Question

What is the value of p in this proportion?

710p=2538

Enter your answer as a simplified fraction in the box.

p =

Answer 1

To solve for \( p \) in the equation \( 710p = 2538 \), we need to isolate \( p \).

We can do this by dividing both sides of the equation by 710:

\[
p = \frac{2538}{710}
\]

Now, we simplify the fraction \( \frac{2538}{710} \).

First, we find the greatest common divisor (GCD) of 2538 and 710.

To do that, we can perform prime factorization or use the Euclidean algorithm:

1. Divide 2538 by 710:
- \( 2538 \div 710 \approx 3.57 \) (whole part is 3)
- \( 710 \times 3 = 2130 \)
- \( 2538 - 2130 = 408 \)

2. Now, apply the same process between 710 and 408:
- \( 710 \div 408 \approx 1.74 \) (whole part is 1)
- \( 408 \times 1 = 408 \)
- \( 710 - 408 = 302 \)

3. Apply the same process between 408 and 302:
- \( 408 \div 302 \approx 1.35 \) (whole part is 1)
- \( 302 \times 1 = 302 \)
- \( 408 - 302 = 106 \)

4. Now, apply between 302 and 106:
- \( 302 \div 106 \approx 2.84 \) (whole part is 2)
- \( 106 \times 2 = 212 \)
- \( 302 - 212 = 90 \)

5. Now, between 106 and 90:
- \( 106 - 90 = 16 \)

6. Now, between 90 and 16:
- \( 90 \div 16 \approx 5.625 \) (whole part is 5)
- \( 16 \times 5 = 80 \)
- \( 90 - 80 = 10 \)

7. Now, between 16 and 10:
- \( 16 - 10 = 6 \)

8. Now, between 10 and 6:
- \( 10 - 6 = 4 \)

9. Now, between 6 and 4:
- \( 6 - 4 = 2 \)

10. Now, between 4 and 2:
- \( 4 - 2 = 2 \)

11. Finally, between 2 and 2:
- \( 2 - 2 = 0 \)

So the GCD is \( 2 \).

Now, we can simplify the fraction:

\[
p = \frac{2538 \div 2}{710 \div 2} = \frac{1269}{355}
\]

Since 1269 and 355 do not have any common factors, this fraction is in its simplest form.

Thus, the value of \( p \) is:

\[
\boxed{\frac{1269}{355}}
\]