In a cricket match, Team A scored 232 runs without losing a wicket. The score consisted of byes, wides and runs scored by two opening batsmen, Ram and Shyam. The runs scored by the two batsmen are 26 times wides. There are 8 more byes than wides. If the ratio of runs scored by Ram and Shyam is 6: 7, then the runs scored by Ram is

### Concept & Strategy This problem tests the application of **linear equations** combined with **ratios**. The strategy is to express all components of the total score (byes, wides, and batsmen runs) in terms of a single variable (wides), solve for that variable using the total score, and then divide the total batsmen runs using the given ratio. ### Step-by-Step Solution **Given:** * Total Score = 232 * Batsmen runs ($R + S$) = $26 \times Wides$ * Byes = $Wides + 8$ * Ratio of Ram's runs to Shyam's runs = $6:7$ **Calculation:** Step 1: Set up the total score equation. Total Score = Byes + Wides + Batsmen runs $232 = (Wides + 8) + Wides + 26(Wides)$ Step 2: Solve for Wides ($W$). $232 = 28W + 8$ $224 = 28W$ $W = \frac{224}{28} = 8$ Step 3: Calculate the total runs scored by the batsmen. Batsmen runs = $26 \times 8 = 208$ Step 4: Divide the runs using the ratio $6:7$. Total parts in ratio = $6 + 7 = 13$ parts. Value of 1 part = $\frac{208}{13} = 16$ runs. Ram's runs (6 parts) = $6 \times 16 = 96$. ### Exam Strategy & Shortcut **The Multiplicity Check (Ratio Method):** Ram's runs correspond to 6 parts in the ratio. Therefore, Ram's score *must* be a multiple of 6. Looking at the options: (a) 88 is not a multiple of 6. (b) 96 is a multiple of 6 ($16 \times 6$). (c) 102 is a multiple of 6 ($17 \times 6$). (d) 112 is not a multiple of 6. This eliminates (a) and (d) instantly. While you still need to solve the equation to choose between 96 and 102, this check is incredibly useful for verifying your final calculation. ### Common Pitfall Students often forget to include the "wides" variable itself in the total sum, writing the equation as $Total = Byes + Batsmen runs$, completely missing that wides also contribute directly to the 232 total. Always list all individual components before summing. ### Final Answer **Therefore, the correct answer is 96.**