Mixed Numbers and the Least Common Denominator | Grade 6 Math | LevelKite

You are baking, and your only clean scoop holds $\frac{1}{4}$ cup. The recipe takes 11 scoops of flour. How many cups is that?

Here the whole is 1 cup. Each scoop is one fourth: 1 of the 4 equal parts that fill the cup. Eleven scoops are 11 fourths, written $\frac{11}{4}$ cups.

Every 4 scoops fill 1 whole cup. So 11 scoops fill 2 whole cups, with 3 scoops left over. That is 2 cups plus $\frac{3}{4}$ cup, written $2\frac{3}{4}$ cups.

Both names describe the same amount of flour. The first counts only fourths. The second counts whole cups first, then the fourths that are left.

These two kinds of names have math terms. Recall that the denominator of a fraction tells how many equal parts make 1 whole. The numerator counts how many of those parts you have.

An improper fraction is a fraction whose numerator is greater than or equal to its denominator, such as $\frac{11}{4}$ or $\frac{4}{4}$ . It names an amount of 1 whole or more by counting parts only.

A mixed number has a whole-number part of 1 or more and a fraction part between 0 and 1, such as $2\frac{3}{4}$ . It means a sum: $2\frac{3}{4} = 2 + \frac{3}{4}$ . The 2 counts wholes. The $\frac{3}{4}$ counts parts of one more whole of the same size.

The same amount of flour, named two ways. The whole is 1 cup.

Quarter-cup scoops	Improper fraction (cups)	Grouped into wholes	Mixed or whole number (cups)
5	$\frac{5}{4}$	1 whole and 1 fourth	$1\frac{1}{4}$
7	$\frac{7}{4}$	1 whole and 3 fourths	$1\frac{3}{4}$
8	$\frac{8}{4}$	2 wholes and no fourths	$2$
11	$\frac{11}{4}$	2 wholes and 3 fourths	$2\frac{3}{4}$

Key concept

Rename the number, keep the amount

A fraction counts equal parts of a whole. You can rename a number without changing its amount, as long as the whole stays the same.

Group parts into wholes: an improper fraction becomes a mixed number, or a whole number when no parts are left over.
Cut wholes into parts: a mixed number becomes an improper fraction.
Cut parts into smaller equal parts: two fractions can share one denominator.

Matching denominators mean the same number of parts in each whole. The parts are equal in size only when the fractions are about the same whole.

To turn an improper fraction into a mixed number, group the parts into wholes. The denominator tells you how many parts fill 1 whole, so divide the numerator by the denominator.

The quotient is the number of full groups. Each full group makes 1 whole.
The remainder is the number of parts left over. It becomes the numerator of the fraction part.
The denominator stays the same. The left-over parts are still parts of the same whole, so their size has not changed.

When the remainder is 0, there is no fraction part, and the answer is a whole number. For example, $\frac{8}{4} = 2$ . When the fraction part is not in lowest terms, simplify it as you learned earlier. For example, $\frac{22}{8} = 2\frac{6}{8} = 2\frac{3}{4}$ .

Worked example

From an improper fraction to a mixed number

Write $\frac{17}{5}$ as a mixed number. Every part is a fifth of the same-sized whole.

Read the denominator. It is 5, so 5 fifths make 1 whole. Divide 17 by 5 to find how many full groups of 5 fifths fit into 17 fifths.
$17 \div 5 = 3 \text{ remainder } 2$
Say what each result means. The quotient 3 means 3 full groups, which use 15 fifths. The remainder 2 means 2 fifths are left over.
$\frac{17}{5} = \frac{15}{5} + \frac{2}{5}$
Write the 15 fifths as 3 wholes, because each group of 5 fifths is 1 whole. The left-over parts are still fifths of the same whole, so the denominator stays 5. Then write the sum of 3 wholes and 2 fifths as one mixed number.
$\frac{15}{5} + \frac{2}{5} = 3 + \frac{2}{5} = 3\frac{2}{5}$
Check by going back. Three wholes are 3 groups of 5 fifths, which is 15 fifths. Add the 2 left-over fifths to get 17 fifths, the amount you started with.
$3 \times 5 + 2 = 17$

Answer

As a mixed number, $\frac{17}{5}$ is $3\frac{2}{5}$ .

To turn a mixed number into an improper fraction, go the other way: cut every whole into parts.

Take $2\frac{3}{4}$ . The denominator is 4, so each whole is 4 fourths. The 2 wholes are $2 \times 4 = 8$ fourths. Add the 3 fourths from the fraction part: $8 + 3 = 11$ fourths. So $2\frac{3}{4} = \frac{8}{4} + \frac{3}{4} = \frac{11}{4}$ .

In short: multiply the whole number by the denominator, add the numerator, and keep the same denominator. This works because the whole-number part and the fraction part use the same-sized whole. Once the wholes are cut up, every piece you count is the same size.

The two conversions undo each other. Multiplying by the denominator cuts wholes into parts. Dividing by the denominator groups the parts back into wholes. Neither one adds or takes away any amount, so the value stays the same.

Worked example

From a mixed number to an improper fraction

Write $4\frac{2}{3}$ as an improper fraction. The 4 counts wholes, and the $\frac{2}{3}$ is part of one more whole of the same size.

Read the denominator. It is 3, so each whole is 3 thirds. Cut all 4 wholes into thirds: 4 groups of 3 thirds are 12 thirds.
$4 = \frac{4 \times 3}{3} = \frac{12}{3}$
Add the 2 thirds from the fraction part. You can add these counts because every piece is now a third of the same-sized whole.
$4\frac{2}{3} = \frac{12}{3} + \frac{2}{3} = \frac{12 + 2}{3} = \frac{14}{3}$
Check by going back. Fourteen thirds make 4 full groups of 3 thirds, with 2 thirds left over. That matches the mixed number you started with.
$14 \div 3 = 4 \text{ remainder } 2$

Answer

As an improper fraction, $4\frac{2}{3}$ is $\frac{14}{3}$ .

Now take two fractions of the same whole: $\frac{5}{6}$ and $\frac{3}{8}$ . The first counts sixths. The second counts eighths. Sixths and eighths of the same whole are different sizes, so the numerators 5 and 3 count different things. Both numbers are written as fractions, but that alone does not make their parts the same size.

To count both amounts in parts of one size, rename both fractions so they share a denominator. You already know how to write an equivalent fraction, one with the same value: multiply the numerator and the denominator by the same nonzero whole number. So you need one denominator that both 6 and 8 can reach by multiplying.

A multiple of a number is that number times 1, 2, 3, and so on. The multiples of 6 are 6, 12, 18, 24, and so on. A common multiple of two numbers is a multiple of both. The smallest one is their least common multiple.

A common denominator of two fractions is a common multiple of their denominators. The least common denominator, or LCD, is the least common multiple of their denominators.

To find the LCD, list the multiples of each denominator from smallest to largest. The first number that appears in both lists is the LCD. Any common denominator works for renaming, but the LCD keeps the numbers smallest.

Keep one condition in mind. The LCD gives both fractions the same number of parts per whole. Those parts are the same size only when both fractions refer to the same whole. Renaming alone does not make them match.

Finding the LCD by listing multiples

Denominators	Multiples of the first	Multiples of the second	LCD	What to notice
$4$ and $12$	$4, 8, 12, \dots$	$12, 24, \dots$	$12$	$12$ is already a multiple of $4$ , so the larger denominator is the LCD.
$3$ and $5$	$3, 6, 9, 12, 15, \dots$	$5, 10, 15, \dots$	$15$	The first match is $3 \times 5 = 15$ .
$4$ and $6$	$4, 8, 12, \dots$	$6, 12, \dots$	$12$	$4 \times 6 = 24$ is a common denominator, but it is not the least.

Worked example

Find the LCD and rename both fractions

Two fractions describe parts of the same whole: $\frac{5}{6}$ and $\frac{3}{8}$ . Find their least common denominator. Then rewrite both fractions with it.

List the multiples of each denominator from smallest to largest. Stop when the same number appears in both lists.
$\text{Multiples of 6: } 6, 12, 18, 24 \qquad \text{Multiples of 8: } 8, 16, 24$
The first number in both lists is 24. So 24 is the least common multiple of 6 and 8, and that makes it the least common denominator.
$\text{LCD} = 24$
Rename five sixths with the denominator 24. Since 6 times 4 is 24, multiply the numerator and the denominator by 4. Each sixth is cut into 4 equal smaller parts. There are 4 times as many parts, and each is one fourth as big, so the amount stays the same.
$\frac{5}{6} = \frac{5 \times 4}{6 \times 4} = \frac{20}{24}$
Rename three eighths the same way. Since 8 times 3 is 24, multiply the numerator and the denominator by 3.
$\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}$
Read the result. Both fractions now count twenty-fourths. The twenty-fourths are the same size because both fractions refer to the same whole, not just because the denominators match.

Answer

The least common denominator of $\frac{5}{6}$ and $\frac{3}{8}$ is $24$ . With it, $\frac{5}{6} = \frac{20}{24}$ and $\frac{3}{8} = \frac{9}{24}$ .

Here is the idea to keep: you can rename a number without changing its amount, as long as the whole stays the same. Group parts into wholes to write an improper fraction as a mixed number or a whole number. Cut wholes into parts to go back. To count two fractions of the same whole in equal-sized parts, rename them with the least common denominator, which is the least common multiple of their denominators. Next, you will use these skills to compare fractions, put fractions and mixed numbers in order, and see fractions as division.