Fractions, Decimals & Rational Numbers by Grade

What are fractions, decimals, and rational numbers? Half of a pizza can be written as 1/2, 0.5, or 50%; the notation changes, but the amount does not. That single idea connects years of school math—from third-grade fractions to eighth-grade rational and irrational numbers. For children, the difficulty is often not arithmetic. It is knowing when each idea is supposed to make sense. A child may know that 1/2 equals 0.5 but still struggle to explain why.

Based on official standards, WuKong Education has compiled a Grade 3–8 Common Core roadmap, step-by-step methods for converting fractions to decimals and decimals to fractions, side-by-side fraction operations, repeating decimals, rational numbers, and grade-based practice.

If conversions feel confusing, start with place value. “Zero point eight” is often a language problem before it is a math problem: 0.8 means eight tenths.

By WuKong Math

Common Core references in this guide are based on the official standards. Verify standards at thecorestandards.org.

Why Fractions, Decimals, and Percents Are the Same Numbers

Fractions, decimals, and percents are not three unrelated topics.

They are three ways to describe the same quantity.

Think about half of a pizza:

1/2 = 0.5 = 50%

All three expressions locate the same point on a number line.

You can think of them as three mathematical languages:

This is why conversion should not be taught as a collection of tricks.

A child who understands equivalent quantities can reason:

1/2 = 5/10 = 0.5

and:

0.5 = 50/100 = 50%

The notation changes because different forms are useful in different situations.

You may use:

• fractions in recipes;
• decimals in money and measurement;
• percents for discounts, test scores, and statistics.

The key question is not:

“Which form is correct?”

It is:

“Which form is most useful right now?”

That idea starts with fractions as numbers in Grade 3 and develops into rational-number reasoning through middle school. Common Core Grade 3 explicitly treats fractions as numbers that can be located on a number line.

The Common Core Roadmap: Grades 3–8

Parents often search for a single fractions-to-decimals rule, but the skill is built across several years.

A student struggling in Grade 6 may actually have a missing Grade 4 place-value connection or a Grade 5 fraction-operation gap.

The table below shows how the ideas develop.

Grade 3 — Fractions Become Numbers | 3.NF.A.2

Long-tail focus: Grade 3 fractions on a number line

In Grade 3, the most important conceptual shift is simple:

A fraction is a number.

Students represent fractions such as 1/4, 2/4, and 3/4 on a number line rather than seeing fractions only as shaded parts of circles or rectangles.

Under 3.NF.A.2, students locate unit fractions and general fractions on number-line diagrams.

A child who understands:

3/4

as a point between 0 and 1 is better prepared later to understand that:

3/4 = 0.75

Parent tip: Before teaching fraction-decimal conversion, check whether your child can place common fractions correctly on a number line.

Grade 4 — Decimals as Fractions | 4.NF.C.6–7

Long-tail focus: Grade 4 fractions to decimals worksheet

Grade 4 is where the fraction-decimal bridge becomes explicit.

Under 4.NF.C.6, students use decimal notation for fractions with denominators 10 or 100. For example:

62/100 = 0.62

Under 4.NF.C.7, students compare decimals to hundredths by reasoning about their size.

This is the right time to connect spoken place-value language:

0.8 = eight tenths = 8/10 = 4/5

and:

0.08 = eight hundredths = 8/100 = 2/25

The difference between 0.8 and 0.08 is not a small detail. It is place value.

Parent tip: Ask your child to read the decimal using place-value words before converting it.

Grade 5 — Subtracting, Multiplying, and Dividing Fractions

Long-tail focus: Grade 5 subtracting fractions, multiplying fractions, dividing unit fractions

Grade 5 is the major fraction-operations year.

Students add and subtract fractions with unlike denominators under 5.NF.A.1. They interpret a fraction as division under 5.NF.B.3, multiply a fraction by another fraction under 5.NF.B.4, and divide unit fractions by whole numbers or whole numbers by unit fractions under 5.NF.B.7.

They also read, write, and compare decimals through thousandths under 5.NBT.A.3.

Subtracting Fractions: 1/2 − 1/4

Step 1: Find a common denominator.

1/2 = 2/4

Step 2: Subtract the numerators.

2/4 - 1/4 = 1/4

Step 3: Simplify if needed.

1/4 is already simplified.

Memory hook:

Same-sized pieces first.

You cannot directly subtract halves and fourths until both fractions describe pieces of the same size.

Multiplying Fractions: 1/2 × 1/4

Step 1: Multiply numerators.

1 × 1 = 1

Step 2: Multiply denominators.

2 × 4 = 8

Step 3: Simplify.

1/8

So:

1/2 × 1/4 = 1/8

Memory hook:

Multiply straight across.

Conceptually, this means “one-half of one-fourth.”

Dividing Fractions: 1/2 ÷ 1/4

A common classroom memory device is:

Keep – Change – Flip

Keep the first fraction:

1/2

Change division to multiplication:

÷ → ×

Flip the second fraction:

1/4 → 4/1

Then:

1/2 × 4/1 = 4/2 = 2

So:

1/2 ÷ 1/4 = 2

But the meaning matters more than the rhyme.

The question asks:

How many one-fourth fits inside one-half?

The answer is 2.

This meaning helps children remember why the reciprocal method works instead of treating it as magic.

Parent tip: If fraction operations are weak, fix them before expecting rational-number fluency later.

Grade 6 — Dividing Fractions, Percents, and Negative Numbers

Long-tail focus: Grade 6 dividing fractions and fraction-decimal-percent conversion

Under 6.NS.A.1, students divide fractions by fractions.

For example:

2/3 ÷ 4/5

becomes:

2/3 × 5/4 = 10/12 = 5/6

Grade 6 also introduces percent reasoning through 6.RP.A.3.c, where students understand a percent as a rate per 100.

So:

0.25 = 25/100 = 25% = 1/4

becomes one connected idea.

Negative rational numbers also become explicit on the number line in Grade 6 through 6.NS.C.6.

Students begin locating values such as:

-2.5

-3/4

and:

1.2

on the same number line.

Parent tip: Grade 6 is where separate elementary topics begin merging into one rational-number system.

Grade 7 — Rational Number Operations and Long Division

Long-tail focus: Grade 7 rational number conversion

Grade 7 extends all four operations to signed rational numbers.

Students learn to add, subtract, multiply, and divide positive and negative rational numbers under 7.NS.A.1–2.

They also use long division to convert rational numbers to decimal form under 7.NS.A.2.d.

The standard emphasizes that the decimal form of a rational number either:

• terminates, or
• eventually repeats.

Examples:

3/8 = 0.375

1/3 = 0.333...

2/11 = 0.181818...

A useful Grade 7 extension is learning how repeating patterns can later be converted back into fractions.

By Grade 7, students operate with both positive and negative rational numbers.

A rational number can be:

• positive;
• negative;
• zero.

Examples:

3/4

-3/4

2.5

-2.5

0

All are rational.

For multiplication and division, sign rules are especially important.

A useful memory pattern is:

Same signs → positiveDifferent signs → negative

Examples:

(-3/4) × (2/5) = -6/20 = -3/10

(-2/3) ÷ (-4/5)

Keep–Change–Flip:

(-2/3) × (-5/4)

= 10/12

= 5/6

The result is positive because both numbers were negative.

Grade 7 Common Core extends multiplication and division rules to rational numbers and explicitly includes signed-number products and quotients.

Parent tip: If your child thinks 0.333... is only “approximately” 1/3, this is a conceptual gap worth fixing.

Grade 8 — Rational vs. Irrational Numbers | 8.NS.A.1

Long-tail focus: rational vs irrational numbers Grade 8

Grade 8 broadens the number system.

Under 8.NS.A.1, students learn that numbers that cannot be written as ratios of integers are irrational. The standard also connects rational numbers with decimal expansions that terminate or eventually repeat and includes converting repeating decimal expansions into rational numbers.

Examples of rational numbers:

3/4

-2

0.125

0.333...

Examples of irrational numbers:

√2

π

A rational decimal either ends or repeats.

An irrational decimal continues without a repeating pattern.

Repeating Decimals to Fractions

Students often first encounter repeating decimals before they formally learn how to convert them back into fractions.

For example:

1/3 = 0.333...

The dots mean the digit continues forever.

In Grade 7, Common Core explicitly expects students to use long division to convert rational numbers to decimals and recognize that the result terminates or eventually repeats.

The reverse direction—showing that an eventually repeating decimal can be converted into a rational number—is made explicit in Grade 8 under 8.NS.A.1.

Example: Convert 0.363636… to a Fraction

Let:

x = 0.363636...

The repeating block has two digits, so multiply by 100:

100x = 36.363636...

Now subtract the original equation:

100x - x = 36.363636... - 0.363636...

The repeating parts cancel:

99x = 36

Divide by 99:

x = 36/99

Simplify by 9:

x = 4/11

Therefore:

0.363636... = 4/11

Check the Answer

Divide:

4 ÷ 11 = 0.363636...

The repeating pattern returns exactly.

So the fraction is correct.

Why Multiply by 100?

Because the repeating block has two digits: 36.

For a one-digit repeating block, multiply by 10.

Example:

x = 0.777...

10x = 7.777...

Subtract:

9x = 7

Therefore:

x = 7/9

For a three-digit repeating block, you would typically multiply by 1000.

The goal is always the same:

Move one full repeating block to the left of the decimal so subtraction cancels the infinite repeating part.

Parent tip: By Grade 8, the goal is no longer just conversion. Students need to understand what kind of number they are looking at.

Common Mistakes

1. Thinking 3/8 Means 0.38

It does not.

A fraction bar means division, not a decimal point.

3/8 = 3 ÷ 8 = 0.375

not:

0.38

2. Forgetting to Simplify

Example:

0.75 = 75/100

This is correct, but not simplified.

Divide numerator and denominator by 25:

75/100 = 3/4

The simplest form is:

3/4

3. Mixing Up Conversion Directions

For fraction → decimal:

Divide numerator by denominator.

For decimal → fraction:

Use place value, write the number over 10, 100, 1000, and simplify.

If a student keeps reversing the processes, write these two directions side by side.

4. Writing 0.08 as 8/10

0.08

means:

eight hundredths

So:

0.08 = 8/100 = 2/25

By contrast:

0.8 = 8/10 = 4/5

The zero after the decimal matters because it changes the place value.

5. Identifying the Wrong Repeating Block

Consider:

0.272727...

The repeating block is:

27

not just 7.

That means an algebraic conversion should shift two digits:

100x

rather than:

10x

Correctly identifying the repeating block is the first step.

Practice by Grade

Use these problems as a quick diagnostic rather than a speed test.

Grade 4 Practice — Fractions to Decimals

Standards focus: 4.NF.C.6–7 Long-tail: grade 4 fractions to decimals worksheet

1. Write 7/10 as a decimal.
2. Write 43/100 as a decimal.
3. Write 0.6 as a fraction with denominator 10.
4. Which is greater: 0.58 or 0.6?

Answers

1. 0.7
2. 0.43
3. 6/10, which simplifies to 3/5
4. 0.6, because 0.60 > 0.58

Grade 5 Practice — Fraction Operations

Standards focus: 5.NF.A.1, 5.NF.B.3–4, 5.NF.B.7 Long-tail: Grade 5 subtracting and multiplying fractions worksheet

1. 3/4 - 1/6
2. 2/3 × 3/5
3. Write 7/8 as a division expression.
4. Convert 0.625 to a fraction.

Answers

1. Common denominator 12: 9/12 - 2/12 = 7/12
2. 6/15 = 2/5
3. 7 ÷ 8
4. 625/1000 = 5/8

Check Question 4:

5 ÷ 8 = 0.625

Grade 6 Practice — Dividing Fractions and Percents

Standards focus: 6.NS.A.1, 6.RP.A.3.c Long-tail: Grade 6 dividing fractions worksheet

1. 3/4 ÷ 2/5
2. Write 35% as a fraction in simplest form.
3. Write 0.45 as a percent.
4. Find 20% of 60.

Answers

1. 3/4 × 5/2 = 15/8 = 1 7/8
2. 35/100 = 7/20
3. 45%
4. 12

Grade 7 Practice — Rational Numbers and Repeating Decimals

Standards focus: 7.NS.A.1–2, 7.NS.A.2.d Long-tail: Grade 7 rational numbers conversion worksheet

1. Convert 5/8 to a decimal.
2. Does 2/9 terminate or repeat?
3. (-3/5) × (10/9)
4. (-4/7) ÷ (2/3)
5. Convert 0.454545... to a fraction as an extension problem.

Answers

1. 0.625
2. Repeating: 0.222...
3. -30/45 = -2/3
4. (-4/7) × (3/2) = -12/14 = -6/7
5. Let x = 0.454545...; then 100x - x = 45, so 99x = 45, giving x = 45/99 = 5/11

Check:

5 ÷ 11 = 0.454545...

Free worksheet PDF: Download the Fractions, Decimals & Rational Numbers practice pack organized by grade level.

FAQs

Q1: Is a repeating decimal a rational number?

Yes.

A decimal that terminates or eventually repeats represents a rational number. Grade 7 connects rational numbers to terminating or repeating decimal expansions through long division, while Grade 8 explicitly develops the rational-versus-irrational distinction.

Q2: What is the difference between terminating and repeating decimals?

A terminating decimal ends:

0.25

A repeating decimal continues forever in a repeating pattern:

0.333...

Both are rational numbers.

Q3: Why do we convert between fractions and decimals?

Different forms are useful in different situations.

Fractions often show exact relationships clearly. Decimals work well for money, measurement, and calculators. Percents are useful when comparing values out of 100.

Demi | WuKong Math Teacher

I am an educator from Yale University with ten years of experience in mathematics education, including extensive hands-on experience preparing students for the SAT Math section. I believe that my professional expertise and refined teaching approach will allow me to make a meaningful contribution to the growth of Wukong Education. Within this community, I look forward to sharing insights on children’s educational psychology and effective learning strategies, with the hope of providing quality learning resources that help more children grow into confident, capable learners.