Calculating with fractions

Let the question choose the operation

Joining two lengths suggests addition; finding the change between them suggests subtraction. Taking two thirds of a measured amount suggests multiplication. Asking how many half-litre portions fit into a given volume suggests division. Before working with the fraction symbols, say what the operation means in the question and name the quantity you want to find.

Keep the reference unit clear. To add 1/2 metre and 1/3 metre, both fractions refer to the same metre. Half of a 20 cm ribbon and one third of a 30 cm ribbon instead mean 10 cm and 10 cm: convert their lengths to a shared measurement unit before adding. Fractions as numbers have a fixed numerical unit; fractions of physical wholes also require the sizes of those wholes.

Add or subtract counts of equal-sized parts

Three eighths plus two eighths make five eighths: 3/8 + 2/8 = 5/8. We combined counts of parts of size 1/8; we did not change the size of those parts. For integer numerators a and c and a positive denominator b, a/b + c/b = (a + c)/b. The denominator stays b because the fractional unit stays 1/b. Adding the denominators would name a different unit and give a different number.

Subtraction uses the same shared unit: 5/8 − 2/8 = 3/8, and 2/8 − 5/8 = −3/8. A signed result can describe a change, a balance or a point on the number line. If a question concerns an ordinary remaining ribbon length, a negative answer means that removing the requested length exceeds the available ribbon. The subtraction is valid; check whether the physical action is possible. Subtracting a negative also follows the signed-number rule: 1/4 − (−1/2) = 1/4 + 1/2 = 3/4.

Rewrite different parts before combining them

Thirds and quarters are different-sized parts. To calculate 2/3 + 3/4, use twelfths: 2/3 = 8/12 and 3/4 = 9/12. Then 8/12 + 9/12 = 17/12 = 1 5/12. The sum can exceed one; the reference unit still stays fixed. For subtraction, 2/3 − 3/4 = 8/12 − 9/12 = −1/12. A common denominator makes the part sizes agree without changing either starting value.

Choose any positive common multiple of the denominators; the least common multiple often keeps the counts smaller. For positive b and d, the product b × d always works, giving a/b ± c/d = (a × d ± c × b)/(b × d). Within each starting fraction, scale both numerator and denominator by the same factor. Combine the resulting numerators, keep the common denominator, and simplify the complete result if it has a common factor.

A fraction can scale a quantity

Two thirds of nine litres means divide nine litres into three equal amounts, then take two: (2/3) × 9 L = 6 L. The fraction is a dimensionless scale factor, while the result keeps the volume unit. The same idea works when the starting amount is already fractional: half of three quarters of a litre is (1/2) × (3/4 L) = 3/8 L.

For a positive starting quantity, multiplying by a factor between zero and one reduces the amount; multiplying by a factor greater than one enlarges it. Multiplying by one preserves it, and multiplying by zero gives zero. Thus multiplication does not always make a number larger. For signed quantities, a negative factor also reverses the sign: (−2/3) × (1/2) = −1/3, while (−2/3) × (−1/2) = 1/3. These are the same sign rules used with integers.

An area model explains the product rule

Take a unit square with width one unit and height one unit. Split its width into thirds and its height into quarters. This creates twelve equal small rectangles. Select two thirds of the width and three quarters of the height; their overlap contains 2 × 3 = 6 small rectangles. The overlap occupies 6/12 = 1/2 of the unit square, so (2/3) × (3/4) = 1/2.

For nonnegative fractions no greater than one, partitions into b columns and d rows produce b × d equal cells; an overlap spanning a columns and c rows contains a × c cells. This explains (a/b) × (c/d) = (a × c)/(b × d). The rule also works for improper and signed fractions, using whole units and the signed multiplication rules. A bounded unit-square picture alone does not show a negative area. If the actual side lengths are 2/3 metre and 3/4 metre, their area is 1/2 square metre: length multiplied by length produces area, so the units also multiply.

Cancel common factors, with the whole expression visible

In (2/3) × (9/4), the product is (2 × 9)/(3 × 4). A factor 2 appears in the numerator and denominator, and so does a factor 3. Dividing both complete products by those common factors gives (1 × 3)/(1 × 2) = 3/2. You can simplify these factor pairs before multiplying because you are preserving the value of one overall fraction. Multiplying first gives 18/12, which simplifies to the same 3/2.

This reasoning applies to factors joined by multiplication. In (2 + 3)/3, the numerator is a sum, not a product with 3 as a factor. Deleting the two matching 3s would not preserve its value: (2 + 3)/3 = 5/3. If every term shares a factor, first show it explicitly: (6 + 9)/3 = 3 × (2 + 3)/3 = 5. Check that a proposed cancellation divides the entire numerator and the entire denominator by the same nonzero factor.

Division can ask how many groups an amount contains

How many half-litre portions are contained in three quarters of a litre? Express both volumes in quarters of a litre. A half-litre portion contains two quarters, and the available volume contains three quarters. The portion count is 3 ÷ 2 = 3/2, so (3/4) ÷ (1/2) = 3/2. This means one complete portion plus half of another portion, not three halves of a litre.

The question determines how to report the answer. If only completely filled half-litre bottles count, three quarters of a litre fills one bottle and leaves one quarter of a litre; it does not fill two whole bottles. Dividing a volume by a volume in the same unit gives a dimensionless group count. Division also expresses sharing: (3/4 L) ÷ 3 people = 1/4 L per person. Keep the divisor’s meaning visible so that a group count and an amount per person do not get confused.

A nonzero reciprocal reverses multiplication

The reciprocal of a nonzero fraction c/d is d/c, because (c/d) × (d/c) = 1. For example, multiplying by 1/2 is undone by multiplying by 2. Therefore dividing by 1/2 is multiplying by 2: (3/4) ÷ (1/2) = (3/4) × 2 = 3/2. Both the original denominator d and the divisor’s numerator c must be nonzero. A negative reciprocal keeps its sign; the reciprocal of −2/3 is −3/2.

Division finds the number which, multiplied by the divisor, gives the dividend. If q = (a/b) ÷ (c/d), then q × (c/d) = a/b. Multiplying both sides by the reciprocal d/c gives q = (a/b) × (d/c) = (a × d)/(b × c). This is why the divisor is inverted while the dividend is kept. Check the example by multiplying back: (3/2) × (1/2) = 3/4. For positive numbers, division by a factor less than one increases the result relative to the dividend, while division by a factor greater than one decreases it.

A mixed number joins a whole and a fractional remainder

The mixed number 1 2/3 means 1 + 2/3 = 5/3; the space does not mean multiplication. Converting to an improper fraction lets one fraction represent the complete number during an operation. For example, (1 2/3) × (3/4) = (5/3) × (3/4) = 5/4 = 1 1/4. You can also add whole parts and fractional parts separately, but any borrowed whole or carried whole must be expressed in the same fractional unit.

A leading minus sign applies to the entire mixed magnitude: −1 1/2 = −(1 + 1/2) = −3/2. It does not mean −1 + 1/2, which equals −1/2. When combining signed fractions, retain these whole values before choosing a common denominator or applying multiplication and division. For example, −1 1/2 + 2/3 = −3/2 + 2/3 = −9/6 + 4/6 = −5/6. Use parentheses whenever the scope of a sign could otherwise be unclear.

Change the operation while keeping the two numbers

Begin with 2/3 and 3/4 in the explorer. Select addition, subtraction, multiplication and division in turn. Addition and subtraction first give the operands a shared fractional unit; multiplication can show their overlap in a unit square; division asks what factor multiplying the second number gives the first. Explain why the same two operands give different answers under different operations. The symbols select different mathematical questions.

Choose the pair 3/4 and 1/2 and inspect the division result 3/2: it counts one complete group and half of another. Try the signed pair, the pair containing improper fractions, a zero dividend with a nonzero divisor, and a zero divisor. Track the complete-whole count, any fractional remainder and the sign of each displayed number. The tool uses selected examples to expose the reasoning; verify each result by rewriting equivalent fractions, applying an inverse operation or checking its expected size.

Keep zero restrictions and operation order explicit

For any defined fraction x, x + 0 = x, x − 0 = x, x × 0 = 0, x × 1 = x and x ÷ 1 = x. Also 0 ÷ x = 0 and x ÷ x = 1 when x is nonzero. Division by zero is undefined, including 0 ÷ 0. For a nonzero dividend, no number multiplied by zero returns that dividend; for a zero dividend, every number multiplied by zero returns zero, so there is no unique quotient. A fraction with denominator zero is already undefined before any further calculation.

Fraction expressions follow the usual operation order: parentheses first, then powers and roots, then multiplication and division from left to right, then addition and subtraction from left to right. Thus 1/2 + (2/3) × (3/4) = 1/2 + 1/2 = 1. Changing the grouping changes the question: (1/2 + 2/3) × (3/4) = (7/6) × (3/4) = 7/8. Read a fraction bar as grouping its complete numerator and denominator; (2 + 3)/4 means divide their sum by 4.

Check the value, then return to the quantity

Use an estimate before or after the exact calculation. Since 2/3 is between 1/2 and 1, adding 3/4 gives a sum between 1 1/4 and 1 3/4; 17/12 fits that interval. Subtracting the slightly larger 3/4 from 2/3 gives a small negative result, consistent with −1/12. Their positive product must be smaller than either factor, consistent with 1/2. Their quotient is less than one because 2/3 is smaller than the divisor 3/4; 8/9 fits.

Check subtraction by adding back, and check division by multiplying back. Confirm that an equivalent rewrite or simplification kept the value, that every denominator and divisor used was allowed, and that the final measurement unit or group label answers the original question. Exact fractions can remain the final answer; a decimal conversion is not required for these checks. The next lesson connects fractions with decimal place value and explains what terminating and recurring decimal forms mean.

Worked examples

Try it yourself

1. Calculate 3/8 + 2/8, 2/8 − 5/8 and 1/4 − (−1/2). Explain why the first answer does not have denominator 16.

   Show the answer and reasoning

   5/8, −3/8 and 3/4. The first sum still counts eighths.

   Addition combines three eighth-units with two eighth-units; it does not halve their size to make sixteenths. Subtraction gives a signed difference of −3 eighth-units. Subtracting −1/2 is adding 1/2 = 2/4, so 1/4 + 2/4 = 3/4.

2. Calculate 5/6 + 3/4 and 1/3 − 3/4 using twelfths. Give a size or sign check for each answer.

   Show the answer and reasoning

   5/6 + 3/4 = 19/12 = 1 7/12; 1/3 − 3/4 = −5/12.

   Rewrite the first pair as 10/12 and 9/12, then add the counts. Both starting values exceed 1/2 and are less than 1, so their sum should be between 1 and 2. For the second pair, 1/3 = 4/12 and 3/4 = 9/12, giving −5/12. Subtracting a larger positive value from a smaller one must give a negative result.

3. A rectangle has sides 2/3 metre and 3/4 metre. Find its area and describe a twelve-cell model. Separately, find 3/2 of 2/3 litre and explain whether multiplying always makes an amount smaller.

   Show the answer and reasoning

   The area is 1/2 square metre, represented by six of twelve equal cells in a square-metre model. The scaled volume is 1 litre; multiplication can enlarge or reduce a positive amount.

   A unit square divided into three columns and four rows has twelve cells. The two-column, three-row overlap has six, so the product is 6/12 = 1/2 and the units are square metres. For volume, (3/2) × (2/3 L) = 1 L. The factor 3/2 is greater than one, so it enlarges the starting 2/3 litre; a positive factor between zero and one would reduce it.

4. Simplify (2/3) × (9/4). Can a matching 3 be cancelled directly in (2 + 3)/3? Explain how (6 + 9)/3 can instead be simplified by factoring.

   Show the answer and reasoning

   The product is 3/2. Direct cancellation in (2 + 3)/3 is invalid; its value is 5/3. Factoring gives (6 + 9)/3 = 3 × (2 + 3)/3 = 5.

   In (2 × 9)/(3 × 4), dividing the complete numerator and denominator by common factors preserves their quotient. A single term of a sum is not the complete numerator, so deleting it does not perform that same division. In the last expression, both numerator terms contain a factor 3; writing their sum as 3 × (2 + 3) makes the common factor explicit.

5. Calculate (5/6) ÷ (2/3) and check by multiplying back. Then classify 0 ÷ (2/3), (2/3) ÷ 0 and 0 ÷ 0 as a value or an undefined expression.

   Show the answer and reasoning

   (5/6) ÷ (2/3) = 5/4. The zero cases are 0, undefined and undefined, respectively.

   The nonzero divisor 2/3 has reciprocal 3/2, so (5/6) × (3/2) = 15/12 = 5/4. Multiplying back gives (5/4) × (2/3) = 10/12 = 5/6. Zero divided by a nonzero number is zero. Division by zero has no unique quotient: no number times zero gives 2/3, while every number times zero gives zero.

6. Calculate −1 1/2 + 2/3. Then compare 1/2 + (2/3) × (3/4) with (1/2 + 2/3) × (3/4). Explain the roles of the minus sign and parentheses.

   Show the answer and reasoning

   −1 1/2 + 2/3 = −5/6. The two grouped expressions are 1 and 7/8, respectively.

   The negative mixed number is −(1 + 1/2) = −3/2, giving −9/6 + 4/6 = −5/6. In the expression without an outer sum grouping, multiply first: (2/3) × (3/4) = 1/2, then add 1/2. In the other expression, calculate the parenthesised sum first: 1/2 + 2/3 = 7/6, then multiply by 3/4 to get 7/8.

7. One piece is half of a 20 cm ribbon; another is one third of a 30 cm ribbon. Find their combined length. Does adding 1/2 + 1/3 = 5/6 directly establish their combined physical length? What does 3/4 litre divided by 1/2 litre count?

   Show the answer and reasoning

   The pieces total 20 cm. The numerical sum 5/6 alone does not establish that length because the reference ribbons differ. The volume division counts 3/2 half-litre portion equivalents.

   Half of 20 cm is 10 cm, and one third of 30 cm is also 10 cm; add these lengths in centimetres. The fractions 1/2 and 1/3 alone do not specify lengths: each reference whole must also be known. Here their different fractions and different reference lengths happen to give equal pieces. Dividing volumes measured in the same unit cancels that unit and gives a group count, which may include a fraction of a group.

What to carry forward

• Addition and subtraction combine signed counts of the same fractional unit; rewrite unlike denominators using equivalent fractions first.
• Multiplication scales a quantity. An area partition explains why numerators multiply and denominators multiply; physical units also follow the operation.
• Division by a nonzero fraction is multiplication by its reciprocal. Explain whether the result counts groups or gives an amount per group, and multiply back to check.
• Simplify common factors of a complete numerator and denominator. A mixed number is a whole plus a fraction, with a leading minus sign applying to its entire magnitude.
• Keep operation order, reference wholes and measurement units clear. Zero divided by a nonzero value is zero; a zero denominator or divisor makes division undefined.