Chapter 2 · Fraction foundations review
Round Mixed Numbers and Put Fractions in Order
Round a mixed number by comparing its fraction part with one half. Then order fractions and mixed numbers with benchmarks or a common denominator, and check your order.
You already know how to compare two fractions. In this lesson you will round mixed numbers, put several numbers in order, and check your order. Each job starts the same way: find where the number sits on a number line.
On one number line, the distance from 0 to 1 is the whole, and every 1 is that same size. That matters. You can compare fractions as amounts only when they refer to the same whole, or to wholes of equal size. Every number in this lesson refers to that one whole.
A whole number is one of the numbers 0, 1, 2, 3 and so on. A mixed number is a whole number together with a fraction that is less than 1. Take . You say three and three eighths, and it means . The 3 is its whole number part. The is its fraction part.
On the number line, sits between 3 and 4. To round it to the nearest whole number means to pick the whole number it is closer to.
Why does matter? Halfway between 3 and 4 is .
- A number between 3 and is closer to 3.
- A number between and 4 is closer to 4.
So the fraction part decides. You only need to compare it with .
Try it with . The denominator (the bottom number) is 8. It tells you that 8 equal parts make 1 whole, so 4 of those parts make one half: . The numerator (the top number) is 3. It counts 3 of those parts. Three parts are fewer than four parts of the same size, so .
That puts below the halfway point. The distances agree. It is above 3 and below 4, because . So is closer to 3. Here is that position in symbols.
| Fraction part compared with one half | Where the number sits | Example | Rounds to |
|---|---|---|---|
Less than | Below halfway, so closer to 3 | , because | , the whole number part |
Equal to | Exactly halfway, so neither 3 nor 4 is closer | , because | , by the usual rule that halfway rounds up |
Greater than | Above halfway, so closer to 4 | , because | , the next whole number |
Worked example
Round a mixed number
Round to the nearest whole number.
Find the two whole numbers it sits between. Six and four sevenths is 6 plus a fraction less than 1, so it is between 6 and 7. The halfway point is six and one half.
Compare the fraction part with one half of the same whole. The denominator is 7, so 7 equal parts make 1 whole. Half of 7 is three and a half, so three and a half of those parts make one half. The numerator is 4, and 4 parts are more than three and a half parts.
A fraction part greater than one half puts the number above the halfway point, so it is closer to 7. The distances agree: it is 4 sevenths above 6 and only 3 sevenths below 7.
Answer
Rounded to the nearest whole number, is .
Now for ordering. To put numbers in order from least to greatest means to list them the way they sit on the number line, from left to right.
You already know two tools for comparing two numbers.
- A benchmark is a familiar number, such as , 1 or , that you compare other numbers with. If one number is less than a benchmark and another is greater than that same benchmark, the first number is the smaller of the two.
- A common denominator is one denominator that two or more fractions are rewritten to share. Then the parts are the same size, so the fraction with more parts is greater.
Both tools compare amounts. So both need the numbers to refer to the same whole, or to wholes of equal size.
Which tool fits? Try benchmarks first, because they are quick. Rounding gave you practice: you compared a fraction part with . A common denominator works for any pair of fractions, but it takes more steps. Save it for pairs that benchmarks cannot separate.
| What you notice about the pair | Tool that fits | Example |
|---|---|---|
One is less than and the other is greater | The benchmark | and , so |
One is less than a whole number and the other is greater | A whole number benchmark | and , so |
Both sit between the same two benchmarks, such as and | A common denominator | and , so |
Key concept
Whole numbers and halves are your benchmarks
To round a number or to order numbers, first find where each one sits on the number line. Whole numbers and halfway points, such as and , are the benchmarks that tell you.
- Rounding a mixed number: compare its fraction part with . Less than rounds to the whole number part. Greater than rounds up to the next whole number. Exactly is halfway, and the usual rule is to round up too.
- Ordering: sort the numbers by benchmarks first. For numbers that sit between the same benchmarks, use a common denominator.
These comparisons tell you about amounts only when every number refers to the same whole, or to wholes of equal size.
Worked example
Order three numbers with benchmarks
Put , and in order from least to greatest. All three refer to the same whole.
Choose a tool. Three eighths is a small part of 1, five sixths is nearly 1, and one and one fourth is past 1. The numbers look spread out, so try the benchmarks one half and 1 first.
Compare the two fractions with one half. For eighths, 4 parts make one half, and 3 parts are fewer, so three eighths is less than one half. For sixths, 3 parts make one half, and 5 parts are more, so five sixths is greater than one half.
Compare with 1. Five sixths is 5 of the 6 equal parts that make 1 whole, so it is less than 1. One and one fourth is 1 whole plus one fourth more, so it is greater than 1.
A benchmark falls between each pair, so you do not need a common denominator. List the numbers from left to right on the number line.
Answer
From least to greatest, the order is , , .
Before you trust an order, check the neighbors. Go along your list and compare each number with the one right after it.
In a list from least to greatest, each number must be less than the next one, or equal to it. If every pair of neighbors passes, the list never steps down from the first number to the last, so the entire list is in order. This check compares amounts, so the numbers must refer to the same whole, or to wholes of equal size.
Watch the check catch a slip. Someone writes , , . The first two both sit between and 1, so compare them in fifteenths: and . Ten fifteenths is more than nine fifteenths, so that pair is out of order.
Swap the two and check again: , , . Now , because . And , so the benchmark 1 falls between the second pair. Both pairs pass, so this order is right.
Worked example
Order four numbers and check
Put , , and in order from least to greatest. All four refer to the same whole.
Start with the benchmark 1. Seven eighths is 7 of the 8 equal parts that make 1 whole, so it is less than 1. Each of the other numbers is 1 whole plus some more, so each is greater than 1. That makes seven eighths the least.
Sort the three mixed numbers with the benchmark one and one half. They share the whole number part 1, so compare each fraction part with one half. One third is less, because thirds are smaller than halves of the same whole. Two fourths make one half, and three fourths is more. Three sixths make one half, and five sixths is more.
Two numbers now sit between the same benchmarks, one and one half and 2, so those benchmarks cannot separate them. Use a common denominator for the fraction parts. Each fourth is 3 twelfths and each sixth is 2 twelfths, so rewrite both as twelfths. The parts are now the same size, and 9 parts are fewer than 10 parts. One and three fourths is the smaller number.
List the numbers from left to right on the number line: first the one below 1, next the one below one and one half, then the last two in the order the twelfths gave.
Check each number against the one next to it. The benchmark 1 falls between the first pair. The benchmark one and one half falls between the second pair. The twelfths settle the third pair. All three pairs of neighbors are in order, so the entire list is in order.
Answer
From least to greatest, the order is , , , .
Here is the idea to keep. Whole numbers and halves are benchmarks that show where a number sits on the number line. To round a mixed number, compare its fraction part with . To order numbers, use benchmarks first, and use a common denominator for any pair they cannot separate. Then check each number against the one next to it. All of this compares amounts, so the numbers must refer to the same whole, or to wholes of equal size.