The cost of 6 pens and 3 pencils is ₹ 84. One-third of the cost of one pen is equal to the cost of one pencil. What is the total cost of 4 pens and 5 pencils?

### Concept & Strategy This is a classic simultaneous linear equation problem. First, translate the word problem into two algebraic equations. Use substitution to find the individual cost of a pen and a pencil, then evaluate the final expression. ### Step-by-Step Solution Let the cost of one pen be $x$. Let the cost of one pencil be $y$. **Step 1: Form the equations** From the first sentence: $6x + 3y = 84$ --- (Equation 1) From the second sentence ("One-third of the cost of one pen is equal to the cost of one pencil"): $\frac{1}{3}x = y$ Rearranging this to avoid fractions: $x = 3y$ --- (Equation 2) **Step 2: Solve for $y$ (cost of a pencil)** Substitute Equation 2 into Equation 1: $6(3y) + 3y = 84$ $18y + 3y = 84$ $21y = 84$ $y = \frac{84}{21}$ $y = 4$ So, the cost of one pencil is ₹ 4. **Step 3: Solve for $x$ (cost of a pen)** Substitute the value of $y$ back into Equation 2: $x = 3(4)$ $x = 12$ So, the cost of one pen is ₹ 12. **Step 4: Calculate the target cost** The question asks for the total cost of 4 pens and 5 pencils. Total Cost = $4x + 5y$ Total Cost = $4(12) + 5(4)$ Total Cost = $48 + 20$ Total Cost = $68$ ### Exam Strategy & Shortcut **Direct Ratio Substitution:** "One-third of a pen = one pencil" means 1 pen costs the same as 3 pencils. The first purchase is 6 pens and 3 pencils. Convert pens to pencils: 6 pens = 18 pencils. So, 18 pencils + 3 pencils = 21 pencils total. 21 pencils cost ₹ 84. Thus, 1 pencil = ₹ 4. The target purchase is 4 pens and 5 pencils. Convert pens to pencils: 4 pens = 12 pencils. Total target = 12 pencils + 5 pencils = 17 pencils. Target Cost = $17 \times 4 = \text{₹ } 68$. This completely avoids setting up $x$ and $y$ algebra. ### Common Pitfall A common translation error is writing the ratio equation backward, such as $x = \frac{1}{3}y$ or $3x = y$. Carefully re-read "One-third of... one pen... is equal to... one pencil" $\implies \frac{1}{3} \cdot \text{Pen} = \text{Pencil}$. ### Final Answer Therefore, the correct answer is **₹ 68**.