Question

\( \frac{0.0027 \times 10^x}{0.09 \times 10^y} = 3 \times 10^8 \)
What is the value of y less than x?

• A. 9
• B. 10
• C. 11
• D. 12
• E. 13

Hint:

To avoid errors with decimals and exponents, always convert all numbers into standard scientific notation (a single non-zero digit before the decimal point) before you begin simplifying the expression.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem involves simplifying an expression with scientific notation and solving for the difference between two exponents, \(x-y\). The phrase "y less than x" translates to the expression \(x-y\).
Step 2: Key Formula or Approach:
We will use the rules of exponents:
\( a^m \times a^n = a^{m+n} \)
\( \frac{a^m}{a^n} = a^{m-n} \)
The first step is to convert the decimal coefficients into a consistent format, preferably standard scientific notation.
Step 3: Detailed Explanation:
Let's rewrite the decimal numbers in the expression: \[ 0.0027 = 2.7 \times 10^{-3} \] \[ 0.09 = 9 \times 10^{-2} \] Substitute these back into the original equation: \[ \frac{(2.7 \times 10^{-3}) \times 10^x}{(9 \times 10^{-2}) \times 10^y} = 3 \times 10^8 \] Now, group the coefficients and the powers of 10 separately: \[ \left( \frac{2.7}{9} \right) \times \left( \frac{10^{-3} \times 10^x}{10^{-2} \times 10^y} \right) = 3 \times 10^8 \] Simplify the coefficient part: \[ \frac{2.7}{9} = \frac{27}{90} = \frac{3}{10} = 0.3 \] Simplify the powers of 10 using exponent rules: \[ \frac{10^{-3+x}}{10^{-2+y}} = 10^{(-3+x) - (-2+y)} = 10^{x - y - 3 + 2} = 10^{x - y - 1} \] Now the equation becomes: \[ 0.3 \times 10^{x - y - 1} = 3 \times 10^8 \] To solve this, we need the coefficient on the left to match the coefficient on the right. Let's write 0.3 in scientific notation as \(3 \times 10^{-1}\): \[ (3 \times 10^{-1}) \times 10^{x - y - 1} = 3 \times 10^8 \] Combine the powers of 10 on the left side: \[ 3 \times 10^{-1 + (x - y - 1)} = 3 \times 10^8 \] \[ 3 \times 10^{x - y - 2} = 3 \times 10^8 \] Since the bases (3 and 10) are equal on both sides, the exponents of 10 must also be equal: \[ x - y - 2 = 8 \] Solve for \(x-y\): \[ x - y = 8 + 2 \] \[ x - y = 10 \] Step 4: Final Answer
The value of y less than x, which is \(x-y\), is 10.