When 1 is added to each of the two given numbers, their ratio becomes 3 : 4 and when 5 is subtracted from each, the ratio becomes 7 : 10. One of the numbers is

### Concept & Simultaneous Equations from Ratios Ratios represent simplified fractions. By setting up equations based on modifications to unknown variables (adding or subtracting constants), we create a system of linear equations that can be solved simultaneously. ### Step-by-Step Solution 1. Let the two original numbers be $a$ and $b$. 2. Condition 1: When 1 is added to each, the ratio is $3 : 4$. $$ \frac{a + 1}{b + 1} = \frac{3}{4} $$ $$ 4(a + 1) = 3(b + 1) \Rightarrow 4a + 4 = 3b + 3 \Rightarrow 4a - 3b = -1 \quad \text{--- (Eq 1)} $$ 3. Condition 2: When 5 is subtracted from each, the ratio is $7 : 10$. $$ \frac{a - 5}{b - 5} = \frac{7}{10} $$ $$ 10(a - 5) = 7(b - 5) \Rightarrow 10a - 50 = 7b - 35 \Rightarrow 10a - 7b = 15 \quad \text{--- (Eq 2)} $$ 4. Solve the system of equations. Multiply (Eq 1) by 7 and (Eq 2) by 3 to eliminate $b$: $$ 28a - 21b = -7 $$ $$ 30a - 21b = 45 $$ 5. Subtract the first from the second: $$ 2a = 52 \Rightarrow a = 26 $$ 6. Substitute $a = 26$ back into (Eq 1): $$ 4(26) - 3b = -1 \Rightarrow 104 - 3b = -1 \Rightarrow 3b = 105 \Rightarrow b = 35 $$ 7. The numbers are 26 and 35. Looking at the options, 26 is present. ### Exam Strategy & Shortcut **Back-solving using options:** Check option (c) 26. Assume one number is 26. If we add 1, it becomes 27. The new ratio has parts 3:4. Since 27 is a multiple of 3 (it is $9 \times 3$), the other number after adding 1 would be $9 \times 4 = 36$, meaning the original number is 35. Now test the second condition: subtract 5 from both (26 - 5 = 21; 35 - 5 = 30). The new ratio is $21 : 30$, which simplifies to $7 : 10$. This perfectly matches the prompt, validating 26 immediately. ### Common Pitfall Setting up the fractions incorrectly, such as applying the operation (adding 1 or subtracting 5) to only the numerator or mixing up the order of the ratio terms. ### Final Answer Therefore, the correct answer is **26**.