The distributive property is one of the first key algebra skills students learn in middle school Pre-Algebra class. While it starts simply—multiplying across parentheses—it becomes essential for simplifying expressions, solving equations, and handling rational numbers like fractions and decimals.

Whether you’re preparing for school math or aiming to build early confidence for contests like AMC 8, mastering this property lays a strong foundation.

In this guide, we’ll review how it works, apply it to rational numbers, and explore examples from classroom practice to competition-level problems.

What Is the Distributive Property?

The distributive property connects multiplication with addition and subtraction. It tells us that multiplying a number by a sum is the same as multiplying the number by each term inside the parentheses, then adding the results:

a(b + c) = ab + ac

This property applies to all real numbers—including whole numbers, fractions, decimals, and variables—and is frequently used in:

• Simplifying algebraic expressions
• Solving linear equations
• Expanding polynomial terms
• Word problems involving cost, distance, or quantity

School-Level Distributive Property Examples

We’ve reviewed the basics, let’s see how the distributive property is applied in real problems. We’ll start with straightforward in-school examples, then move on to AMC 8-style questions that require deeper reasoning.

Distributive Property Example 1 in Number Operations

Problem

Simplify using the distributive property: 10(12 + 15)

Solution

The distributive property states:

a × (b + c) = (a × b) + (a × c)

Apply it to the expression:

10(12 + 15)

= (10 × 12) + (10 × 15)

= 120 + 150

= 270

Answer: 270

Distributive Property Example 2 in Algebraic Expressions

Problem

Which expression is equivalent to: 4(3x − 1.25) − 2(3.5x + 2)?

(A) −4x (B) −2x (C) 5x − 1 (D) 5x − 9

Solution

Apply the distributive property to both terms:

First expression:

4(3x − 1.25)

= 4 × 3x − 4 × 1.25

= 12x − 5

Second expression:

−2(3.5x + 2)

= −2 × 3.5x − 2 × 2

= −7x − 4

Now combine both:

(12x − 5) − (7x + 4)

= 12x − 5 − 7x − 4

= (12x − 7x) + (−5 − 4)

= 5x − 9

Answer: (D) 5x − 9

AMC 8-Style Application of Distributive Property

Distributive Property Example 3

Problem (2013 AMC 8 Question 3)

What is the value of:

4 × (−1 + 2 − 3 + 4 − 5 + 6 − 7 + ⋯ + 1000)?

(A) −10 (B) 0 (C) 1 (D) 500 (E) 2000

Solution

We group the terms inside the parentheses in pairs:

(−1 + 2) + (−3 + 4) + (−5 + 6) + … + (−999 + 1000)

Each pair equals 1:

(−1 + 2) = 1

(−3 + 4) = 1

…

(−999 + 1000) = 1

There are 500 such pairs between 1 and 1000.

So, the value inside the parentheses is:

1 + 1 + 1 + … + 1 = 500

Then multiply by 4:

4 × 500 = 2000

Answer: (E) 2000

This question challenges students to recognize patterns in alternating sums and use grouping and the distributive property to simplify repeated terms for efficient calculation.

Distributive Property Example 4

Problem (2023 AMC 8 Question 13)

Along the route of a bicycle race, 7 water stations are evenly spaced between the start and finish lines. There are also 2 repair stations evenly spaced between the start and finish lines. The 3rd water station is located 2 miles after the 1st repair station.

How long is the race in miles?

(A) 8 (B) 16 (C) 24 (D) 48 (E) 96

Solution

Let x be the distance between each water station.

There are 6 intervals between the 7 water stations, plus 1 interval before the first water station (from the start line) and 1 after the last (to the finish line).

So the total race length is:

Total race length = 8x

There is 1 interval between the 2 repair stations, plus 2 more intervals between the start and finish lines.

So in total, the race is divided into 3 equal intervals by the repair stations.

We are told that the 3rd water station is located 2 miles after the 1st repair station.

So the distance from the start line to the 1st repair station is:

• x (from start to 1st water station)
• 2x (from 1st to 3rd water station, 2 intervals)
• − 2 (because the 3rd water station is 2 miles after the 1st repair station)

Which gives:

Distance from start to 1st repair station = 2x + x − 2 = 3x − 2

So the total race length can also be written as:

Total race length = 3 × (3x − 2)

Now set up the equation:

8x = 3(3x − 2)

8x = 9x − 6

−x = −6

x = 6

Now plug back in:

Total race length = 8 × 6 = 48

Answer: (D) 48 miles

This problem tests students’ understanding of algebraic reasoning and the distributive property, requiring them to represent a real-world scenario with multiple expressions and set up an equation.

What’s the Difference Between In-School and AMC 8 Distributive Property Problems?

The distributive property of rational numbers is a foundational skill—but it’s far from the end goal.

In both classroom learning and math competitions, students will continue to use it to simplify algebraic expressions, solve equations, and tackle inequalities.

In-school problems focus on mastering the basics:

• You’re usually asked to apply the property directly to a numerical or algebraic expression.
• The structure is straightforward, and the purpose is skill-building.

AMC 8 problems, by contrast, involve more critical thinking:

• The distributive property may be one step in a multi-part problem.
• You often need to identify when and how to apply it, possibly alongside number patterns, logical reasoning, or estimation.

Understanding the property is just the beginning—what matters most is learning to use it flexibly and strategically as problems grow more complex.

Ready to Practice?

• 🎯 Download your free Pre-Algebra worksheets – Open the worksheets page and choose the [G6 and above] Tab, find the practice for Grade 7.
• ✅ Score 100%? Take the Think Academy Pre-Algebra assessment for personalized 1-on-1 feedback.

• 💡 Interested in math competitions? Try our AMC 8 Evaluation Test to see where you stand.

About Think Academy

Think Academy, a top math education brand under TAL Education Group, offers specialized training for the prestigious AMC 8 competition. In 2025, 672 Think Academy students earned national awards—65% of them placed on the Achievement, Honor, or Distinguished Honor Rolls. From 2022 to 2025, our results have grown rapidly, setting records in award counts and perfect scores. Backed by expert teachers and a proven curriculum, Think Academy prepares middle schoolers for lasting success in competitive math. Explore our AMC 8 programs and start your child’s winning journey.

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