Express each of the following ratios in simplest form :
95 : 57

Answer

To express the ratio 95 : 57 in its simplest form, we will follow these steps:

### Step 1: Write the ratio as a fraction
We start by expressing the ratio 95 : 57 as a fraction:
\[ \frac{95}{57} \]

### Step 2: Find the Highest Common Factor (HCF)
Next, we need to find the HCF of 95 and 57. We can use the common division method (also known as the Euclidean algorithm) to find the HCF.

1. Divide 95 by 57:
- 95 ÷ 57 = 1 (remainder 38)
- So, we write: 95 = 57 × 1 + 38

2. Now, take the divisor (57) and the remainder (38) and repeat the process:
- 57 ÷ 38 = 1 (remainder 19)
- So, we write: 57 = 38 × 1 + 19

3. Now, take the last divisor (38) and the last remainder (19):
- 38 ÷ 19 = 2 (remainder 0)
- So, we write: 38 = 19 × 2 + 0

Since we have reached a remainder of 0, the last non-zero remainder is the HCF. Thus, the HCF of 95 and 57 is 19.

### Step 3: Divide both numbers by the HCF
Now, we will divide both the numerator and the denominator of the fraction by the HCF (19):
\[ \frac{95 \div 19}{57 \div 19} = \frac{5}{3} \]

### Step 4: Write the simplest form of the ratio
Now we can express the simplified ratio:
\[ 5 : 3 \]

Thus, the simplest form of the ratio 95 : 57 is **5 : 3**.

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