Find r, if 6r7$_8$ = 511$_9$

1. Analyze the question

We are given an equation involving numbers in different bases: $6r7_8 = 511_9$. We need to find the value of the digit $r$.

2. Explain the concept

To solve problems involving different bases, we convert both sides of the equation to Base 10 (decimal system). The formula to convert a number $d_n...d_2d_1d_0$ in base $x$ to base 10 is: $(d_n \times x^n) + ... + (d_2 \times x^2) + (d_1 \times x^1) + (d_0 \times x^0)$

3. Show the solution

Step 1: Convert the left side ($6r7_8$) to base 10 $6r7_8 = (6 \times 8^2) + (r \times 8^1) + (7 \times 8^0)$ $= (6 \times 64) + (8r) + (7 \times 1)$ $= 384 + 8r + 7$ $= 391 + 8r$

Step 2: Convert the right side ($511_9$) to base 10 $511_9 = (5 \times 9^2) + (1 \times 9^1) + (1 \times 9^0)$ $= (5 \times 81) + (1 \times 9) + (1 \times 1)$ $= 405 + 9 + 1$ $= 415$

Step 3: Equate both sides and solve for $r$ $391 + 8r = 415$ Subtract 391 from both sides: $8r = 415 - 391$ $8r = 24$ Divide by 8: $r = \frac{24}{8}$ $r = 3$

4. Identify the answer

Comparing our result with the given options, $r = 3$ corresponds to option A.

5. Tip

When dealing with base conversion, always remember that the digit in the base must be smaller than the base itself. Since the number is in base 8, the digit $r$ must be less than 8. Our answer, 3, satisfies this condition perfectly!