P, Q and R are three typists who working simultaneously can type 216 pages in 4 hours. In one hour, R can type as many pages more than Q as Q can type more than P. During a period of five hours, R can type as many pages as P can during seven hours. How many pages does each of them type per hour ?

### Concept & Linear Equations Translate the descriptive relationships between their work rates into a system of linear equations. ### Step-by-Step Solution * **Establish Hourly Rates:** Let the number of pages typed per hour by P, Q, and R be $P$, $Q$, and $R$ respectively. * **Equation 1 (Combined Rate):** Together, they type $216$ pages in $4$ hours. Combined hourly rate: $P + Q + R = \frac{216}{4} = 54$. * **Equation 2 (Rate Differences):** "R can type as many pages more than Q as Q can type more than P." $R - Q = Q - P$ Rearranging gives: $P + R = 2Q$. * **Solve for Q:** Substitute $(P + R)$ with $2Q$ in Equation 1: $(P + R) + Q = 54$ $2Q + Q = 54$ $3Q = 54$ $Q = 18$ pages/hour. * **Equation 3 (Time/Work Equivalency):** "In 5 hours, R types as many pages as P does in 7 hours." $5R = 7P$ $R = \frac{7}{5}P$. * **Solve for P and R:** We know $P + R = 2Q$, and $Q = 18$, so $P + R = 36$. Substitute $R = \frac{7}{5}P$: $P + \frac{7}{5}P = 36$ $\frac{12P}{5} = 36$ $P = \frac{36 \times 5}{12} = 3 \times 5 = 15$ pages/hour. Now, calculate $R$: $R = \frac{7}{5} \times 15 = 7 \times 3 = 21$ pages/hour. The rates are P = $15$, Q = $18$, R = $21$. ### Exam Strategy & Shortcut Use Option Elimination based on the first extracted rule: $R - Q = Q - P$. This implies the rates must be in an arithmetic progression. Check differences in options: (a) $14, 17, 20$ (diff is 3, valid) (b) $15, 17, 22$ (diffs are 2 and 5, invalid) (c) $15, 18, 21$ (diff is 3, valid) (d) $16, 18, 22$ (diffs are 2 and 4, invalid) Now check the "5 hrs of R = 7 hrs of P" rule ($5 \times R = 7 \times P$) on the remaining valid options: Test (a): $5 \times 20 = 100$; $7 \times 14 = 98$ (Not equal). Test (c): $5 \times 21 = 105$; $7 \times 15 = 105$ (Equal!). ### Common Pitfall Setting up the relative rate equation incorrectly. Interpreting "R types as many more than Q" as $R + Q$ instead of $R - Q$ will completely break the algebraic flow. ### Final Answer Therefore, the correct answer is **15, 18, 21**.