Work out $\frac{2}{5} \times \frac{6}{7}$. Give your answer in its simplest form.
Answer (1)
Multiply the numerators: 2 × 6 = 12 .
Multiply the denominators: 5 × 7 = 35 .
Form the fraction: 35 12 .
The fraction is already in its simplest form: 35 12 .
Explanation
Problem Analysis We are asked to multiply two fractions, 5 2 and 7 6 , and express the answer in its simplest form. This involves multiplying the numerators and the denominators separately.
Multiplying Numerators First, we multiply the numerators: 2 × 6 = 12 .
Multiplying Denominators Next, we multiply the denominators: 5 × 7 = 35 .
Forming the Fraction Now, we write the resulting fraction: 35 12 .
Checking for Simplification To ensure the fraction is in its simplest form, we need to check if the numerator and denominator have any common factors other than 1. We can find the greatest common divisor (GCD) of 12 and 35.
Finding the GCD The prime factors of 12 are 2, 2, and 3 ( 2 2 × 3 ). The prime factors of 35 are 5 and 7 ( 5 × 7 ). Since there are no common prime factors between 12 and 35, their greatest common divisor is 1.
Simplifying the Fraction Since the GCD of 12 and 35 is 1, the fraction 35 12 is already in its simplest form.
Final Answer Therefore, the final answer is 35 12 .
Examples
In baking, if a recipe calls for 5 2 of a cup of flour and you want to make 7 6 of the recipe, you would multiply these fractions to determine how much flour you need. So, you would calculate 5 2 × 7 6 = 35 12 of a cup of flour. This ensures you scale the ingredients correctly to maintain the recipe's proportions, which is essential for consistent results in baking.
