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7th Grade · Math

Mixed Operations with Rational Numbers

Free sample questions, a clear explanation, and 5 practice skills with an AI tutor that guides without giving the answer away.

Concept Review

Mixed Operations with Rational Numbers: The Mathematical Recipe

What if I told you that -3 + 2 × 5² ÷ (-2) could equal either -15.5 or 361? Same numbers, same operations, but wildly different answers. The secret lies in following the correct mathematical "recipe" — the order of operations.

Just like baking a cake, mathematics has a specific sequence that must be followed. You can't frost before baking, and you can't add before multiplying (unless parentheses say otherwise). This becomes especially crucial when working with rational numbers — positive and negative fractions, decimals, and integers all mixed together.

PEMDAS: Your Mathematical GPS

Remember PEMDAS (Please Excuse My Dear Aunt Sally) — but now we're navigating through positive and negative territories:

P - Parentheses

Handle what's inside first

E - Exponents

Powers and roots

MD - Multiply/Divide

Left to right

AS - Add/Subtract

Left to right

Let's Solve That Mystery

Now let's properly evaluate -3 + 2 × 5² ÷ (-2):

-3 + 2 × 5² ÷ (-2)

= -3 + 2 × 25 ÷ (-2) [Exponents first: 5² = 25]

= -3 + 50 ÷ (-2) [Multiply: 2 × 25 = 50]

= -3 + (-25) [Divide: 50 ÷ (-2) = -25]

= -28 [Add: -3 + (-25) = -28]

🔍 The Sign Detective

Here's what trips up most students: negative signs aren't always subtraction!

-3 + 2 means "negative three plus two" = -1

0 - 3 + 2 means "zero minus three plus two" = -1

Same answer, but the first negative sign describes the number itself, while the second is an operation.

Whether you're working with fractions like ½ × (-¾), decimals like -2.5 + 1.8 × 3, or mixed numbers, the order of operations remains your reliable guide. Each rational number keeps its positive or negative identity throughout the journey, but the sequence of operations determines where that journey leads.

🔑 Key Takeaway

Mathematics isn't about memorizing tricks — it's about following a universal language. When you respect the order of operations with rational numbers, that mysterious expression -3 + 2 × 5² ÷ (-2) has only one correct answer: -28. The recipe never lies.

Sample questions

1. Evaluate: -3 + 5 × 2

✓ 7

○ 4

○ -7

○ -4

Answer: 7 — Order of operations: multiply first: 5×2=10, then -3+10=7.

2. Calculate: 12 ÷ (-3) - 4

○ 0

○ 8

✓ -8

○ -1

Answer: -8 — 12 ÷ (-3) = -4, then -4 - 4 = -8.

3. Find: (-2) × 5 + (-3) × 4

○ 2

○ 22

○ -2

✓ -22

Answer: -22 — (-2)×5 = -10, (-3)×4 = -12, -10 + (-12) = -22.

Skills in this topic

Evaluate expressions with rational numbers using the order of operations
Simplify complex fractions containing positive and negative numbers
Identify the properties of operations (Commutative, Associative, Distributive)
Evaluate multi-step formulas with negative variables
Solve multi-step real-world word problems with rational numbers

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