A Practical Guide to Fractions

Arithmetic is the math of counting, sharing, and comparing amounts. It covers the four operations plus fractions, decimals, percents, and ratios.

A Practical Guide to Fractions: the central idea

A fraction can represent a number, ratio, quotient, or operator. Its denominator names the part size and its numerator counts those parts.

Words you need

Numerator

The numerator is the top number of a fraction, and it counts how many equal parts you have.

Denominator

The denominator is the bottom number of a fraction, and it names how many equal parts make one whole.

Equivalent fractions

Equivalent fractions are fractions that are written differently but name the same amount, such as $\tfrac24$ and $\tfrac12$.

Improper fraction

An improper fraction is a fraction whose numerator is at least as large as its denominator, such as $\tfrac{17}{12}$, so it is worth 1 or more.

Mixed number

A mixed number is a whole number written beside a proper fraction, such as $1\tfrac{5}{12}$, and it means the two parts added together.

Reciprocal

A reciprocal is the fraction you get by swapping the numerator and the denominator, so the reciprocal of $\tfrac34$ is $\tfrac43$ and the two multiply to 1.

What to know before this lesson

Understand equal parts, multiplication facts, common factors, and why division by zero is undefined.

If one of those prerequisites is uncertain, use the Arithmetic subject guide to locate the earlier concept before memorizing a procedure.

Fractions: a worked example

Every step, with the arithmetic

1. Step 1 - Write the problem$\tfrac23+\tfrac34$
2. Step 2 - List multiples to find the least common denominator3, 6, 9, 12 and 4, 8, 12, so the LCD is 12
3. Step 3 - Rename the first fraction$\tfrac23\times\tfrac44=\tfrac{8}{12}$
4. Step 4 - Rename the second fraction$\tfrac34\times\tfrac33=\tfrac{9}{12}$
5. Step 5 - Add the numerators and keep the denominator$\tfrac{8}{12}+\tfrac{9}{12}=\tfrac{17}{12}$
6. Step 6 - Rewrite the improper fraction as a mixed number$17\div12=1$ remainder 5, so $\tfrac{17}{12}=1\tfrac{5}{12}$
7. Step 7 - Estimate to check$\tfrac23$ and $\tfrac34$ are each more than $\tfrac12$, so a total a little above 1 is exactly what you expect

$1/3+1/4$ becomes $4/12+3/12=7/12$ because twelfths are the smallest convenient equal part size.

Fractions, decimals, and percents worth knowing by heart

The step-by-step method for fractions

1. For addition or subtraction, create equal-sized parts with a common denominator.
2. For multiplication, multiply numerators and denominators; for division, multiply by the reciprocal of a nonzero divisor.
3. Simplify with common factors and estimate against benchmarks such as $0$, $1/2$, and $1$.

How to check your answer

Convert $7/12$ to a decimal or compare it with $1/2=6/12$; the result should be slightly larger than one-half.

Use rounding, inverse operations, and benchmark fractions such as one-half or one-quarter to verify the sign and size of the answer.

A mistake that changes the mathematics

Numerators can be added only after denominators describe the same-sized parts. $1/3+1/4$ is not $2/7$.

Where you will actually use this

Cutting a recipe down

Halving a recipe that calls for $\tfrac34$ cup of milk means $\tfrac34\times\tfrac12=\tfrac38$ cup. Notice the answer is smaller, because multiplying by a number below 1 shrinks the amount instead of growing it.

Reading a tape measure or a wrench

Inches are split into sixteenths, so a mark at $\tfrac{6}{16}$ inch is the same as $\tfrac38$ inch. A $\tfrac12$-inch wrench is bigger than a $\tfrac{7}{16}$-inch wrench because $\tfrac12=\tfrac{8}{16}$ and 8 sixteenths beat 7.

Sharing food evenly

Five friends splitting three pizzas get $3\div5=\tfrac35$ of a pizza each. If each pizza is cut into 8 slices, that is $\tfrac35\times8=4.8$ slices per person, so roughly five slices each.

How fractions connects to the rest of arithmetic

• Percentages — A percent is a fraction whose denominator is 100, so $\tfrac34$, 0.75, and 75 percent are three spellings of one number.
• Ratios and proportions — A ratio of 3 to 4 is written as the fraction $\tfrac34$, and a proportion is simply the claim that two such fractions are equivalent.
• Order of operations — A fraction bar is a grouping symbol, so everything on top and everything on the bottom is finished before the division happens.

Try a transfer problem

A recipe needs $3/4$ cup per batch. Find the amount for $2\tfrac12$ batches and explain the units in the product.

Show the worked answer

Rewrite the mixed number as an improper fraction: $2\tfrac12=\tfrac52$. Then multiply: $\tfrac34\times\tfrac52=\tfrac{15}{8}$, which is $1\tfrac78$ cups. The units explain why multiplying is the right move: $\tfrac34$ is cups per batch and $\tfrac52$ is batches, so batches cancel and cups are left. Check it another way: two full batches take $\tfrac34+\tfrac34=\tfrac{12}{8}$ cups, half a batch takes $\tfrac38$ cup, and $\tfrac{12}{8}+\tfrac38=\tfrac{15}{8}$. In measuring cups that is 1 cup plus $\tfrac34$ cup plus $\tfrac18$ cup.

Once you have an answer, check it with the fraction calculator. Work the problem yourself first: the calculator confirms the arithmetic, but choosing the method is the part that transfers to the next question.

Work without copying the example. When finished, use the relevant focused calculator or formula reference to check the setup and result, then correct the first line where your reasoning changed. When the method feels reliable, move to arithmetic practice questions.

Questions about fractions

Why do I flip the second fraction when I divide?

Because dividing by $\tfrac34$ asks how many three-quarters fit inside the first number, and multiplying by $\tfrac43$ answers exactly that. It works because $\tfrac34\times\tfrac43=1$, so multiplying by the reciprocal undoes the division. For example $\tfrac12\div\tfrac34=\tfrac12\times\tfrac43=\tfrac46=\tfrac23$.

Do I have to use the least common denominator?

No, any common denominator works. Adding $\tfrac23+\tfrac34$ with a denominator of 24 gives $\tfrac{16}{24}+\tfrac{18}{24}=\tfrac{34}{24}$, which simplifies to $\tfrac{17}{12}$, the same answer. The least common denominator just keeps the numbers small and saves you a simplification step at the end.

What is the fastest way to compare two fractions?

Cross multiply and compare the two products. For $\tfrac58$ against $\tfrac{7}{11}$, compute $5\times11=55$ and $7\times8=56$. Since 56 is larger and it sits above the 11, the fraction $\tfrac{7}{11}$ is bigger. This is quicker than converting both to a common denominator of 88.

Can a fraction have zero on the bottom?

No. A fraction like $\tfrac50$ would ask what number times 0 gives 5, and nothing does, so it is undefined. Zero on top is perfectly fine: $\tfrac05=0$, because zero parts of anything is nothing.