Perform operations with whole numbers, fractions, and decimals - TACHS Math

SECTION 1

Where Did These Number Types Come From?

People have used numbers for thousands of years. At first, we only needed whole numbers (like 1, 2, 3) to count things such as sheep or bags of grain. But soon people needed to share things equally. What if two farmers had to split three loaves of bread? That is where fractions came in.

Later, traders and scientists wanted a faster way to write fractions. They invented decimals (numbers with a dot, like 3.5). Today you use all three kinds every day—prices at a store, recipes in a kitchen, and scores on a test.

3000 BCE • Ancient Egypt Uses Fractions

Egyptian scribes wrote fractions to divide food, land, and building materials fairly.

300 BCE • Greek Mathematicians Study Ratios

Euclid and other Greek thinkers explored how numbers relate to each other through ratios and proportions.

1585 CE • Decimals Are Introduced in Europe

Simon Stevin published a book showing merchants how to use decimal notation instead of complicated fractions.

Today • All Three Number Types on the TACHS

State tests like the TACHS ask you to add, subtract, multiply, and divide whole numbers, fractions, and decimals confidently.

The big question this lesson answers is: How do you add, subtract, multiply, and divide whole numbers, fractions, and decimals—and how do these operations connect to each other?

SECTION 2

Core Principles & Definitions

Before you dive into calculations, you need to know a few key ideas. These four principles apply no matter which type of number you are working with.

Place Value Matters

Every digit in a number has a value based on its position. In 345, the 3 means 300. In 3.45, the 4 means four tenths. Lining up place values is the secret to adding and subtracting correctly.

Common Denominators for Fractions

You can only add or subtract fractions when they share the same denominator (bottom number). Think of it like making sure puzzle pieces are the same size before you combine them.

Multiply Straight Across

When you multiply fractions, you multiply the tops together and the bottoms together. No common denominator is needed! For decimals, ignore the dots, multiply, then count decimal places.

Dividing Means Flipping (for Fractions)

Dividing by a fraction is the same as multiplying by its reciprocal (the fraction flipped upside down). For decimals, move the decimal point to turn the divisor into a whole number first.

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KEY TAKEAWAY

Think of whole numbers, fractions, and decimals as three different languages for the same idea. 1/2, 0.5, and "half of 1" all mean the same thing. Learning to switch between these forms is like being able to translate between languages—it lets you pick the easiest way to solve any problem.

SECTION 3

Seeing the Number System

The diagram below shows the same value expressed as a whole number with fractions and as a decimal. It also shows how the four operations (add, subtract, multiply, divide) connect to each other.

The top row shows that 3/4 and 0.75 represent the same value. The bottom row shows the four operations and how they pair up as inverses (opposites). Knowing these pairs helps you check your work.

Notice that addition and subtraction are inverse operations—one undoes the other. The same is true for multiplication and division. If 4 × 5 = 20, then 20 ÷ 5 = 4. Keeping these pairs in mind lets you check answers quickly.

SECTION 4

Rules and Formulas for Each Operation

Each operation has simple rules you can follow. Let's look at the most important ones for fractions and decimals. Whole-number operations follow the same patterns you have used since elementary school.

ADDING FRACTIONS

a/c + b/c = (a + b)/c

When the denominators (bottom numbers) are the same, just add the numerators (top numbers). If the denominators differ, find the least common denominator (LCD) first.

SUBTRACTING FRACTIONS

a/c − b/c = (a − b)/c

Same rule as addition—keep the common denominator, subtract the numerators.

MULTIPLYING FRACTIONS

(a/b) × (c/d) = (a × c) / (b × d)

Multiply the numerators together and the denominators together. No common denominator needed! Simplify the result if possible.

DIVIDING FRACTIONS

(a/b) ÷ (c/d) = (a/b) × (d/c)

Keep the first fraction, change ÷ to ×, and flip the second fraction (use its reciprocal). Then multiply straight across.

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Decimal Tip

When you add or subtract decimals, line up the decimal points vertically. When you multiply, count the total decimal places in both numbers and put that many places in your answer. For example, 1.2 × 0.3 → 12 × 3 = 36 → two decimal places → 0.36.

SECTION 5

Detailed Breakdown by Number Type

The diagram below shows a step-by-step decision tree. When you see a problem on the TACHS, ask yourself two questions: What type of numbers am I working with? and Which operation do I need? Then follow the matching path.

Use this decision tree on test day. First identify the number type, then follow the steps for your operation. Remember, you can always convert between fractions and decimals if one form feels easier.

Operation	Whole Numbers	Fractions	Decimals
Add / Subtract	Line up ones, tens, hundreds…	Find the LCD, then add/subtract numerators	Line up the decimal points, add/subtract
Multiply	Standard algorithm (or area model)	Multiply numerators, multiply denominators	Multiply as whole numbers, count total decimal places
Divide	Long division	Keep-Change-Flip, then multiply	Move decimal to make divisor whole, then divide

Quick-reference table of rules for every operation and number type.

SECTION 6

Worked Example: A Multi-Step Problem

Let's work through a problem that mixes fractions and decimals. This is the kind of multi-step question you might see on the TACHS.

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Problem

Maria has 2.5 pounds of flour. She uses 3/4 of a pound for a cake. How much flour does she have left? Give your answer as a decimal.

Solution

1
Step 1 — Identify What You Know

Maria starts with 2.5 pounds (a decimal). She uses 3/4 pound (a fraction). The operation is subtraction because flour is being removed.
2
Step 2 — Convert to the Same Form

Convert 3/4 to a decimal: 3 ÷ 4 = 0.75. Now both numbers are decimals.

3/4 = 0.75
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Step 3 — Line Up Decimal Points and Subtract

Write the subtraction vertically: 2.50 − 0.75. Start from the right. 0 − 5: borrow to get 10 − 5 = 5. Then 4 − 7: borrow to get 14 − 7 = 7. Finally, 1 − 0 = 1. The result is 1.75.

2.50 − 0.75 = 1.75 pounds
4
Step 4 — Check Your Answer

Use the inverse operation: 1.75 + 0.75 = 2.50 ✓. The answer checks out!

Maria has 1.75 pounds of flour left.

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STRATEGY TIP

When a problem mixes fractions and decimals, convert them all to the same form before you start calculating. It is like making sure everyone on a team speaks the same language before you begin a project.

SECTION 7

Strengths and Pitfalls of Each Number Form

Each number form has strengths and weaknesses. Knowing them helps you pick the best path on test day.

Number Form	Strengths	Common Pitfalls
Whole Numbers	Easiest to work with; no extra steps needed.	Students sometimes forget to carry or borrow in large numbers.
Fractions	Exact values (1/3 is exact, 0.333… is not). Great for multiplication and division.	Forgetting to find a common denominator before adding or subtracting. Not simplifying the final answer.
Decimals	Easy to compare sizes (0.75 > 0.5). Familiar from money. Great for addition and subtraction.	Misaligning the decimal point. Miscounting decimal places when multiplying.

Choose the form that makes the problem easiest for you.

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KEY TAKEAWAY

Think of fractions and decimals as different tools in a toolbox. A screwdriver and a wrench can both tighten a bolt, but one might be easier for a particular bolt. Pick the number form that makes the math simplest, then convert your answer if the question asks for a specific form.

SECTION 8

Connection to Algebra and Beyond

The skills you are building now are the same ones you will use in algebra and higher math. The table below shows how today's skills connect to what you will learn next.

Skill You Learn Now	How It Appears Later
Finding a common denominator	Adding algebraic fractions like x/3 + x/5
Multiplying decimals	Working with scientific notation (3.2 × 10⁴)
Converting fractions ↔ decimals	Converting between percents, fractions, and decimals in statistics
Checking with inverse operations	Solving equations by "undoing" operations (inverse functions)

Every time you practice adding fractions or multiplying decimals, you are also training your brain for algebra. Think of this lesson as building the foundation of a house—without strong walls on the ground floor, you cannot add a second story.

SECTION 9

Practice Problems

Try these five problems on your own. They start easy and get harder. Write your work on paper before checking the answers!

Question 1: CONCEPTUAL

True or false: To add 2/5 + 1/3, you can just add the numerators (2 + 1 = 3) and the denominators (5 + 3 = 8) to get 3/8. Explain your reasoning.

Show answer

False. You cannot add fractions by adding the numerators and denominators separately. The fractions 2/5 and 1/3 have different-sized pieces (fifths and thirds). You must first find the LCD, which is 15. Rewrite: 6/15 + 5/15 = 11/15. Adding tops and bottoms directly would give the wrong answer because it ignores the size of each piece.

Question 2: BASIC CALCULATION

Calculate: 4.8 + 3.25

Show answer

Line up the decimal points: 4.80 + 3.25. Add from right to left: 0 + 5 = 5, 8 + 2 = 10 (write 0, carry 1), 4 + 3 + 1 = 8. The answer is 8.05. You can check by estimating: 5 + 3 = 8, so 8.05 is reasonable.

Question 3: INTERMEDIATE

Calculate: 3/4 × 2/5, then express your answer as a decimal.

Show answer

Multiply the numerators: 3 × 2 = 6. Multiply the denominators: 4 × 5 = 20. The fraction is 6/20. Simplify by dividing both by 2: 3/10. Convert to a decimal: 3 ÷ 10 = 0.3. You could also notice that any fraction with 10 in the denominator easily becomes a decimal—just move the decimal point one place.

Question 4: APPLIED

Jake buys 3 notebooks at $2.75 each and pays with a $10 bill. How much change does he receive?

Show answer

First, multiply: 3 × $2.75. Think of 2.75 as 275 cents. 275 × 3 = 825 cents = $8.25. Next, subtract from $10.00: $10.00 − $8.25 = $1.75. Jake receives $1.75 in change. You can check: $8.25 + $1.75 = $10.00 ✓.

Question 5: CRITICAL THINKING

A recipe calls for 2 1/3 cups of sugar. You want to make 1.5 times the recipe. How much sugar do you need? Give your answer as a mixed number in simplest form.

Show answer

Convert the mixed number to an improper fraction: 2 1/3 = 7/3. Convert 1.5 to a fraction: 1.5 = 3/2. Multiply: (7/3) × (3/2) = 21/6. Simplify 21/6 by dividing both by 3: 7/2. Convert to a mixed number: 7 ÷ 2 = 3 remainder 1, so 7/2 = 3 1/2. You need 3 1/2 cups of sugar. Notice that the 3 in the numerator and denominator cancelled—this is called cross-cancelling, and it can save you time.

SUMMARY

Lesson Summary

Summary

In this lesson you learned to perform the four basic operations—addition, subtraction, multiplication, and division—with whole numbers, fractions, and decimals. For fractions, remember to find a common denominator before adding or subtracting, multiply straight across, and use Keep-Change-Flip when dividing.

For decimals, always line up the decimal points for addition and subtraction, count total decimal places when multiplying, and move the decimal to create a whole-number divisor when dividing. You can always convert between fractions and decimals to pick whichever form makes the problem easiest. Finally, check your work using inverse operations.