Grade 4 · 4.NF.A.2 · 5.NF.A.1

Common denominators and the least common denominator

One ribbon is measured in fifths of a yard and another in tenths. How can both lengths be written in the same size of piece without changing either one?

You need

Greatest common factor and least common multipleGrades 5 to 6 Making an equivalent fraction by multiplyingGrade 4

Unlocks

Adding and subtracting fractions with unlike denominatorsGrade 5 Comparing by finding a common denominatorGrade 5 Dividing fractions: how many fitGrade 5

Where you are 4 in Equivalent fractions

Fractions, ratio and percent3

Fractions15

Equivalent fractions4

Seeing equivalent fractions in pictures and on the number line Making an equivalent fraction by multiplying

Common denominators and the least common denominator

Simplifying a fraction to simplest form

When you reach for it

Nil, Roksolana and Stojan share a sewing kit. It holds ribbons and fabric squares, and every length is written as a fraction of a yard. Nil’s red ribbon is \tfrac{2}{5} yard long. Roksolana’s blue ribbon is \tfrac{3}{10} yard long.

Those two fractions count different pieces. The first counts fifths of a yard and the second counts tenths. A fifth is bigger than a tenth, so “2 pieces” and “3 pieces” do not say which ribbon is longer, and they cannot be put together into one count. It is like saying “2 feet and 3 inches”: the numbers are fine, but the units do not match.

You reach for this move whenever two or more fractions have to be set side by side: to compare them, to add or subtract them, or just to see both on the same ruler. You rename every fraction so that all of them use the same size of piece. Nothing gets longer or shorter. Only the names change.

You already have the tool. From Making an equivalent fraction by multiplying: multiplying the top and the bottom of a fraction by the same counting number cuts every piece smaller and gives an equivalent fraction. This lesson asks which size of piece to cut to, so that two fractions land on the same one. It has three parts, one for each day:

Part 1: why the pieces must match, and what a common denominator is.
Part 2: when one denominator is a multiple of the other, and when neither is.
Part 3: the least common denominator, found by listing multiples, and renaming three fractions at once.

The procedure

Part 1: matching pieces, and what a common denominator is

Lay a yardstick marked in fifths above a yardstick marked in tenths. Put Nil’s ribbon on the top one and Roksolana’s on the bottom one.

Figure 1Every fifth of a yard is 2 tenths. So \tfrac{2}{5} yard is the same length as \tfrac{4}{10} yard, and now it sits on the same ruler as \tfrac{3}{10}.

Every fifth of the top yardstick sits exactly over 2 tenths of the bottom one, because the yard was cut into 5 parts on top and into 5 \times 2 = 10 parts below. So Nil’s 2 fifths reach the same point as 2 \times 2 = 4 tenths:

\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10}

Now both ribbons are measured in tenths of a yard: \tfrac{4}{10} yard and \tfrac{3}{10} yard. The pieces match, and the two counts, 4 and 3, can be set side by side.

The bottom number of a fraction, the denominator, names the size of the piece. So “the same size of piece” means “the same denominator”. A number that can be the denominator of both fractions is called a common denominator of the two fractions. Writing both fractions as equivalent fractions over that number is called renaming them to a common denominator.

Which numbers can be a common denominator? Multiplying never changes the amount, but it can only reach certain denominators. A fraction in fifths can be renamed in fifths, tenths, fifteenths, twentieths, and so on: the bottom becomes 5 \times 1, 5 \times 2, 5 \times 3, 5 \times 4, which are the multiples of 5. A fraction in tenths can be renamed only over the multiples of 10. A common denominator has to be on both lists:

Multiples of 5	5	10	15	20	25	30
Multiples of 10	10	20	30	40	50	60

The numbers on both rows, 10, 20 and 30, are common denominators of \tfrac{2}{5} and \tfrac{3}{10}, and the list goes on forever. So a common denominator is a common multiple of the two denominators: a number each denominator divides into with no remainder. Any one of them works. In twentieths, for example, every fifth becomes 4 pieces and every tenth becomes 2:

\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20} \qquad \frac{3}{10} = \frac{3 \times 2}{10 \times 2} = \frac{6}{20}

Each fraction gets its own multiplier, the number that turns its denominator into the common one. The top of each fraction is multiplied by that same number, because one cut goes through every piece of that fraction.

Part 2: one denominator a multiple of the other, or neither

When one denominator is a multiple of the other. The pair \tfrac{2}{5} and \tfrac{3}{10} is the easy kind. The number 10 is itself a multiple of 5, because 5 \times 2 = 10. So 10 is already on both lists, and only the fifths need cutting. The tenths stay as they are.

To spot this case, divide the larger denominator by the smaller one. If the answer is a whole number, the larger denominator is a common denominator, and that whole number is the multiplier for the other fraction. Here 10 \div 5 = 2, so \tfrac{2}{5} is multiplied top and bottom by 2, as in Part 1.

When neither denominator is a multiple of the other. Stojan cuts two fabric strips from the same bolt, each one yard long. He sews lace along \tfrac{3}{4} of the first strip and along \tfrac{5}{6} of the second. Quarters and sixths do not fit inside each other: 6 \div 4 is not a whole number. So both fractions have to be cut.

The product always works. Multiply the two denominators: 4 \times 6 = 24. This number is a multiple of 4 (it is 4 groups of 6) and a multiple of 6 (it is 6 groups of 4, by the commutative property of multiplication). So 24 is always on both lists. Each fraction is multiplied, top and bottom, by the other fraction’s denominator:

\frac{3}{4} = \frac{3 \times 6}{4 \times 6} = \frac{18}{24} \qquad \frac{5}{6} = \frac{5 \times 4}{6 \times 4} = \frac{20}{24}

A smaller shared multiple often works too. The product is safe, but it is not always the smallest choice. The multiples of 6 are 6, 12, 18, 24, \ldots, and 12 is already a multiple of 4. In twelfths, every quarter becomes 3 pieces and every sixth becomes 2 pieces:

Figure 2Cut every quarter into 3 and every sixth into 2. Both strips are now in twelfths, so \tfrac{3}{4} = \tfrac{9}{12} and \tfrac{5}{6} = \tfrac{10}{12}.

\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \qquad \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}

Both renamings are correct. Twenty-fourths and twelfths are both common denominators of \tfrac{3}{4} and \tfrac{5}{6}. In fact each twenty-fourth is half of a twelfth, so \tfrac{18}{24} is \tfrac{9}{12} with every piece cut in two, and \tfrac{20}{24} is \tfrac{10}{12} cut the same way. The twelfths simply use smaller numbers, which is easier to work with.

Part 3: the least common denominator, and three fractions

Of all the common denominators, the smallest one is called the least common denominator. It is the least common multiple of the denominators, the smallest number that is a multiple of each of them (Greatest common factor and least common multiple). For \tfrac{3}{4} and \tfrac{5}{6} it is 12.

Finding it by listing multiples. List the multiples of the larger denominator, one at a time, and stop at the first one the smaller denominator divides into (Least common multiple by listing does this for whole numbers). Starting with the larger number is quicker, because its list climbs faster. For 4 and 6:

6: is it a multiple of 4? No, 6 \div 4 leaves a remainder.
12: is it a multiple of 4? Yes, 4 \times 3 = 12.

So the least common denominator of \tfrac{3}{4} and \tfrac{5}{6} is 12. The list cannot miss a smaller one, because it checks every multiple of 6 in order, and any common denominator has to be a multiple of 6.

Keep listing and you find every other common denominator: 24, 36, 48, and so on. Each of them is a multiple of 12. That is not luck: for every pair of denominators, each common denominator is a multiple of the least one (the reason belongs to Greatest common factor and least common multiple). So renaming over a bigger common denominator is the same as renaming over the least one and then cutting every piece again.

Two special cases come straight from Part 2.

If one denominator is a multiple of the other, the larger denominator is the least common denominator. For \tfrac{2}{5} and \tfrac{3}{10} it is 10, the first number on the list of multiples of 10.
If the two denominators share no factor except 1, the least common denominator is their product. For \tfrac{1}{3} and \tfrac{1}{5}, the multiples of 5 are 5, 10, 15, and 15 = 3 \times 5 is the first one that 3 divides.

Three fractions. The kit has three spools of thread with \tfrac{1}{2}, \tfrac{2}{5} and \tfrac{3}{4} yard left on them. A common denominator of three fractions must be a multiple of all three denominators. List the multiples of the largest, 5, and test each one against 2 and 4:

Multiple of 5	5	10	15	20
A multiple of 2?	no	yes	no	yes
A multiple of 4?	no	no	no	yes

The first number with two yeses is 20, so the least common denominator is 20. Now rename each fraction with its own multiplier: 2 \times 10 = 20, 5 \times 4 = 20 and 4 \times 5 = 20.

\frac{1}{2} = \frac{10}{20} \qquad \frac{2}{5} = \frac{8}{20} \qquad \frac{3}{4} = \frac{15}{20}

All three lengths are now counted in twentieths of a yard. Ordering them, or adding them, is the work of the lessons that come next; renaming is the step that makes that work possible.

Worked examples

Examples 1 and 2 belong to Part 1. Examples 3 to 5 belong to Part 2. Examples 6 and 7 belong to Part 3.

Example 1

Three ways to match thirds and fifths

Stojan has \tfrac{1}{3} yard of green ribbon and \tfrac{4}{5} yard of gold ribbon. Find three common denominators, and rename both fractions using two of them.

Solution

A common denominator must be a multiple of 3 and a multiple of 5. List both:

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45

The numbers on both lists are 15, 30 and 45. Each is a common denominator.

In fifteenths: 3 \times 5 = 15, so every third is cut into 5, and 5 \times 3 = 15, so every fifth is cut into 3.

\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15} \qquad \frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}

In thirtieths: 3 \times 10 = 30 and 5 \times 6 = 30.

\frac{1}{3} = \frac{1 \times 10}{3 \times 10} = \frac{10}{30} \qquad \frac{4}{5} = \frac{4 \times 6}{5 \times 6} = \frac{24}{30}

Check: \tfrac{10}{30} is \tfrac{5}{15} with every piece cut in two, and \tfrac{24}{30} is \tfrac{12}{15} cut the same way. Both pairs name the same two ribbons.

Example 2

A number that fits only one denominator

Nil wants to write \tfrac{3}{8} and \tfrac{1}{3} over the denominator 12, because 12 is a multiple of 3. Does 12 work? If not, find a denominator that does, and rename both.

Solution

For \tfrac{1}{3}, twelfths are fine: 3 \times 4 = 12, so \tfrac{1}{3} = \tfrac{4}{12}.

For \tfrac{3}{8} we would need \tfrac{3}{8} = \tfrac{\square}{12}. That means cutting every eighth into the same whole number of pieces so that 8 pieces become 12. But 8 \times 1 = 8 and 8 \times 2 = 16, and no whole number lies between 1 and 2. So 12 is not a multiple of 8, and it is not a common denominator. A number on only one list does not work.

Look for a number on both lists. The multiples of 8 are 8, 16, 24, \ldots, and 24 is a multiple of 3, since 3 \times 8 = 24. So use 24:

\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24} \qquad \frac{1}{3} = \frac{1 \times 8}{3 \times 8} = \frac{8}{24}

Example 3

Eighths already hold quarters

Roksolana has a fabric square that is \tfrac{3}{4} yard on each side and a strip of trim that is \tfrac{7}{8} yard long. Rename both lengths to a common denominator, changing as little as possible.

Solution

Divide the larger denominator by the smaller: 8 \div 4 = 2, a whole number. So 8 is a multiple of 4, and eighths are a common denominator. The trim is already in eighths and stays \tfrac{7}{8}. Each quarter of a yard is cut into 2 eighths:

\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}

So the two lengths are \tfrac{6}{8} yard and \tfrac{7}{8} yard. Sixteenths would also work, but they would mean cutting both fractions, with bigger numbers and no gain.

Example 4

When the product is the smallest choice

Stojan needs \tfrac{2}{3} yard of felt and \tfrac{3}{10} yard of elastic. Rename both to a common denominator.

Solution

Neither denominator is a multiple of the other: 10 \div 3 is not a whole number. The product 3 \times 10 = 30 is always a common denominator. Multiply each fraction by the other one’s denominator:

\frac{2}{3} = \frac{2 \times 10}{3 \times 10} = \frac{20}{30} \qquad \frac{3}{10} = \frac{3 \times 3}{10 \times 3} = \frac{9}{30}

Is there anything smaller? A common denominator must be a multiple of 10, and the only multiples of 10 below 30 are 10 and 20. Neither is a multiple of 3. So 30 is the smallest common denominator here. The product is the best you can do whenever the two denominators share no factor except 1.

Example 5

The product, and a smaller shared multiple

Nil has \tfrac{5}{8} yard of satin ribbon and \tfrac{3}{10} yard of cotton ribbon. Rename both using the product of the denominators. Then find a smaller common denominator and rename them again.

Solution

The product is 8 \times 10 = 80. Each fraction is multiplied by the other denominator:

\frac{5}{8} = \frac{5 \times 10}{8 \times 10} = \frac{50}{80} \qquad \frac{3}{10} = \frac{3 \times 8}{10 \times 8} = \frac{24}{80}

For a smaller choice, try the multiples of 10 below 80: 10, 20, 30, 40. The first one that 8 divides is 40, since 8 \times 5 = 40. Every eighth becomes 5 pieces and every tenth becomes 4:

\frac{5}{8} = \frac{5 \times 5}{8 \times 5} = \frac{25}{40} \qquad \frac{3}{10} = \frac{3 \times 4}{10 \times 4} = \frac{12}{40}

Check: \tfrac{50}{80} is \tfrac{25}{40} with every piece cut in two, and so is \tfrac{24}{80} from \tfrac{12}{40}. The fortieths do the same job with half the numbers.

Example 6

Listing multiples for the least common denominator

Roksolana’s pattern calls for \tfrac{7}{10} yard of lining and \tfrac{5}{12} yard of piping. Find the least common denominator and rename both.

Solution

List the multiples of the larger denominator, 12, and test each one against 10. A multiple of 10 ends in 0.

12, 24, 36, 48: none ends in 0.
60: ends in 0, and 10 \times 6 = 60.

So the least common denominator is 60. The multipliers are 60 \div 10 = 6 and 60 \div 12 = 5:

\frac{7}{10} = \frac{7 \times 6}{10 \times 6} = \frac{42}{60} \qquad \frac{5}{12} = \frac{5 \times 5}{12 \times 5} = \frac{25}{60}

The product 10 \times 12 = 120 would also work, but its numbers are twice as large. Here the listing saves real work.

Example 7

Three ribbons in one size of piece

Nil’s drawer of the sewing kit has three ribbons, \tfrac{1}{2}, \tfrac{2}{3} and \tfrac{3}{8} yard long. Rename all three to their least common denominator.

Solution

The common denominator must be a multiple of 2, 3 and 8. List the multiples of the largest, 8, and test each against 2 and 3:

Multiple of 8	8	16	24
A multiple of 2?	yes	yes	yes
A multiple of 3?	no	no	yes

Every multiple of 8 is even, so 2 never stops the search. The first one 3 divides is 24. So the least common denominator is 24, and the multipliers are 24 \div 2 = 12, 24 \div 3 = 8 and 24 \div 8 = 3:

\frac{1}{2} = \frac{12}{24} \qquad \frac{2}{3} = \frac{16}{24} \qquad \frac{3}{8} = \frac{9}{24}

Check one by cutting: a half of a yard cut into 12 pieces, each a twenty-fourth, is 12 of them. All three ribbons are now counted in twenty-fourths of a yard.

Boundaries

Renaming changes names, never amounts. Each new fraction is equivalent to the old one because its top and bottom were multiplied by the same counting number. A different multiplier on top and bottom gives a different amount.
A common denominator must be a multiple of every denominator. A number that is a multiple of only one of them cannot be reached by the other fraction with a whole-number top, as with 12 for eighths in Example 2.
The same whole. Renaming \tfrac{1}{2} yard and \tfrac{1}{4} foot to fourths makes the denominators match, but not the units. Fractions of different wholes have to be written as fractions of one whole first.
Simplifying undoes the match. Dividing \tfrac{9}{12} back down to \tfrac{3}{4} returns to quarters, and the pieces no longer match. The renamed fractions are kept over the common denominator for as long as they need to be compared or combined.
“Least” depends on the fractions as written. For \tfrac{2}{4} and \tfrac{1}{6} the least common denominator of 4 and 6 is 12. But \tfrac{2}{4} = \tfrac{1}{2}, and \tfrac{1}{2} and \tfrac{1}{6} share sixths. Writing each fraction with the smallest numbers first (Simplifying a fraction to simplest form) can give an even smaller common denominator.
The denominators are counting numbers. A denominator of 0 names no fraction at all, so it has no multiples to list.

Where this goes next

Once two fractions use the same size of piece, comparing them is counting pieces. That is the work of Comparing by finding a common denominator; another route, measuring each fraction against a friendly one such as \tfrac{1}{2}, is Comparing against a benchmark. Adding and subtracting fractions with different denominators starts with exactly this renaming (Adding and subtracting fractions with unlike denominators), and so does one way to divide one fraction by another, asking how many times one length fits in another (Dividing fractions: how many fit).

Much later the same move works on fractions with letters in them. To add \tfrac{1}{x} + \tfrac{1}{x+1} you rename both over a common denominator, found the same way, as a shared multiple of the bottoms (Adding and subtracting rational expressions).

Exercises

Part 1: common denominators

Rename \tfrac{1}{2} and \tfrac{3}{7} using the denominator 14.
Which of 6, 9, 12, 15 and 18 are common denominators of \tfrac{2}{3} and \tfrac{1}{6}?
Rename \tfrac{3}{4} and \tfrac{1}{5} using the denominator 20.
Find the first three common denominators of \tfrac{4}{5} and \tfrac{1}{6}.
Roksolana says 15 is a common denominator of \tfrac{2}{3} and \tfrac{1}{6}. Is she right? Explain.
A pattern calls for \tfrac{2}{3} yard of fabric and \tfrac{1}{2} yard of lining. Write both amounts in sixths of a yard.
Rename \tfrac{5}{8} and \tfrac{1}{6} using the denominator 48.

Part 2: one a multiple of the other, or neither

Rename \tfrac{1}{2} and \tfrac{3}{8} to a common denominator, changing only one fraction.
Rename \tfrac{3}{5} and \tfrac{9}{100} to hundredths.
Rename \tfrac{5}{6} and \tfrac{1}{12} to a common denominator, changing only one fraction.
Rename \tfrac{1}{4} and \tfrac{2}{5} using the product of the denominators.
Rename \tfrac{2}{3} and \tfrac{1}{8} using the product of the denominators.
Rename \tfrac{3}{10} and \tfrac{1}{4} using the product of the denominators. Then find a smaller common denominator and rename them again.
Stojan has \tfrac{7}{4} yards of denim and \tfrac{1}{6} yard of ribbon. Rename both lengths using a common denominator smaller than 24.
Nil has \tfrac{2}{3} yard of red ribbon and \tfrac{4}{9} yard of blue ribbon. Rename both to a common denominator, changing only one fraction.

Part 3: the least common denominator

Find the least common denominator, then rename the fractions.

\tfrac{5}{6} and \tfrac{2}{15}
\tfrac{3}{4} and \tfrac{1}{10}
\tfrac{1}{6} and \tfrac{3}{8}
\tfrac{3}{8} and \tfrac{7}{10}
\tfrac{1}{3}, \tfrac{3}{4} and \tfrac{1}{6}
\tfrac{3}{5}, \tfrac{1}{2} and \tfrac{7}{10}
Roksolana’s three fabric squares measure \tfrac{1}{2}, \tfrac{3}{5} and \tfrac{2}{3} yard on a side. Rename all three lengths to their least common denominator.

Reasoning

Explain why the product of two denominators is always a common denominator.
Nil says the least common denominator of \tfrac{4}{9} and \tfrac{5}{12} is 108, because 9 \times 12 = 108. Is 108 a common denominator? Is it the least one? Rename the fractions to the least common denominator.
Explain why, when one denominator is a multiple of the other, the larger denominator is the least common denominator.

Answers

\tfrac{7}{14} and \tfrac{6}{14}. 2 \times 7 = 14, so \tfrac{1}{2} = \tfrac{1 \times 7}{2 \times 7}; 7 \times 2 = 14, so \tfrac{3}{7} = \tfrac{3 \times 2}{7 \times 2}.
6, 12 and 18. Each is a multiple of both 3 and 6. The numbers 9 and 15 are multiples of 3 but not of 6.
\tfrac{15}{20} and \tfrac{4}{20}. The multipliers are 20 \div 4 = 5 and 20 \div 5 = 4.
30, 60 and 90. The multiples of 6 are 6, 12, 18, 24, 30, \ldots, and the first one that ends in 0 or 5 is 30. The next common ones are 60 and 90.
No. 15 is a multiple of 3 but not of 6: 6 \times 2 = 12 and 6 \times 3 = 18. So \tfrac{1}{6} = \tfrac{\square}{15} has no whole number that fits.
\tfrac{4}{6} yard of fabric and \tfrac{3}{6} yard of lining. 3 \times 2 = 6 and 2 \times 3 = 6, so \tfrac{2}{3} = \tfrac{2 \times 2}{3 \times 2} and \tfrac{1}{2} = \tfrac{1 \times 3}{2 \times 3}.
\tfrac{30}{48} and \tfrac{8}{48}. The multipliers are 48 \div 8 = 6 and 48 \div 6 = 8.
\tfrac{4}{8} and \tfrac{3}{8}. 8 \div 2 = 4, so only \tfrac{1}{2} changes: \tfrac{1 \times 4}{2 \times 4} = \tfrac{4}{8}.
\tfrac{60}{100} and \tfrac{9}{100}. 100 \div 5 = 20, so \tfrac{3}{5} = \tfrac{3 \times 20}{5 \times 20}.
\tfrac{10}{12} and \tfrac{1}{12}. 12 \div 6 = 2, so \tfrac{5}{6} = \tfrac{5 \times 2}{6 \times 2}.
\tfrac{5}{20} and \tfrac{8}{20}. The product is 4 \times 5 = 20; \tfrac{1 \times 5}{4 \times 5} and \tfrac{2 \times 4}{5 \times 4}.
\tfrac{16}{24} and \tfrac{3}{24}. The product is 3 \times 8 = 24; \tfrac{2 \times 8}{3 \times 8} and \tfrac{1 \times 3}{8 \times 3}.
Product 40: \tfrac{12}{40} and \tfrac{10}{40}. Smaller, 20: \tfrac{6}{20} and \tfrac{5}{20}. The multiples of 10 are 10, 20, \ldots, and 20 = 4 \times 5.
\tfrac{21}{12} and \tfrac{2}{12}. The multiples of 6 are 6, 12, \ldots, and 12 = 4 \times 3. So \tfrac{7}{4} = \tfrac{7 \times 3}{4 \times 3} and \tfrac{1}{6} = \tfrac{1 \times 2}{6 \times 2}.
\tfrac{6}{9} and \tfrac{4}{9}. 9 \div 3 = 3, so \tfrac{2}{3} = \tfrac{2 \times 3}{3 \times 3}.
30: \tfrac{25}{30} and \tfrac{4}{30}. The multiples of 15 are 15, 30, and 30 = 6 \times 5. Multipliers 5 and 2.
20: \tfrac{15}{20} and \tfrac{2}{20}. The multiples of 10 are 10, 20, and 20 = 4 \times 5. Multipliers 5 and 2.
24: \tfrac{4}{24} and \tfrac{9}{24}. The multiples of 8 are 8, 16, 24, and 24 = 6 \times 4. Multipliers 4 and 3.
40: \tfrac{15}{40} and \tfrac{28}{40}. The multiples of 10 are 10, 20, 30, 40, and 40 = 8 \times 5. Multipliers 5 and 4.
12: \tfrac{4}{12}, \tfrac{9}{12} and \tfrac{2}{12}. The multiples of 6 are 6, 12; 6 is not a multiple of 4, and 12 is a multiple of 3, 4 and 6.
10: \tfrac{6}{10}, \tfrac{5}{10} and \tfrac{7}{10}. 10 is a multiple of 5 and of 2, so only the first two fractions change, by 2 and by 5.
30: \tfrac{15}{30}, \tfrac{18}{30} and \tfrac{20}{30}. The multiples of 5 are 5, 10, 15, 20, 25, 30, and 30 is the first that both 2 and 3 divide. Multipliers 15, 6 and 10.
Call the denominators b and d. The product b \times d is d groups of b, so it is on the list of multiples of b. By the commutative property it is also d \times b, which is b groups of d, so it is on the list of multiples of d too. A number on both lists is a common denominator: the first fraction is multiplied top and bottom by d, and the second by b.
108 is a common denominator, because it is 9 \times 12, but it is not the least. The multiples of 12 are 12, 24, 36, and 36 = 9 \times 4, so the least common denominator is 36: \tfrac{4}{9} = \tfrac{16}{36} and \tfrac{5}{12} = \tfrac{15}{36}.
Say the larger denominator is a multiple of the smaller one. Then the larger denominator is on both lists of multiples, so it is a common denominator. Nothing smaller can be: every common denominator is a multiple of the larger denominator, and the smallest multiple of a counting number is the number itself. So the larger denominator is the least.