Use long division to megthod to delermine the followning \( 2.5 \div 7.1560 \div 15 \) 2.5.2. \( 502 \div 6 \)

**Problem 1. Compute \(2.5 \div 7.1560 \div 15\)** We interpret the expression as a left‐to‐right division: \[ \frac{\left(\frac{2.5}{7.1560}\right)}{15} = \frac{2.5}{7.1560 \times 15}. \] **Step 1. Eliminate decimals by scaling.** It can be easier to work with integers. Write the numbers as fractions: \[ 2.5 = \frac{25}{10}\quad,\quad 7.1560 = \frac{71560}{10000}. \] Then \[ \frac{2.5}{7.1560} = \frac{\frac{25}{10}}{\frac{71560}{10000}} = \frac{25}{10} \times \frac{10000}{71560} = \frac{25 \times 10000}{10 \times 71560}. \] Simplify by canceling the common factor \(10\): \[ = \frac{25 \times 1000}{71560} = \frac{25000}{71560}. \] **Step 2. Divide the result by 15.** Dividing by 15 is the same as multiplying by \(\frac{1}{15}\): \[ \left(\frac{25000}{71560}\right) \div 15 = \frac{25000}{71560 \times 15}. \] Compute the product in the denominator: \[ 71560 \times 15 = 71560 \times (10+5) = 715600 + 357800 = 1073400. \] Thus, the expression becomes: \[ \frac{25000}{1073400}. \] **Step 3. Simplify and/or convert to decimal (using long division).** Divide numerator and denominator by 10: \[ \frac{2500}{107340}. \] It is not immediately clear if further cancellation is possible by inspection. We now perform long division to approximate the value: \[ 2.5 \div 7.1560 \div 15 = \frac{25000}{1073400} \approx 0.0233. \] A brief outline of the long division approach: 1. Divide \(25000\) by \(1073400\). Since \(25000 \ll 1073400\) the quotient starts with \(0.\) 2. Multiply the remainder by 10 repeatedly and determine each digit. 3. Continuing this process yields an approximate quotient of \(0.0233\) (to 4 decimal places). Thus, \[ 2.5 \div 7.1560 \div 15 \approx 0.0233. \] --- **Problem 2. Compute \(502 \div 6\)** We set up the long division of \(502\) by \(6\). **Step 1. Divide digit by digit.** 1. **Divide 6 into 5:** Since \(6\) does not go into \(5\), consider the first two digits: \(50\). 2. **Divide 6 into 50:** \[ 6 \times 8 = 48 \quad (\text{since } 6 \times 9 = 54 \text{ is too large}). \] Write \(8\) in the quotient. Remainder: \(50 - 48 = 2\). 3. **Bring down the next digit:** Append the next digit \(2\) (from \(502\)) to the remainder to form \(22\). 4. **Divide 6 into 22:** \[ 6 \times 3 = 18 \quad (\text{since } 6 \times 4 = 24 \text{ is too large}). \] Write \(3\) in the quotient. Remainder: \(22 - 18 = 4\). The long division shows: \[ 502 \div 6 = 83 \text{ remainder } 4. \] **Step 2. Convert the remainder to a decimal.** To express the answer as a decimal, continue the division: 1. Append a decimal point and a zero to the remainder \(4\) to make \(40\). 2. **Divide 6 into 40:** \[ 6 \times 6 = 36, \] giving a new remainder \(40 - 36 = 4\). Since the remainder is again \(4\), the digit \(6\) will repeat. Thus, in decimal form: \[ 502 \div 6 = 83.6666\ldots = 83.\overline{6}. \] --- **Final Answers:** \[ \boxed{2.5 \div 7.1560 \div 15 \approx 0.0233,\quad 502 \div 6 = 83.\overline{6}} \]