Multiplying by multiples of 10 Knowledge-Based Study

Study notes

1. A multiple of 10 is any number that can be formed by multiplying an integer by 10. Examples include 10, 20, 30, 40, and so on. Every multiple of 10 ends with at least one zero in the units place.

   When we multiply a whole number by 10, each digit shifts one place to the left on the place value chart, which results in placing a zero at the end of the number:

   • $6 \times 10 = 60$
   • $14 \times 10 = 140$
   • $235 \times 10 = 2350$

   Understanding this place-value shift allows us to break down more complex multiplications into simpler, familiar steps.

   To multiply a single-digit number by a multiple of 10, we can use the associative property of multiplication. This property states that the way factors are grouped does not change the product.

   Consider the calculation $4 \times 30$:

   1. Decompose 30 into $3 \times 10$:
      $4 \times 30 = 4 \times (3 \times 10)$
   2. Regroup the non-zero digits using parentheses:
      $(4 \times 3) \times 10$
   3. Multiply the basic single-digit facts first:
      $12 \times 10$
   4. Apply the place-value shift (attach one zero):
      120

   Therefore, $4 \times 30 = 120$. In practice, you simply multiply the non-zero digits ($4 \times 3 = 12$) and append one zero to the product.

   When both factors are multiples of 10, the same associative strategy applies to both numbers.

   Consider $60 \times 40$:

   • Write each factor as a product of a single digit and 10:
     $(6 \times 10) \times (4 \times 10)$
   • Rearrange the factors so that the non-zero digits are multiplied together and the tens are multiplied together:
     $(6 \times 4) \times (10 \times 10)$
   • Calculate each part:
     $24 \times 100 = 2400$

   Rule of Thumb: Count the total number of trailing zeros in the factors. Multiply the base digits, then append that total number of zeros to the result. In $60 \times 40$, there is one zero in 60 and one zero in 40 (two zeros total). Since $6 \times 4 = 24$, the final product is 2400.

   This strategy also extends to multiplying multi-digit numbers by multiples of 10, as well as multiples of 100 or 1000.

   Example with a Two-Digit Factor

   Calculate $23 \times 40$:

   • Multiply the non-zero parts: $23 \times 4 = 92$
   • Place the zero from 40 at the end: $920$
   • Thus, $23 \times 40 = 920$.

   A Common Pitfall: Built-in Zeros

   Be careful when the multiplication of the non-zero digits produces a number that already ends in zero. For example, in $50 \times 60$:

   • Non-zero multiplication: $5 \times 6 = 30$
   • Count of trailing zeros in factors: two zeros (one from 50, one from 60)
   • Append the two trailing zeros to 30: 3000

   Do not confuse the zero in 30 with one of the zeros brought over from the factors; the product has three zeros in total.