same – they are three’s – so we move to the tens place. Comparing the digits in the tens place, 1302 contains a zero and 1349 contains a 4. Since zero is smaller than 4, 1302 is less than 1349, making 1302 the third number in our ordering and 1349 the largest number.

rounding2

So the ordering of these four numbers from least to greatest is: first 891, then 910, then 1302, and finally 1349

 

Example 3

Round to the nearest hundred: 1492

 

For step one, we’ll underline the digit in the hundreds place – it is a 4

math 203

For step 2, the digit to the right is a 9. Following step 3, since the value to the right is 5 or greater, we’ll round UP by adding one to the 4 and changing all digits to the right to zero.

math 205

So the 4 becomes a 5 and the 9 and 2 become zeros. The final, rounded, result is: 1500

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