The numbers $x, y, z$ are proportional to 2, 3, 5. The sum of $x, y, z$ is 100. If $y = px - 10$, then $p$ is equal to

### Concept & Direct Proportions When numbers are proportional to a given set of values, they can be represented as multiples of a common variable constant. Find the exact values using their provided sum, then substitute them into the given algebraic equation to solve for the unknown variable. ### Step-by-Step Solution * Since $x$, $y$, and $z$ are proportional to 2, 3, and 5, let $x = 2k$, $y = 3k$, and $z = 5k$. * We are given that their sum is 100: $x + y + z = 100$. * $2k + 3k + 5k = 100$ * $10k = 100 \implies k = 10$ * Substitute $k$ back to find the actual values: * $x = 2(10) = 20$ * $y = 3(10) = 30$ * $z = 5(10) = 50$ * We are given the equation $y = px - 10$. Substitute the values of $x$ and $y$: * $30 = p(20) - 10$ * $30 + 10 = 20p$ * $40 = 20p \implies p = 2$ ### Exam Strategy & Shortcut Observe that the sum of the basic ratio units is $2 + 3 + 5 = 10$. The total sum given is 100, which is exactly 10 times the sum of the ratio units. Thus, the actual values are simply 10 times the ratio numbers ($x = 20$, $y = 30$). Plug these quickly into the equation $30 = 20p - 10$ to see $20p = 40$, meaning $p = 2$. ### Common Pitfall Making algebraic sign errors when moving terms across the equals sign, such as calculating $30 - 10 = 20p$ instead of correctly adding the 10 to both sides ($30 + 10 = 20p$). ### Final Answer Therefore, the correct answer is **2**.