\( 1, \overline{23}-1,4= \) Hint: first convert \( 1, \overline{23} \) to its simplest ractional form.

To solve the expression \( 1, \overline{23} - 1.4 \), we first need to convert \( 1, \overline{23} \) into its simplest fractional form. ### Step 1: Convert \( 1, \overline{23} \) to a Fraction Let \( x = 1, \overline{23} \). This means that \( x = 1.232323...\). To eliminate the repeating decimal, we can multiply \( x \) by 100 (since the repeating part has two digits): \[ 100x = 123.232323... \] Now, we can set up the equation: \[ 100x - x = 123.232323... - 1.232323... \] This simplifies to: \[ 99x = 122 \] Now, we can solve for \( x \): \[ x = \frac{122}{99} \] ### Step 2: Simplify the Fraction Next, we need to simplify \( \frac{122}{99} \). We can find the greatest common divisor (GCD) of 122 and 99. The prime factorization of 122 is \( 2 \times 61 \), and the prime factorization of 99 is \( 3^2 \times 11 \). Since they have no common factors, the GCD is 1. Thus, \( \frac{122}{99} \) is already in its simplest form. ### Step 3: Convert \( 1.4 \) to a Fraction Now, we convert \( 1.4 \) to a fraction: \[ 1.4 = \frac{14}{10} = \frac{7}{5} \] ### Step 4: Perform the Subtraction Now we need to perform the subtraction: \[ 1, \overline{23} - 1.4 = \frac{122}{99} - \frac{7}{5} \] To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 99 and 5 is \( 495 \). Now we convert both fractions: \[ \frac{122}{99} = \frac{122 \times 5}{99 \times 5} = \frac{610}{495} \] \[ \frac{7}{5} = \frac{7 \times 99}{5 \times 99} = \frac{693}{495} \] Now we can subtract: \[ \frac{610}{495} - \frac{693}{495} = \frac{610 - 693}{495} = \frac{-83}{495} \] ### Final Answer Thus, the result of \( 1, \overline{23} - 1.4 \) is: \[ \frac{-83}{495} \]