When you hear the word product, what comes to mind?
Maybe it’s something you see on a store shelf, something made in a factory, or even something you’ve created yourself, like a finished art project or a recipe you followed to perfection.
No matter how you think about it, a product is usually the result of putting things together to make something new.
That idea carries over to math, too. In fact, a product is one of those concepts that follows you throughout your math journey. You’ll come across it in early multiplication, in fractions and decimals, and even in algebra and beyond.
In this guide, we’ll break down what a product is and how to find it. We’ll explore how it works with different kinds of numbers, answer common questions students have, and give you a chance to practice what you’ve learned.
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At Mathnasium, we like to say that a product is the answer to a multiplication question.
So, in a × b = c, c is the product.
And if the product is the answer, what makes up the question?
It consists of a multiplicand and a multiplier.
The multiplicand is the number we multiply.
The multiplier is the number we multiply by.
In this case, a is the multiplicand, and b is the multiplier.
When you multiply a and b, you get the product c.
Here’s another way to think about a product:
Imagine you're baking cookies.
The multiplicand is how many cookies you make in one batch—let’s say 12.
The multiplier is how many batches you decide to bake—maybe 3.
To find out how many cookies you’ll have in total, you multiply:
12 cookies × 3 batches = 36 cookies
That final number, 36, is your product.
To find the product, we simply multiply numbers.
For instance, to find the product of 7 and 3, we multiply:
7 × 3 = 21
And what if there are more than two numbers, say, 2, 5, and 4?
We still multiply:
2 × 5 × 4 = 40
The idea is as simple as it gets. The only part that gets tougher is the size or type of numbers you’ll work with throughout your math journey.
See how Mathnasium’s proprietary teaching approach, the Mathnasium Method™, helps students learn and master any math topic, including multiplication.
To find the product of two fractions, we multiply them, but this time, we multiply twice: once across the top and once across the bottom.
So, for instance, to multiply \(\displaystyle \frac{2}{3}\) by \(\displaystyle \frac{4}{5}\), we need to:
Step 1: Multiply the numerators (top numbers)
2 × 4 = 8
Step 2: Multiply the denominators (bottom numbers)
3 × 5 = 15
So, \(\displaystyle \frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\)
That’s our product!
One extra step we can (but don’t always need to) take is to check if we can simplify the fraction. We do this by seeing if the top and bottom numbers share a greatest common factor (GCF).
But since 8 and 15 don’t share any common factor greater than 1, \(\displaystyle \frac{8}{15}\) is already in its simplest form.
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To find the product of decimals, we multiply, just like with whole numbers, with one extra step at the end.
So, say we want to find the product of 0.4 and 0.3. To do that, we:
Step 1: Ignore the decimals for now
Think of 0.4 as 4 and 0.3 as 3 and multiply them.
4 × 3 = 12
Step 2: Count how many digits come after the decimals in the original numbers:
0.4 has 1 digit after the decimal.
0.3 has 1 digit after the decimal.
That’s 2 digits total.
Step 3: Place the decimal in your answer
Start from the right and move 2 places to the left in 12:
12 \(\rightarrow\) 0.12
So, the product of 0.4 and 0.3 is 0.12.
If finding the product means we’re multiplying, it helps to understand how math properties (also called number properties) affect multiplication and the product we get.
These properties are like rules that work behind the scenes in math. They don’t change the numbers themselves, but they can change how we look at or work with those numbers to find the product more easily.
We’ll take a closer look at these four properties:
Distributive Property
Identity Property
The commutative property of multiplication shows we can change the order of numbers when we multiply them and still get the same product.
We can express the commutative property like so:
a x b = b x a
Let’s see how it works:
Say we want to find the product of 4 and 5. To do so, we can multiply in any order and still get the same result:
4 × 5 = 20
and also
5 × 4 = 20
So, we can change the order without changing the product.
The associative property of multiplication shows us that when we multiply numbers, we can group them with parentheses in any way we want, and we’ll still get the same product.
(a × b) × c = a × (b × c)
Let’s say we want to find the product of 2, 3, and 4.
We can group the first two numbers:
(2 × 3) × 4 = 6 × 4 = 24
Or we can group the last two numbers:
2 × (3 × 4) = 2 × 12 = 24
In both cases, the product is 24.
So, we can change the grouping without changing the product.
An identity in math is a number that, when used in an operation with another number, keeps that number the same.
For multiplication, that number is 1.
That’s why we call 1 the identity element for multiplication (or the multiplicative identity).
When you multiply any number by 1, the product is that number—nothing changes.
Let’s see it in action:
6 × 1 = 6
1 × 9 = 9
25 × 1 = 25
No matter what number you multiply by 1, the product stays the same.
The distributive property is a rule in math that lets us multiply a number outside parentheses by each term inside the parentheses.
We can write it like this:
a × (b + c) = a × b + a × c
Let’s see it in action:
Say we want to find the product of 3 × (4 + 5).
Step 1: Use the distributive property to break it apart
3 × 4 + 3 × 5
Step 2: Multiply:
12 + 15 = 27
Now let’s check by solving inside the parentheses first
4 + 5 = 9
3 × 9 = 27
Same product either way!
The distributive property helps us break apart expressions to make multiplication easier or to better understand how numbers work together.
Ready to practice what you’ve learned? Try working out the tasks below. When you’re finished, check how you did at the bottom of the section.
What is the product of 6 and 7?
Find the product of 4 × 3 × 2
What is the product of \(\frac{1}{2} \times \frac{1}{3}\)?
What is the product of 0.6 × 0.4?
Each of these equations shows a math property that helps us understand or find products more easily.
Can you name the property?
5 × 1 = 5
(2 × 6) × 3 = 2 × (6 × 3)
7 × 4 = 4 × 7
3 × (2 + 5) = 3 × 2 + 3 × 5
Learning about products often brings up questions, both from students who are curious and from parents who want to support their child’s learning.
At Mathnasium of Richardson West, we hear these questions very often. So we’ve gathered a few of the most common ones and included the answers here to clear up the dilemmas that can get in the way of true understanding.
Most students are introduced to the idea of a product in Grade 2 or 3, when they begin learning multiplication. From there, the concept grows with them into fractions, decimals, and algebra.
Understanding products early on gives students a strong foundation for nearly every future math topic.
Think of a product as the result of a multiplication question, and factors as the numbers you multiply to get there.
For example, in 6 × 4 = 24:
6 and 4 are the factors
24 is the product
You can think of them as working in opposite directions:
Factors are the "ingredients" you use.
The product is what comes out when you multiply.
The product is always 0.
It doesn’t matter how big or small the other number is, once you multiply it by 0, the product becomes 0. In math, we often call this zero multiplication property.
Yes, this can happen when you multiply fractions or decimals that are less than 1.
For example:
\(\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}\)
Here, both numbers are less than 1, and their product is even smaller.
This is an important concept in middle school math, and it’s one we focus on a lot at Mathnasium because it helps students better understand the behavior of numbers.
At Mathnasium of Richardson West, questions are always welcome! Our tutors encourage students to speak up, explore ideas, and build a deep understanding of any topic, including products.
Mathnasium of Richardson West is a math-only learning center for K–12 students of all skill levels in Richardson, TX.
Using our proprietary teaching approach, the Mathnasium Method™, our specially trained math tutors provide face-to-face instruction in a caring and engaging group environment to help students truly understand and master multiplication, including how to find products with whole numbers, fractions, and decimals.
Students begin their Mathnasium journey with a diagnostic assessment that helps us identify their specific strengths and knowledge gaps. Using these insights, we create a personalized learning plan that puts each student on the best path toward math mastery.
Whether your student is looking to catch up, keep up, or get ahead in math, schedule an assessment and enroll at Mathnasium of Richardson West today!
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If you’ve given our practice exercises a try, check your answers here.
1. What is the product of 6 and 7?
Answer: 42
2. Find the product of 4 × 3 × 2
Answer: 24
3. What is the product of \(\frac{1}{2} \times \frac{1}{3}\)?
Answer: 16
4. What is the product of 0.6 × 0.4?
Answer: 0.24
Each of these shows a multiplication property that helps us understand or find products more easily.
5. 5 × 1 = 5
Answer: Identity Property of Multiplication
6. (2 × 6) × 3 = 2 × (6 × 3)
Answer: Associative Property of Multiplication
7. 7 × 4 = 4 × 7
Answer: Commutative Property of Multiplication
8. 3 × (2 + 5) = 3 × 2 + 3 × 5
Answer: Distributive Property of Multiplication
Mathnasium of Richardson West is a math-only learning center for K-12 students in Richardson, TX. Trusted by over a million parents, Mathnasium uses personalized learning plans and the proprietary Mathnasium Method™ to help students catch up, keep up, and get ahead on their math journey.
Our specially trained tutors deliver face-to-face instruction in a supportive and fun small-group environment, working with students both in center and online to develop a deep understanding of math, build confidence, and improve academic performance.
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