Express each of the following ratios in simplest form :
`(2)/(3) : (5)/(6)`

Answer

To express the ratio \( \frac{2}{3} : \frac{5}{6} \) in simplest form, follow these steps:

### Step 1: Rewrite the Ratio as a Fraction
We can express the ratio \( \frac{2}{3} : \frac{5}{6} \) as a fraction:
\[ \frac{2/3}{5/6} \]

### Step 2: Apply the Rule of Division
When dividing fractions, we multiply by the reciprocal of the divisor. Thus, we can rewrite the expression as:
\[ \frac{2}{3} \times \frac{6}{5} \]

### Step 3: Multiply the Numerators and the Denominators
Now, we multiply the numerators together and the denominators together:
\[ \frac{2 \times 6}{3 \times 5} = \frac{12}{15} \]

### Step 4: Simplify the Fraction
Next, we simplify \( \frac{12}{15} \) by finding the greatest common divisor (GCD) of 12 and 15. The GCD is 3. We divide both the numerator and the denominator by 3:
\[ \frac{12 \div 3}{15 \div 3} = \frac{4}{5} \]

### Step 5: Write the Simplified Ratio
Finally, we express the simplified ratio:
\[ 4 : 5 \]

Thus, the simplest form of the ratio \( \frac{2}{3} : \frac{5}{6} \) is \( 4 : 5 \).

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