The H.C.F. of $1.75$, $5.6$ and $7$ is

### Concept & Strategy To find the Highest Common Factor (HCF) of decimals, avoid working with them directly as it can cause alignment errors. **Strategy:** Convert all decimals to whole numbers by multiplying them by a power of $10$ (based on the maximum decimal places). Find the HCF of the new whole numbers, then divide the result by that same power of $10$. ### Step-by-Step Solution * **Given:** $1.75$, $5.6$, and $7$. * **Step 1: Equalize decimal places.** The maximum number of decimal places is two (in $1.75$). Rewrite all numbers with two decimal places: $1.75$, $5.60$, $7.00$ * **Step 2: Convert to whole numbers.** Multiply by $100$: $175$, $560$, $700$ * **Step 3: Find the H.C.F. of $175, 560, 700$.** Prime factorization: $175 = 5^2 \times 7$ $560 = 2^4 \times 5 \times 7$ $700 = 2^2 \times 5^2 \times 7$ The common prime factors are $5$ and $7$. The lowest powers are $5^1$ and $7^1$. $$\text{H.C.F.} = 5 \times 7 = 35$$ * **Step 4: Convert back to decimal.** Divide the H.C.F. by $100$: $$35 \div 100 = 0.35$$ ### Exam Strategy & Shortcut Look at $1.75$ and $7$. You know $1.75 \times 4 = 7$. So, $1.75$ perfectly divides $7$. The H.C.F. must be a factor of $1.75$. Option (c) $3.5$ is larger than $1.75$, so it's eliminated. Now check if $0.35$ works for $5.6$. Since $35 \times 16 = 560$, it works perfectly. ### Common Pitfall The most fatal mistake is multiplying by different powers of 10 for each number (e.g., $1.75 \times 100$, $5.6 \times 10$, $7 \times 1$) resulting in $175, 56, 7$, which gives a completely incorrect H.C.F. You **must** equalize the decimal places first before removing the decimal. ### Final Answer **Therefore, the correct answer is 0.35.**