How to Master Basic Operations with Integers, Fractions & Decimals | ACT Math Grade 9-10 Practice
Welcome to IrfanEdu.com’s comprehensive ACT prep resources! If you’re preparing for the ACT Math section, understanding how to confidently work with integers, fractions, and decimals in mixed operation problems is absolutely essential. Today, we’re tackling a medium-difficulty practice problem that combines all these elements, and I’ll show you exactly how to approach it with the speed and accuracy needed for test day success. 🎯
ACT SCORE BOOSTER: Master This Topic for 2-3 Extra Points!
This topic appears in most tests (5-10 questions) on the ACT Math section. Understanding it thoroughly can add 2-3 points to your composite score. Let’s break it down with proven strategies that work!
🚀 Jump to ACT Strategy →⚡ Quick Answer (TL;DR)
Problem: Calculate: $$(-12) + 18 \div 3 – 2.5 \times 4 + \frac{3}{4} = ?$$
Answer: $$-15.25$$ or $$-15\frac{1}{4}$$
💡 Key Strategy: Follow PEMDAS strictly! Division and multiplication first (left to right), then addition and subtraction (left to right). Convert the fraction to decimal for easier calculation.
📋 Practice Problem 1: Mixed Operations Challenge
ACT Math | Pre-Algebra | Difficulty: Medium
Calculate: $$(-12) + 18 \div 3 – 2.5 \times 4 + \frac{3}{4} = ?$$
This problem tests your ability to handle multiple types of numbers (negative integers, positive integers, decimals, and fractions) while correctly applying the order of operations. It’s exactly the kind of question you’ll encounter in the early-to-middle section of the ACT Math test, where one small mistake can cost you valuable points. According to the official ACT website, basic operations questions like this form the foundation of the Pre-Algebra section. Let’s break it down together! 📊
🎯 ACT Strategy: The 30-Second Approach
⏱️ Time Target: 30-45 seconds
For ACT Math, you have an average of 60 seconds per question. Basic operation problems like this should take you half that time if you’re well-practiced. Here’s your game plan:
- Scan for PEMDAS violations (5 seconds) – Identify what operations need to happen first
- Convert fraction to decimal (5 seconds) – $$\frac{3}{4} = 0.75$$ (memorize common fractions!)
- Calculate multiplication/division (10 seconds) – Do these operations left to right
- Calculate addition/subtraction (10 seconds) – Work left to right with your results
- Double-check signs (5 seconds) – Negative number errors are the #1 mistake!
🚨 ACT Trap Warning: The test makers LOVE to include problems where students forget order of operations and work strictly left to right. If you calculated $$(-12) + 18 = 6$$, then $$6 \div 3 = 2$$, you’ve fallen into the trap! Always identify multiplication and division first.
🎥 Video Explanation
Watch this detailed video explanation to understand the concept better with visual demonstrations and step-by-step guidance.
🎨 Visual Solution Breakdown
Let’s visualize the order of operations with a clear diagram showing exactly what happens at each step:
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🚀 Start ACT Practice Test Now →✅ Complete Step-by-Step Solution
Now let’s walk through this problem with detailed explanations for each step. This is the methodology you should internalize for test day:
1 Write Down the Problem Clearly
$$(-12) + 18 \div 3 – 2.5 \times 4 + \frac{3}{4}$$
Why this matters: On the ACT, rushing leads to misreading. Take 2 seconds to ensure you’ve captured all numbers and operations correctly.
2 Convert Fractions to Decimals (When Helpful)
$$\frac{3}{4} = 0.75$$
So our problem becomes: $$(-12) + 18 \div 3 – 2.5 \times 4 + 0.75$$
Pro tip: Memorize these common fraction-to-decimal conversions: $$\frac{1}{4} = 0.25$$, $$\frac{1}{2} = 0.5$$, $$\frac{3}{4} = 0.75$$, $$\frac{1}{5} = 0.2$$, $$\frac{1}{8} = 0.125$$
3 Apply PEMDAS: Division First
Identify and calculate: $$18 \div 3 = 6$$
Updated expression: $$(-12) + 6 – 2.5 \times 4 + 0.75$$
Remember: In PEMDAS, Multiplication and Division have equal priority. Work left to right when you encounter them.
4 Apply PEMDAS: Multiplication Next
Calculate: $$2.5 \times 4 = 10$$
Updated expression: $$(-12) + 6 – 10 + 0.75$$
Calculator tip: For $$2.5 \times 4$$, think of it as $$\frac{5}{2} \times 4 = \frac{20}{2} = 10$$ or simply $$2.5 + 2.5 + 2.5 + 2.5 = 10$$
5 Work Left to Right: Addition and Subtraction
Step 5a: $$(-12) + 6 = -6$$
Think: Start at -12 on the number line, move 6 units to the right → land at -6
Step 5b: $$-6 – 10 = -16$$
Think: Subtracting a positive is the same as adding a negative → $$-6 + (-10) = -16$$
Step 5c: $$-16 + 0.75 = -15.25$$
Think: Start at -16, move 0.75 units to the right → land at -15.25
🎉 Final Answer
$$-15.25$$ or $$-15\frac{1}{4}$$ or $$-\frac{61}{4}$$
⚠️ Common Mistakes to Avoid
Let’s look at the most frequent errors students make on problems like this—and how to avoid them:
| ❌ Mistake | Why It Happens | ✅ How to Fix It |
|---|---|---|
| Working strictly left to right | Forgetting PEMDAS order | Always identify mult/div operations first, mark them with circles |
| Sign errors with negatives | Rushing through negative number rules | Use parentheses: treat $$(-12) + 6$$ as $$-12 + 6$$, not $$12 + 6$$ |
| Decimal point errors | Misplacing decimal in $$2.5 \times 4$$ | Double-check: $$2.5 \times 4 = 10$$, not 1.0 or 100 |
| Fraction conversion mistakes | Not memorizing common fractions | Create flashcards for $$\frac{1}{4}$$, $$\frac{1}{2}$$, $$\frac{3}{4}$$, $$\frac{1}{5}$$, $$\frac{1}{8}$$ |
| Calculator input errors | Typing too fast, missing parentheses | Input as: $$(-12) + (18 \div 3) – (2.5 \times 4) + 0.75$$ |
⏱️ Time-Saving Tips for ACT Math
💡 Tip #1: Memorize Common Conversions
Don’t waste 10 seconds dividing $$3 \div 4$$ on your calculator. Know that $$\frac{3}{4} = 0.75$$ instantly. This saves 5-10 seconds per problem!
💡 Tip #2: Use Calculator Parentheses
Input the entire expression with parentheses: $$(-12)+(18\div3)-(2.5\times4)+0.75$$ and let your calculator handle PEMDAS.
💡 Tip #3: Circle Mult/Div Operations
Before calculating anything, physically circle or underline all multiplication and division operations. This prevents PEMDAS violations.
💡 Tip #4: Practice Mental Math
Simple operations like $$18 \div 3$$ or $$2.5 \times 4$$ should be instant. Practice 10 minutes daily to build speed and confidence.
📝 Practice Problems (Try These!)
Ready to test your skills? Try these similar problems. Answers are at the bottom—no peeking! 😊
🎯 Challenge Yourself:
Problem 1 (Easy): $$15 + 20 \div 4 – 3 \times 2 = ?$$
Problem 2 (Medium): $$(-8) + 24 \div 6 – 1.5 \times 4 + \frac{1}{2} = ?$$
Problem 3 (Hard): $$(-15) + 36 \div 4 – 3.25 \times 2 + \frac{3}{8} – 1 = ?$$
🔍 Click to Reveal Answers
Problem 1: $$14$$
Problem 2: $$-1.5$$ or $$-\frac{3}{2}$$
Problem 3: $$-9.125$$ or $$-9\frac{1}{8}$$
❓ Frequently Asked Questions (FAQs)
Q1: What is the order of operations for ACT Math problems?
A: The order of operations follows PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). This is crucial for ACT Math success. Remember that multiplication and division have equal priority, as do addition and subtraction—always work left to right within each priority level.
Q2: How do I handle negative numbers in basic operations?
A: When adding a negative number, subtract its absolute value (e.g., $$5 + (-3) = 5 – 3 = 2$$). When subtracting a negative, add its absolute value (e.g., $$5 – (-3) = 5 + 3 = 8$$). For multiplication/division: same signs give positive results, different signs give negative results. Always use parentheses around negative numbers to avoid confusion: $$(-12) + 6$$ is clearer than $$-12 + 6$$.
Q3: Should I convert fractions to decimals or decimals to fractions?
A: For ACT Math, converting fractions to decimals is usually faster and less error-prone. Use your calculator efficiently. However, keep fractions when they’re simple (like $$\frac{1}{2}$$, $$\frac{1}{4}$$) or when the problem requires exact answers. Memorize common conversions: $$\frac{1}{4} = 0.25$$, $$\frac{1}{2} = 0.5$$, $$\frac{3}{4} = 0.75$$, $$\frac{1}{5} = 0.2$$, $$\frac{1}{8} = 0.125$$. This saves valuable time!
Q4: How much time should I spend on basic operation questions on the ACT?
A: Basic operation questions should take 30-45 seconds maximum. You have 60 minutes for 60 questions on ACT Math (average 60 seconds per question), but simpler problems should be solved faster to give you more time for complex geometry and algebra questions. If you’re spending more than 1 minute on a basic operations problem, you may be overcomplicating it. Practice mental math and calculator efficiency to improve speed.
Q5: What are the most common mistakes in mixed operation problems?
A: The top mistakes are: 1) Ignoring order of operations and working left to right, 2) Sign errors with negative numbers (especially double negatives), 3) Decimal point placement errors, 4) Calculator input mistakes (forgetting parentheses), and 5) Rushing through parentheses. To avoid these, always circle multiplication/division operations first, use parentheses liberally in your calculator, and double-check your signs before finalizing your answer.
🎓 Wrapping Up: Your Path to ACT Math Success
Congratulations! You’ve just mastered a critical ACT Math skill: solving mixed operation problems with integers, fractions, and decimals. While this might seem like a basic topic, it’s the foundation for more complex algebra, geometry, and trigonometry questions you’ll encounter on test day.
Remember these key takeaways:
- PEMDAS is non-negotiable – Always identify multiplication and division operations before doing anything else
- Speed comes from practice – Aim for 30-45 seconds on basic operation problems
- Negative numbers require extra attention – Use parentheses and double-check your signs
- Memorize common conversions – Knowing $$\frac{3}{4} = 0.75$$ instantly saves precious seconds
- Your calculator is your friend – But only if you input expressions correctly with parentheses
- Visual learning helps – Use number lines and flowcharts to understand the process
The path to a higher ACT Math score is built on mastering fundamentals like this, then applying them consistently under time pressure. Practice the three problems above, then find 10 more similar questions and time yourself. Your goal: solve each one correctly in under 45 seconds.
For more comprehensive ACT preparation resources, visit our complete collection of practice problems, video tutorials, and test-taking strategies.
💪 Keep practicing, stay confident, and remember: every point counts on the ACT!
Visit IrfanEdu.com for more ACT Math practice problems and strategies
✍️ Written by Dr. Irfan Mansuri
Educational Content Creator & Competitive Exam Specialist
IrfanEdu.com • United States
Dr. Irfan Mansuri is a distinguished educational content creator and competitive exam specialist with over 15 years of experience spanning high school, undergraduate, and postgraduate levels. As the founder of IrfanEdu.com, he has successfully guided thousands of students through competitive examinations, helping them achieve exceptional results and gain admission to their dream institutions.
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