When required to compute the product 2.38 ÷ 0.0125 by hand, many people move the decimals in both numbers to the right so that the divisor (the number they are dividing by) is an integer, and compute 23,800 ÷ 125, instead.
When dividing one number by another, the number you are dividing into is called the dividend, the number you’re dividing by is called the divisor, and the answer is called the quotient.

1. Carefully explain how you know that 2.38 ÷ 0.0125 = 23,800 ÷ 125 without actually computing either quotient.
2. Carefully explain (and demonstrate) how to compute the following quotients by hand. Be sure to specify what quotient you are actually computing in each case. (For example, to see what 2.38 ÷ 0.0125 equals, you compute 23,800 ÷ 125.)
   1. 0.000585 ÷ 0.18
   2. 0.03312 ÷ 2.4
3. In problem 1, you explained why the “move the decimals in both numbers to the right the same number of places and then compute that quotient instead” method works when computing 2.38 ÷ 0.0125. Now explain why this method works works in all cases.
   One way to start your explanation is to say what is going on when you “move” the decimal to the right a certain number of places.
   
4. Describe a method that allows the user to compute the quotient of two numbers expressed in scientific notation. Demonstrate your method by computing the following quotients by hand.
   1. (5.85 × 10¹¹) ÷ (1.8 × 10⁷)
   2. (1.098 × 10^-4) ÷ (3.05 × 10^-7)
   3. (3.312 × 10^-2) ÷ (2.4 × 10⁷)

Hint to problem 1. What mathematical operation can you use to move the decimals in 2.38 and 0.0125 four places to the right? What effect does that have on the quotient of the new numbers? Can you express 23800 ÷ 125 in terms of 2.38 and 0.0125?

Hint to problem 3. What mathematical operation moves the decimal a given number of places to the right? If you perform this operation on both the dividend and divisor (the number you’re dividing into and the number you’re dividing by), what is the effect on the quotient?

Hint to problem 4. One way to do this is to rewrite the quotient without scientific notation, but that’s not always easy to do. Is there a way to compute the quotient of the decimal numbers first, then deal with the powers of 10? In part (a), what power of 10 completes the equation below?

1. Since 2.38 × 10, 000 = 23,800 and 0.0125 × 10, 000 = 125, the operation that corresponds to moving the decimal 4 places to the right in 2.38 and 0.0125 is multiplication by 10,000. This means that
   
   so the two quotients are equal.
2. 1. Instead of computing 0.000585÷0.18, move the decimal in each number two places to the right and compute 0.0585 ÷ 18, instead:

      

      Since 0.0585 = 0.00585 × 100 and 18 = 0.18 × 100, you can use the strategy from problem 1 to see that

      
      so 0.00585 ÷ 0.18 = 0.00325, too.
   2. Instead of 0.03312 ÷ 2.4, compute 0.3312 ÷ 24:

      

      Therefore, 0.033312 ÷ 2.4 = 0.0138, too.
3. The solution to this problem is a generalization of the one given for problem 1.
   Teacher’s note: The explanation given purposely avoids algebraic symbolism, but such symbolism is convenient when explaining why a ÷ b equals (a × 10ⁿ) ÷ (b × 10ⁿ):
   
   
   Shifting the decimal point in a number to the right a given number of places is mathematically the same as multiplying the given number by 10^{# of places} (10 raised to the power equal to the number of places to be moved). Then (dividend × 10^{# of places}) ÷ (divisor × 10^{# of places}) can be rewritten in fraction notation as , which simplifies as follows:
   
   so the original quotient, dividend ÷ divisor, is equal to the quotient of the numbers you get by shifting their decimals a given number of places to the right, as long as the shift is the same in both numbers.

4. There are many methods for computing the quotient of two numbers expressed using scientific notation. One is demonstrated and explained with the example

   
   Teacher’s note: If your students have taken algebra or are comfortable with algebraic symbolism, you can also explain the method (and the reason it works) as below. (a × 10ⁿ) ÷ (b × 10^m) = , which simplifies to × . By the law of exponents, this equals (a ÷ b) × 10^n-m, so you can compute the quotient by dividing the decimal parts and dividing the powers of 10.
   
   • Rewrite the quotient as a fraction:
   • Rewrite the fraction as a fraction of the decimal parts times a fraction of powers of 10:
   • Compute the quotient of the decimal parts using the method you’ve already used for dividing decimal numbers, then simplify the fraction of powers of 10 by using the law of exponents:
   • Move the decimal and adjust the power of 10 as needed to get the form you prefer for your answer (scientific notation or otherwise): Since the decimal part of the answer was multiplied by 10 (to get from 0.438 to 4.38), the power of 10 was decreased by 1 (that is, 10³⁹ was divided by 10).

   There are other ways to solve these type of problems, as demonstrated in the solutions to parts (a)-(c).

   1. First, use the fact that which, using the “move the decimals” method, is equal to (58.5 ÷ 18) × 10⁴. Since 58.5 ÷ 18 = 3.25, the final answer is 3.25 × 10⁴, or 32500.

      Here’s another method, starting with (58.5÷18)×10⁴:

      which also equals 32500, of course.

      One final method:

      
   2. As in the last example,
      

      Alternatively,

      
      And the last method:
      
   3. Now, for the last quotient,
      
      Alternatively,
      

      And the last method: