- Identify the structure of the expressions as \(a \times (b \times 10) = (a \times b) \times 10\).
- Multiply the single-digit numbers \(a\) and \(b\).
- Multiply the resulting product by 10 by appending a zero.
- The results are \(6 \times 30 = 180\), \(4 \times 50 = 200\), \(8 \times 40 = 320\), \(7 \times 40 = 280\), \(5 \times 60 = 300\), and \(9 \times 20 = 180\).
- \(\boxed{180, 200, 320, 280, 300, 180}\)
Explanation
- Problem Analysis
We are tasked with computing the product of a single-digit number and a multiple of 10 for six distinct pairs. The given expressions are:
- \(6 \times 30\)
- \(4 \times 50\)
- \(8 \times 40\)
- \(7 \times 40\)
- \(5 \times 60\)
- \(9 \times 20\)
Each of these can be expressed in the form \(a \times (b \times 10)\), where \(a\) and \(b\) are single-digit integers. By the associative property of multiplication, we can rewrite each expression as \((a \times b) \times 10\). This allows us to multiply the single-digit numbers first and then multiply the result by 10 by simply appending a zero.
- Step-by-Step Multiplication
We now apply the associative property to calculate each expression step-by-step:
- \(6 \times 30 = 6 \times (3 \times 10) = (6 \times 3) \times 10 = 18 \times 10 = 180\)
- \(4 \times 50 = 4 \times (5 \times 10) = (4 \times 5) \times 10 = 20 \times 10 = 200\)
- \(8 \times 40 = 8 \times (4 \times 10) = (8 \times 4) \times 10 = 32 \times 10 = 320\)
- \(7 \times 40 = 7 \times (4 \times 10) = (7 \times 4) \times 10 = 28 \times 10 = 280\)
- \(5 \times 60 = 5 \times (6 \times 10) = (5 \times 6) \times 10 = 30 \times 10 = 300\)
- \(9 \times 20 = 9 \times (2 \times 10) = (9 \times 2) \times 10 = 18 \times 10 = 180\)
- Summary of Results
Based on the calculations above, we have determined the values for each of the six expressions.
- \(6 \times 30 = 180\)
- \(4 \times 50 = 200\)
- \(8 \times 40 = 320\)
- \(7 \times 40 = 280\)
- \(5 \times 60 = 300\)
- \(9 \times 20 = 180\)
These results are obtained by effectively calculating the product of the non-zero digits and then placing a zero at the end of the product.
Examples
Imagine you are organizing an event and need to buy supplies in packs. If you need 30 chairs and want to order 6 sets, or 40 balloons for 8 different tables, understanding how to multiply by multiples of 10 helps you calculate quantities quickly. For example, if each table setting costs \(\$40\) and you have 7 tables, the total cost is \(7 \times 40 = \$280\). Mastering these mental math techniques allows for faster budgeting and planning in everyday life scenarios.