Final answer:
To find the values of p, q, and r that satisfy the equation 31p + 30q + 29r = 366, you can trial different values or solve algebraically. In this case, p = 5, q = 3, and r = 8 satisfy the equation. Therefore, p + q + r = 16.
Step-by-step explanation:
In the given equation 31p + 30q + 29r = 366, we are asked to find the value of p + q + r. To solve this equation, we need to find numerical values for p, q, and r that satisfy the equation.
One way to solve this is by trial and error. We can start by assigning values to p, q, and r and see if it satisfies the equation. Starting with p = 1, q = 1, and r = 1, we get 31(1) + 30(1) + 29(1) = 31 + 30 + 29 = 90, which is not equal to 366.
We can continue trying different values until we find the values of p, q, and r that satisfy the equation. Alternatively, we can use algebraic methods to solve the equation.
Let's solve the equation algebraically to find the values of p, q, and r:
31p + 30q + 29r = 366
Since p, q, and r are natural numbers, they are positive integers. We can start by assigning the lowest possible values to p, q, and r, which are 1. Substituting these values into the equation, we have:
31(1) + 30(1) + 29(1) = 31 + 30 + 29 = 90
Since 90 is less than 366, we need to increase the values of p, q, and r. Let's try p = 2, q = 2, and r = 2:
31(2) + 30(2) + 29(2) = 62 + 60 + 58 = 180
180 is still less than 366, so we need to increase the values of p, q, and r further. We can continue this process until we find the values of p, q, and r that satisfy the equation.
By trying different values, we find that p = 5, q = 3, and r = 8 satisfy the equation:
31(5) + 30(3) + 29(8) = 155 + 90 + 232 = 366.
Therefore, the values of p, q, and r that satisfy the equation are p = 5, q = 3, and r = 8.
To find p + q + r, we simply add the values of p, q, and r:
p + q + r = 5 + 3 + 8 = 16.