Question

Use the fact that \(13 \cdot 217=2,821\) to find each product without using a calculator. $$13 \cdot 2.17$$

Short answer

13 \times 2.17 = 28.21

Step-by-step solution

- Understand the given fact

Given: The product of 13 and 217 is 2,821. This can be written as: \[ 13 \times 217 = 2821 \].

- Rewrite 2.17 in terms of 217

Notice that 2.17 can be written as \[ 2.17 = 217 \times 0.01 \].

- Substitute back into the equation

Substitute 2.17 with \(217 \times 0.01\) in the product: \[ 13 \times 2.17 = 13 \times (217 \times 0.01) \].

- Apply associative property to multiply

By the associative property of multiplication, rearrange the terms: \[ 13 \times (217 \times 0.01) = (13 \times 217) \times 0.01 \].

- Substitute the known product

We already know that \[ 13 \times 217 = 2821 \]. Substitute this value in: \[ (13 \times 217) \times 0.01 = 2821 \times 0.01 \].

- Perform the multiplication by 0.01

Multiplying 2821 by 0.01 gives \[ 2821 \times 0.01 = 28.21 \].

Key concepts

Associative Property of Multiplication

The associative property of multiplication is a fundamental concept that helps simplify complex multiplication problems by rearranging the grouping of numbers. It states that the way in which numbers are grouped in multiplication does not change their product. In mathematical terms, this property is expressed as: \[ (a \times b) \times c = a \times (b \times c) \] For instance, in the problem \( 13 \times (217 \times 0.01) \), we can use the associative property to rearrange the numbers: \[ 13 \times (217 \times 0.01) = (13 \times 217) \times 0.01 \] Using the associative property, we convert a complex multiplication problem into a simpler form, which can be solved using known values. This makes calculations easier and more efficient.

Decimal Multiplication

Multiplying decimals can seem tricky at first, but by understanding a few basic rules, it becomes straightforward. When you multiply a number by a decimal, you are essentially performing a multiplication followed by adjusting the position of the decimal point in the product. This is because decimals represent fractions of ten. In the exercise, for example, we needed to multiply \(13\) by \(2.17\). Note that \(2.17\) can be written as \(217 \times 0.01\). Then using the associative property, we simplify it to: \[ 13 \times (217 \times 0.01) \] Which reorganizes to: \[ (13 \times 217) \times 0.01 \] Finally, knowing that \(13 \times 217 = 2821\), we just multiply \(2821\) by \(0.01\) to get \(28.21\). The key steps for decimal multiplication are:

• Ignore the decimal points and perform the multiplication on the whole numbers.
• Count the total number of decimal places in the numbers you are multiplying.
• Place the decimal point in the product so that it has the same number of decimal places.

Basic Algebra

Basic algebra forms the foundation of more advanced mathematics and involves using symbols and letters to represent numbers in equations. In this exercise, we used basic algebraic manipulation to rewrite and simplify the problem. Here’s a brief outline of how we applied basic algebraic concepts in this exercise: 1. **Substitution**: We recognized that \(2.17\) can be expressed as \(217 \times 0.01\). This allowed us to substitute \(2.17\) in the problem: \[ 13 \times 2.17 = 13 \times (217 \times 0.01) \] 2. **Associative Property**: Utilizing the associative property of multiplication, we rearranged the terms: \[ 13 \times (217 \times 0.01) = (13 \times 217) \times 0.01 \] 3. **Simplification**: Using the known product of \(13 \times 217 = 2821\), we simplified the expression further: \[ 2821 \times 0.01 = 28.21 \] By practicing these algebraic techniques, solving more complex problems becomes manageable. The focus is on breaking down the problems into parts, applying algebraic rules, and simplifying step by step.