Decimals ↔ Fractions: Tenths, Hundredths, and Number Lines...

SECTION 1

The Story of Parts and Pieces

Long ago, people needed to talk about parts of things. If you had half an apple, how would you write it down? If you wanted to share a dollar among 10 friends, how much would each person get? Different cultures around the world came up with different ways to solve this problem.

3000 BCE

Ancient Fractions

The Egyptians used fractions to divide bread and measure land. They wrote fractions like 1/2 and 1/3 using special symbols.

1200 CE

Decimal Numbers

Arabian mathematicians started using decimal points to show parts of a whole. This made it easier to do math with money and measurements.

1585 CE

Modern Decimals

A Belgian mathematician named Simon Stevin wrote the first book explaining how to use decimal points the way we do today.

Today

Everywhere Around Us

We use decimals and fractions every day - in money ($1.25), sports (batting averages), cooking (3/4 cup), and measuring (2.5 inches).

Both fractions and decimals are ways to show the same thing - parts of a whole. Learning how they connect helps us understand numbers better and makes math easier!

SECTION 2

What Are Decimals and Fractions?

Fractions Show Parts

A fraction like 3/10 means we have 3 parts out of 10 equal pieces. The bottom number tells us how many pieces the whole is split into.

Decimals Use Place Value

A decimal like 0.3 uses places after the decimal point. The first place is tenths, the second is hundredths, just like we count by tens in whole numbers.

They're the Same Thing!

3/10 and 0.3 are exactly the same amount! They're just two different ways to write the same number - like saying 'three-tenths' or 'zero point three.'

Tenths and Hundredths

Tenths split something into 10 pieces (like 7/10 = 0.7). Hundredths split into 100 pieces (like 25/100 = 0.25). More pieces means smaller parts!

✦ KEY TAKEAWAY

Think of fractions and decimals like two languages describing the same pizza slice. If you eat 3 out of 10 slices, you can say "I ate 3/10 of the pizza" or "I ate 0.3 of the pizza" - both mean exactly the same delicious amount!

SECTION 3

Seeing Tenths and Hundredths

This diagram shows how 7/10 equals 0.7 and 25/100 equals 0.25. The number line shows where these decimals live between 0 and 1.

Look at how the tenths rectangle is split into 10 equal pieces. When 7 pieces are colored, that's 7/10 or 0.7. The hundredths square has 100 tiny squares. When 25 are colored, that's 25/100 or 0.25. On the number line, you can see exactly where these numbers fit between 0 and 1.

SECTION 4

How to Change Between Fractions and Decimals

FRACTION TO DECIMAL

top number ÷ bottom number = Decimal

To change a fraction to a decimal, divide the top number (numerator) by the bottom number (denominator). For tenths and hundredths, this is easy because they use 10 and 100!

TENTHS PATTERN

4/10 = 4 ÷ 10 = 0.4

With tenths, the top number becomes the first digit after the decimal point. So 3/10 = 0.3, 8/10 = 0.8, and so on.

HUNDREDTHS PATTERN

37/100 = 37 ÷ 100 = 0.37

With hundredths, the top number becomes the first two digits after the decimal point. So 15/100 = 0.15, 62/100 = 0.62.

DECIMAL TO FRACTION

0.6 = 6/10, 0.43 = 43/100

To change a decimal to a fraction, look at how many places after the decimal point. One place = tenths, two places = hundredths. The digits become the top number!

SECTION 5

Understanding Decimal Place Value

This place value chart shows how each position after the decimal point has a special meaning. The tenths place is highlighted in yellow, and the hundredths place in purple.

The decimal point is like a fence that separates whole numbers from parts of numbers. Everything to the left of the decimal point is a whole number (like 1, 2, 10, 100). Everything to the right shows parts that are smaller than 1.

Just like whole number places get 10 times bigger as you move left (ones, tens, hundreds), decimal places get 10 times smaller as you move right. The first place after the decimal point is tenths (1/10), and the second place is hundredths (1/100).

SECTION 6

Converting Step by Step

Converting 0.75 to a fraction

Step 1 — Count decimal placesLook at 0.75. There are 2 digits after the decimal point (7 and 5). This tells us we're working with hundredths.

2 places = hundredths

Step 2 — Write as fractionSince we have hundredths, the bottom number (denominator) is 100. The digits 75 become the top number (numerator).

0.75 = 75/100

Step 3 — Simplify if possibleBoth 75 and 100 can be divided by 25. So we divide the top and bottom by 25 to make the fraction simpler.

75 ÷ 25 = 3, 100 ÷ 25 = 4

Step 4 — Final answerThe simplest form of our fraction is 3/4. We can check: 3 ÷ 4 = 0.75 ✓

0.75 = 3/4

SECTION 7

Quick Conversion Strategies

Quick reference for converting between decimals and fractions

Conversion Type	Strategy	Example
Decimal → Fraction	Count decimal places. Use 10 for 1 place, 100 for 2 places	0.8 = 8/10, 0.35 = 35/100
Fraction → Decimal	Divide top by bottom. For tenths/hundredths, move the decimal point	6/10 = 0.6, 23/100 = 0.23
Finding on Number Line	Tenths divide 0-1 into 10 parts. Hundredths make 100 tiny parts	0.3 is 3 jumps from 0, 0.27 is 27 tiny jumps

💡 KEY TAKEAWAY

Think of decimals as a shorthand way to write fractions! Just like you might write "u" instead of "you" in a text message, 0.5 is the quick way to write 1/2, and 0.25 is the shortcut for 1/4. Both ways mean exactly the same thing!

SECTION 8

Looking Ahead: More Complex Decimals

What We Know Now	What's Coming Next
Tenths and hundredths (0.7, 0.35)	Thousandths and beyond (0.375, 0.1234)
Simple fractions like 1/2, 3/4	Any fraction like 2/3, 5/8, 7/9
Decimals that end (0.5, 0.25)	Decimals that repeat (0.333..., 0.142857...)
Number line from 0 to 1	Any range of numbers, negative decimals

The skills you're learning now with tenths and hundredths are the building blocks for understanding all decimals and fractions. Soon you'll learn about thousandths (1/1000), how to convert any fraction to a decimal, and even work with decimals bigger than 1. Every mathematician started exactly where you are right now!

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

Look at the decimal 0.6. Explain in your own words what this number means and draw a simple picture to show it.

PROBLEM 2 — BASIC CALCULATION

Convert these decimals to fractions: a) 0.4 b) 0.17 c) 0.09

PROBLEM 3 — INTERMEDIATE

Convert these fractions to decimals and plot them on a number line from 0 to 1: 3/10, 45/100, 1/4

PROBLEM 4 — APPLIED

Maya measured 0.75 cups of flour for a recipe, but her measuring cup only shows fractions. What fraction marking should she look for?

PROBLEM 5 — CRITICAL THINKING

Emma says "0.5 is bigger than 0.25 because 5 is bigger than 25." Explain why Emma is wrong and describe the correct way to compare these decimals.

SUMMARY

Bringing It All Together

Decimals and fractions are two different ways to write the same parts of a whole. Tenths divide something into 10 equal pieces, and hundredths divide it into 100 equal pieces. The decimal point separates whole numbers from parts of numbers, and each place after the decimal point has a special meaning.

To convert from decimal to fraction, count the decimal places to know your denominator (10 for tenths, 100 for hundredths). To go from fraction to decimal, divide the top number by the bottom number. Number lines help us see exactly where these numbers live between 0 and 1, making it easier to compare and understand them.