(a) Write the repeating decimal number 2.718 as an improper fraction (b) Write the repeating decimal number \( 10, \overline{213} \) as an improper fraction

Certainly! Let's tackle both parts of the problem step-by-step. --- ### **Problem (a): Convert the Repeating Decimal \( 2.\overline{718} \) to an Improper Fraction** **Step 1: Let \( x \) represent the repeating decimal.** \[ x = 2.\overline{718} = 2.718718718\ldots \] **Step 2: Multiply \( x \) by \( 1000 \) to shift the decimal point three places to the right (since the repeating block has three digits).** \[ 1000x = 2718.\overline{718} = 2718.718718\ldots \] **Step 3: Subtract the original \( x \) from this new equation to eliminate the repeating part.** \[ 1000x - x = 2718.\overline{718} - 2.\overline{718} \] \[ 999x = 2716 \] **Step 4: Solve for \( x \).** \[ x = \frac{2716}{999} \] **Step 5: Simplify the fraction (if possible).** To check if the fraction can be simplified, find the greatest common divisor (GCD) of 2716 and 999. - Using the Euclidean algorithm: - \( 2716 \div 999 = 2 \) with a remainder of 718 - \( 999 \div 718 = 1 \) with a remainder of 281 - \( 718 \div 281 = 2 \) with a remainder of 156 - \( 281 \div 156 = 1 \) with a remainder of 125 - \( 156 \div 125 = 1 \) with a remainder of 31 - \( 125 \div 31 = 4 \) with a remainder of 1 - \( 31 \div 1 = 31 \) with a remainder of 0 The GCD is **1**, so the fraction is already in its simplest form. **Final Answer for (a):** \[ 2.\overline{718} = \frac{2716}{999} \] --- ### **Problem (b): Convert the Repeating Decimal \( 10.\overline{213} \) to an Improper Fraction** **Step 1: Let \( y \) represent the repeating decimal.** \[ y = 10.\overline{213} = 10.213213213\ldots \] **Step 2: Multiply \( y \) by \( 1000 \) to shift the decimal point three places to the right (since the repeating block has three digits).** \[ 1000y = 10213.\overline{213} = 10213.213213\ldots \] **Step 3: Subtract the original \( y \) from this new equation to eliminate the repeating part.** \[ 1000y - y = 10213.\overline{213} - 10.\overline{213} \] \[ 999y = 10203 \] **Step 4: Solve for \( y \).** \[ y = \frac{10203}{999} \] **Step 5: Simplify the fraction (if possible).** First, divide both numerator and denominator by **3**: \[ \frac{10203 \div 3}{999 \div 3} = \frac{3401}{333} \] Next, check if the fraction can be simplified further by finding the GCD of 3401 and 333. - Using the Euclidean algorithm: - \( 3401 \div 333 = 10 \) with a remainder of 71 - \( 333 \div 71 = 4 \) with a remainder of 49 - \( 71 \div 49 = 1 \) with a remainder of 22 - \( 49 \div 22 = 2 \) with a remainder of 5 - \( 22 \div 5 = 4 \) with a remainder of 2 - \( 5 \div 2 = 2 \) with a remainder of 1 - \( 2 \div 1 = 2 \) with a remainder of 0 The GCD is **1**, so the fraction is already in its simplest form. **Final Answer for (b):** \[ 10.\overline{213} = \frac{3401}{333} \] --- **Summary of Solutions:** - **(a)** \( 2.\overline{718} = \dfrac{2716}{999} \) - **(b)** \( 10.\overline{213} = \dfrac{3401}{333} \)