Question

If $\frac{p}{q}=\frac{19}{5}+\frac{2}{7}-2.4+(\frac{4.8}{2.4})^2$, where p/q is a simplest form, then find the sum of the digits of q.

Answer 1

8

Answer 2

The problem requires us to simplify a given expression to find a fraction $\frac{p}{q}$ in its simplest form and then calculate the sum of the digits of the denominator $q$.

The given expression is: $\frac{p}{q} = \frac{19}{5} + \frac{2}{7} - 2.4 + \left(\frac{4.8}{2.4}\right)^2$

Step 1: Convert decimal numbers to fractions. We convert the decimal numbers into their fractional forms: $2.4 = \frac{24}{10} = \frac{12}{5}$ $4.8 = \frac{48}{10} = \frac{24}{5}$

Step 2: Simplify the term involving the square. The term $\left(\frac{4.8}{2.4}\right)^2$ can be simplified as follows: $\left(\frac{4.8}{2.4}\right)^2 = \left(\frac{24/5}{12/5}\right)^2$ The division of fractions simplifies to: $\left(\frac{24}{12}\right)^2 = (2)^2 = 4$

Step 3: Substitute the simplified terms back into the equation. Now, substitute the fractional forms and the simplified square term back into the original equation: $\frac{p}{q} = \frac{19}{5} + \frac{2}{7} - \frac{12}{5} + 4$

Step 4: Combine like terms. We can group the fractions with the same denominator: $\frac{p}{q} = \left(\frac{19}{5} - \frac{12}{5}\right) + \frac{2}{7} + 4$ $\frac{p}{q} = \frac{19 - 12}{5} + \frac{2}{7} + 4$ $\frac{p}{q} = \frac{7}{5} + \frac{2}{7} + 4$

Step 5: Find a common denominator and add the fractions. To add these rational numbers, we find the least common multiple (LCM) of the denominators 5, 7, and 1 (for the integer 4). The LCM of 5 and 7 is $5 \times 7 = 35$. Convert each term to an equivalent fraction with the denominator 35: $\frac{7}{5} = \frac{7 \times 7}{5 \times 7} = \frac{49}{35}$ $\frac{2}{7} = \frac{2 \times 5}{7 \times 5} = \frac{10}{35}$ $4 = \frac{4}{1} = \frac{4 \times 35}{1 \times 35} = \frac{140}{35}$ Now, add these fractions: $\frac{p}{q} = \frac{49}{35} + \frac{10}{35} + \frac{140}{35}$ $\frac{p}{q} = \frac{49 + 10 + 140}{35}$ $\frac{p}{q} = \frac{59 + 140}{35}$ $\frac{p}{q} = \frac{199}{35}$

Step 6: Verify if the fraction is in simplest form. The fraction obtained is $\frac{199}{35}$. For this to be in simplest form, the greatest common divisor (GCD) of the numerator (199) and the denominator (35) must be 1. The prime factorization of the denominator is $35 = 5 \times 7$. We check if the numerator, 199, is divisible by 5 or 7:

• 199 is not divisible by 5 because its last digit is not 0 or 5.
• For divisibility by 7: $199 \div 7 = 28$ with a remainder of $3$ ($199 = 7 \times 28 + 3$). So, 199 is not divisible by 7. Since 199 is not divisible by any prime factor of 35, the fraction $\frac{199}{35}$ is indeed in its simplest form.

Step 7: Identify q and calculate the sum of its digits. From the simplest form $\frac{p}{q} = \frac{199}{35}$, we have $p = 199$ and $q = 35$. The digits of $q$ are 3 and 5. The sum of the digits of $q$ is $3 + 5 = 8$.