Equivalent fractions and common denominators

Different fraction expressions can name one number

Three quarters and six eighths can describe the same selected length of a fixed bar. The parts have different sizes, and their counts differ, but 3/4 = 6/8. Such expressions are equivalent fractions: they have exactly the same numerical value. They mark the same point on a number line whose unit is fixed.

The equality sign concerns the whole value, not the separate sizes of the numerator and denominator. Six is larger than three, but sixths, eighths and quarters are different units of counting. To decide whether two fractions agree, consider both the number of parts and the size of each part.

Split every part without changing the amount

Start with a bar divided into four equal parts, with three selected. Split every quarter into two equal pieces, including the unselected quarter. The entire bar now has eight equal parts; the selected region has six. The whole and the selected amount stayed fixed, so 3/4 = 6/8. Splitting only selected pieces would leave unequal piece sizes and would not create eighths throughout the whole.

If each quarter is instead split into three equal pieces, the whole contains twelve parts and the selected region contains nine: 3/4 = 9/12. The smaller part size is balanced by a larger part count. On the number line, inserting more subdivision marks leaves the endpoint of the selected length in the same place.

Use the same positive whole-number factor

For an integer numerator a, a positive integer denominator b and a positive whole-number factor k, the rule is a/b = (a × k)/(b × k). For a nonnegative numerator, splitting each original part into k equal pieces turns b parts per whole into b × k parts, and a selected parts into a × k selected parts. For example, 2/5 = (2 × 4)/(5 × 4) = 8/20.

The quotient also stays fixed: both quantities in the division are scaled by the same factor. This explains why the rule applies to negative and improper fractions too: −2/5 = −8/20 and 7/4 = 14/8. The positive k keeps our denominators positive. Using k = 0 would produce a denominator of zero and is not allowed. Changing just one count generally changes the value; adding the same number to both counts does not provide an equivalence rule.

Group smaller parts into larger equal parts

The reverse process groups parts. In 12/18, arrange every set of six eighteenth-parts into one group. Eighteen parts make three such groups across the whole, and twelve selected parts make two groups. Therefore 12/18 = (12 ÷ 6)/(18 ÷ 6) = 2/3. The selected amount remains unchanged; its description uses fewer, larger parts.

To keep integer numerators and denominators, divide both by a positive integer that divides each exactly: a common factor. Dividing 12 and 18 by 2 gives 6/9, which can then be divided by 3 to give 2/3. Both routes preserve the value. Simplifying means dividing the complete numerator and denominator by a common factor; it does not mean deleting matching digits or subtracting the same amount from both.

Lowest terms leave no common factor greater than 1

A fraction is in lowest terms when its integer numerator and positive denominator share no positive factor greater than 1. The greatest common divisor, or GCD, gives the largest factor that divides both. Use the numerator’s absolute value when finding it. For 12/18, gcd(12, 18) = 6; dividing both by 6 gives 2/3 in one step. The numerator 2 and denominator 3 then have GCD 1.

Zero also has a simplest form: 0/7 = (0 ÷ 7)/(7 ÷ 7) = 0/1, because every positive integer divides zero. A negative sign is preserved, so −14/21 simplifies to −2/3. Simplifying an improper fraction need not make it proper: 10/6 becomes 5/3, still greater than 1. Keep the denominator positive, and never divide by zero. Lowest terms change the expression, not the number or its sign.

Give two fractions the same unit of counting

To compare 2/3 with 3/4, thirds and quarters are different-sized fractional units. A twelfth can measure both exactly: one third contains four twelfths, and one quarter contains three twelfths. Thus 2/3 = 8/12 and 3/4 = 9/12. Both now count parts of size 1/12 of the same numerical unit. The denominator 12 is a common denominator.

For positive denominators b and d, choose a positive common multiple D: a number divisible by both. To rewrite a/b with denominator D, multiply both a and b by D ÷ b. To rewrite c/d, multiply both c and d by D ÷ d. The multipliers can differ between fractions; within each fraction, its numerator and denominator must receive the same multiplier. The original values and whole-number unit stay fixed.

A working common denominator need not be the least one

The product b × d is always a common multiple of positive integer denominators b and d, so it always works. The least common multiple, or LCM, is the smallest positive common multiple of those denominators. It uses fewer subdivisions and often smaller numbers. For denominators 6 and 9, their positive multiples begin 6, 12, 18 and 9, 18, 27; their LCM is 18. The product 54 also works. If one denominator divides the other, the larger one is already their LCM, as with 4 and 8.

The connection with common factors is lcm(b, d) = (b × d) ÷ gcd(b, d). Since gcd(6, 9) = 3, their LCM is 54 ÷ 3 = 18. For the expressions 4/6 and 6/9, choosing 18 gives 12/18 and 12/18, while choosing 54 gives 36/54 and 36/54. Both comparisons establish equality. If we simplify the original fractions first, both become 2/3 and already share denominator 3. This is a different starting representation; 18 remains the LCM of the original denominators 6 and 9.

See original parts become a shared partition

Choose two fractions in the explorer, with integer numerators from −8 to 8 and positive denominators from 1 to 8. Each complete bar represents the same numerical unit, and the number line keeps its scale from −8 to 8. Begin with 2/3 and 3/4. Inspect their original parts, then their common-denominator expressions: 8/12 and 9/12. Match the selected amounts with the points on the number line. A larger number of finer subdivision marks must not change either fraction’s value.

Try 4/6 and 2/3, and switch the common-denominator choice between the LCM and the product. The counts and part sizes change, but the equality and number-line positions stay fixed. Then try −2/3 and −3/4, zero with different denominators, and 7/4 with 5/3. Each magnitude is described by its complete-whole count and its remaining shaded fractional part; the bars display only that remainder. The sign applies to the entire magnitude, and the number-line position shows its direction from zero. Describe which parts of the display changed and which numerical quantities stayed the same.

With a positive common denominator, compare the numerators

After 2/3 = 8/12 and 3/4 = 9/12, the parts are the same size. Eight twelfths are less than nine twelfths, so 2/3 < 3/4. Both values are between one half and one, which provides a useful scale check. Comparing the original numerators 2 and 3 alone would leave out their different part sizes.

The same method works beyond one. Rewrite 7/4 and 5/3 using twelfths: 7/4 = 21/12 and 5/3 = 20/12. Since 21 > 20 and the common denominator is positive, 7/4 > 5/3. Both fractions lie between 1 and 2; neither needs to be converted to a decimal or mixed number to make an exact comparison. Equal common numerators would instead establish equal values.

Keep the sign when counting the shared units

Rewrite −2/3 as −8/12 and −3/4 as −9/12. On the number line, −9 twelfth-unit steps lie farther left than −8 such steps, so −3/4 < −2/3. Their distances from zero are ordered the other way: 3/4 > 2/3. A greater magnitude for a negative value therefore does not mean that the signed number is larger.

With any positive common denominator, compare the signed numerators in their usual order: −9 < −8 < 0 < 8 < 9. A negative fraction is less than zero, and a positive fraction is greater than zero. A negative denominator can first be moved into the numerator’s sign, as 2/(−3) = −2/3; with two negative entries, (−2)/(−3) = 2/3. This keeps the shared fractional unit positive.

Cross-products are common-denominator reasoning written briefly

For fractions a/b and c/d with positive denominators, rewrite them as (a × d)/(b × d) and (c × b)/(d × b). Their common denominator is b × d. They are equal exactly when a × d = c × b: their numerators count the same shared fractional units. For example, 2/3 and 4/6 are equal because 2 × 6 = 4 × 3 = 12. The cross-products test follows from equivalent fractions; it is not an independent rule about arranging digits.

The equality test also works for any nonzero denominators: multiplying both values by the same nonzero number b × d gives a × d and c × b. For a size comparison, first make both denominators positive. Then b × d is positive, so a/b < c/d exactly when a × d < c × b. For 2/3 and 3/4, 2 × 4 = 8 < 9 = 3 × 3, confirming the twelfths comparison. A zero denominator invalidates the fraction and the test: equal products cannot make an undefined division valid.

Check the value, the unit and the purpose of the new form

Equivalent fractions are equal as numbers, but equal fractions of different physical wholes need not be equal amounts. Half of a 20 cm ribbon is 10 cm; two quarters of a 30 cm ribbon are 15 cm. Although 1/2 = 2/4, these pieces have different lengths. To compare the lengths, express them in the same measurement unit. To compare the fractions as numbers, use the same numerical unit from 0 to 1.

Choose the form that serves the question. Lowest terms give a compact expression; common denominators give a shared fractional unit for comparison. Check that each rewrite used the same factor in its numerator and denominator, preserved the sign, and kept the expected number-line position. These are the foundations for calculating with fractions: the next lesson will explain how shared units and fraction values guide addition, subtraction, multiplication and division.

Worked examples

Try it yourself

1. A fixed bar has three of five equal parts selected. Split every part into four equal pieces. Write the equivalent fraction and explain why changing 3/5 to 3/20 alone would be different.

   Show the answer and reasoning

   3/5 = 12/20. The expression 3/20 has a smaller value.

   Every fifth becomes four twentieths, so the five original parts become twenty parts and the three selected parts become twelve. Keeping only three selected twentieths would keep the count while making each selected part four times smaller; it would not preserve the selected amount.

2. Someone rewrites 3/4 as 4/5 by adding 1 to both entries. Is this valid? Compare using twentieths. Could multiplying both original entries by zero establish an equivalent fraction?

   Show the answer and reasoning

   No: 3/4 = 15/20 < 16/20 = 4/5. Multiplying both entries by zero is not allowed.

   Adding 1 did not scale the part count and part size in the required relationship. With denominator 20, the unequal values are clear. Multiplying both 3 and 4 by zero would give 0/0, which is undefined because division by zero is not defined.

3. Write 18/24, −14/21, 0/7 and 10/6 in lowest terms with positive denominators. State the common factor used in each case.

   Show the answer and reasoning

   3/4, −2/3, 0/1 and 5/3, using factors 6, 7, 7 and 2 respectively.

   Divide each complete numerator and denominator by its stated greatest common divisor. For the negative fraction, use the magnitude 14 when finding the common factor and preserve the minus sign. Zero divided by 7 is still zero. The simplified 5/3 remains an improper fraction, because simplifying preserves its value greater than 1.

4. Which of 15/18 and 16/18 equals 5/6? Explain with a same-factor rewrite or cross-products. Could a cross-products test establish equality for 1/0 and 2/0?

   Show the answer and reasoning

   15/18 equals 5/6; 16/18 does not. The expressions 1/0 and 2/0 are undefined, so the test cannot establish an equality between fractions.

   Multiplying both entries of 5/6 by 3 gives 15/18. Equivalently, 5 × 18 = 15 × 6 = 90, while 16 × 6 = 96. For the zero-denominator expressions, the products would both be zero, but the prerequisite of nonzero denominators fails.

5. Rewrite 5/6 and 7/9 with the least common multiple of their given denominators, then with the product of those denominators. Which fraction is larger, and must you use the smaller denominator?

   Show the answer and reasoning

   With 18: 15/18 and 14/18. With 54: 45/54 and 42/54. Thus 5/6 > 7/9; either common denominator works.

   The LCM of 6 and 9 is 18, so multiply the entries of the first fraction by 3 and of the second by 2. Their product is 54, requiring factors 9 and 6 respectively. Both positive shared units give the same order. The smaller denominator is efficient, not necessary for a correct comparison.

6. Compare 2/3 with 3/4, −2/3 with −3/4, and 7/4 with 5/3. Rewrite each pair using twelfths and explain the negative pair’s order.

   Show the answer and reasoning

   2/3 < 3/4; −2/3 > −3/4; 7/4 > 5/3. The paired numerators over 12 are 8 and 9, −8 and −9, and 21 and 20.

   With the same positive denominator, compare the signed numerators directly. Since −8 > −9, −2/3 lies to the right of −3/4. The greater distance of −3/4 from zero makes it more negative, not larger. The last pair remains valid even though both values exceed 1.

7. One piece is half of a 20 cm ribbon; another is two quarters of a 30 cm ribbon. Are 1/2 and 2/4 equivalent numbers? Are the pieces equal in length? What should remain fixed in a fraction-bar comparison?

   Show the answer and reasoning

   The fractions are equivalent, but the lengths are 10 cm and 15 cm. Comparable bars must use the same reference whole and the same length scale.

   1/2 = 2/4 describes the same fraction of a unit. Applying that fraction to different ribbon lengths gives different physical amounts. A drawing that rescales unequal ribbons to look identical can show their fractional shares, but cannot establish equal real lengths.

What to carry forward

• Equivalent fractions have exactly the same value and number-line position, even when their part sizes and counts differ.
• Multiply both entries by the same positive whole-number factor to refine a partition; divide both by a positive common factor to simplify. Never create a denominator of zero.
• Lowest terms leave no common factor greater than 1. Zero, negative and improper fractions follow the same value-preserving rules.
• A positive common denominator names a shared fractional unit. The product always works; the LCM of the chosen denominators often gives smaller counts.
• Compare signed numerators after making denominators the same and positive. Cross-products express the same reasoning, while physical amounts also require comparable reference wholes and measurement units.